Patent Publication Number: US-2021164345-A1

Title: A Flow Simulation and Transient Well Analysis Method Based on Generalized Tube Flow and Percolation Coupling

Description:
TECHNICAL FIELD OF THE INVENTION 
     This invention involves various fluid flow simulation and fluid injection and production engineering fields and the like of oil, gas, water, carbon dioxide, chemical agents, microbial agents, water vapors, blood, etc, which is specific to a flow simulation and transient well analysis methods based on generalized tube flow and percolation coupling. 
     BACKGROUND OF THE INVENTION 
     The flow of underground fluids in continuous porous media is called percolation flow, while the free flow of fluids in wellbore, pipes, large fractures, large holes, cavities, caves, fracture caves, karst caves, caverns, channels and so on is called tube flow. In the process of fluid flowing from porous medium area into or out of tube area, the fluid flow laws change from percolation flow to tube flow or from tube flow to percolation flow. The problem of this kind of percolation and tube flow existing at the same time is called tube flow-percolation coupling. It is obvious that the tube flow-percolation coupling is common in the industry of oil and gas exploration and production, groundwater exploitation, underground gas storage, geothermal exploitation. 
     The Darcy&#39;s Law is widely used in the percolation theory. Darcy&#39;s law was created by French engineer Henry Darcy in 1856 through his experiments. It becomes the basic law of percolation mechanics because of its advantages of simple proportional linear function, clear physical concept, and easy solving solutions. It is widely used in underground hydraulics and underground oil and gas percolation mechanics. At present, most of the theoretical models of continuous media in oil and gas reservoirs are based on Darcy&#39;s law. In addition to Darcy&#39;s percolation law, the Hagen-Poisenille laminar-pipe flow formula (1838) is also a linear flow law. However, with the deep practical study, it is found that Darcy percolation law is limited in a certain range of application. The percolation law will become nonlinear beyond the range. The motion equations (or formulas) that are commonly used for describing the nonlinear flow include Irmay (1968)&#39;s low-speed non-Darcy formula, Swartzendruber (1962)&#39;s low-speed non-Darcy formula. Yanzhang Huang (1997)&#39;s low-speed non-Darcy formula; Forchheimer (1901)&#39;s high-speed non-Darcy formula. Barree-Conway (2004)&#39;s high-speed non-Darcy formula; Dexuan Dai (,2019)&#39;s power law non-Newton formula, Yifeng Teng (2016)&#39;s Bingham fluid formula; Jiali Ge (1982)&#39;s stress sensitive index formula; Darcy-Weisbach (1845)&#39;s formula; Navier-Stokes equation for free flow in cavities (Navier, 1827; Stokes, 1845), Stokes-Brinkman equation (Brinkman, 1949), Brinkman-Farchheimer equation (Demin Liu, 2017). 
     On the basis of the actual situations, the above 2 linear and 12 commonly used nonlinear flow laws can be used independently (that is, the whole reservoir follows a single flow law) or they can also be used in combination (different flow laws are applied in different regions or at different scales in the whole reservoirs). The independent application situation is relatively simple, such as using the Darcy percolation model to characterize the entire homogeneous sandstone reservoirs. The homogeneous heavy oil reservoirs can be characterized by power law model. The homogeneous low permeability reservoirs can be characterized by low-speed non-Darcy models. The combined use of above flow laws is relatively complex. For example, three composite distribution areas of polymer, water and oil are formed around an injection well in the reservoir of water drive before polymer drive. The flow laws of the polymer region are characterized by the non-Newtonian fluid models, while the flow laws of the water and oil regions are characterized by the Darcy percolation model, which is a percolation-percolation composite flow between different regions. In the process of oil and gas well production, it is a typical tube flow-percolation coupling flow between formation and wellbore. The spatial structures of naturally fracture-cavern reservoirs with nonhomogeneous vugs and fractures are complex, the medium space scale span is large and the heterogeneity is serious. The fluid flow laws in this kind of oil and gas reservoir are very complicated. It includes both multi-porosity medium percolation and free flow of large holes. The flow of fluids in fracture-caves reservoirs is a complex tube flow (free flow)-percolation coupling. 
     Compared with the percolation-percolation composite flow, the problems of tube flow (free flow)-percolation coupling are much more complicated. Up to now, five kinds of approaches have been proposed for the theoretical study of “tube flow (free flow)-percolation coupling models”. 
     The first category is dual, triple and multi-porosity models based on Darcy&#39;s law, which are mainly used for the reservoirs with relatively uniform distribution and relatively small scales of porous media. In the past 50 years, the industry has continuously published such achievements, and has published the large number of such papers. Representative technical achievements: (1) dual porosity media model with pseudo-steady cross flow created by Warren and Root (1963), which was considered that the fractured reservoir is composed of two groups of parallel continuous systems of fracture and matrix. There are two kinds of pressures of fracture fluid and matrix fluid at every point in the space. In the two flow fields, they meet Darcy&#39;s law and ignore the influence of gravity. The two systems are connected by transfering flow function. (2) A model of unsteady inter porosity flow in dual porous media is established by Jia Yinglan (2013), which divided carbonate reservoir into matrix system and vug system. The vug in the vug system is assumed to be spherical and evenly distributed in the matrix system. The matrix system was connected by the wellbore, and the fluid in the matrix can flow directly into the wellbore. The fluid in the spherical karst vug can flow into the wellbore only with the aid of the matrix. (3) Triple porosity model created by Ciqun Liu (1981), Wu Yushu and Jiali Ge (1983) and Yushu Wu et al (2007) through dividing the types of media into three types: fracture, vug and matrix. (4) Multi-medium model with high speed non Darcy seepage created by Wenguang Feng and Jiali Ge (1985). (5) The dual permeability model of triple porous media constructed by Chang Xuejun (2004), Camacho-V et al. (2005). Gomez (2014) et al., which not only considers the cross flow between the two media, but also considers that both the vugs and natural fractures are connected with the wellbore, and the fluids in the vugs and natural fractures can flow directly into the wellbore, Compared with the triple porosity medium single permeability model constructed by other people above, only the natural fractures are connected with the wellbore, and the fluid in the natural fractures can flow directly into the wellbore, while the fluid in the matrices and the vugs can flow into the wellbore only with the help of the natural fractures, and there is cross flow between the matrices and the vugs, the matrices and the natural fractures, as well as between the vugs and the natural fractures. 
     The second category is continuous-discrete medium composite models established by combining both the permeable flow regions that possess the permeable characteristics and the tube flow regions that possess free flow as the composite regions (blocks) that conforms to the Darcy percolation flow. Its core idea is to treat the tube flow (free flow) region as a permeable area with high permeability and large porosity. Representative technical achievements: (1) Fuxiang Zhang et al. (2009) and yizhao Wan et al. (2015) used irregular regions with high permeability, high porosity (blocks) to describe fractures or caverns with different shapes and large scales, in which the fluid flow in fractures or caverns still belongs to the domain of percolation. (2) Fangfang Chen et al. (2015) established a new numerical well test analysis model with wells drilled outside the holes, in which the flow of caverns and other regions of the reservoirs is all percolation with satisfying Darcy&#39;s law, (3) Kuchuk and Biryukov (2012), Zeng and Zhao (2012) studied regular (or irregular) fracture models with discrete (or continuous) media in heterogeneous reservoirs, in which the matrix blocks are considered as Darcy percolation, whereas fractures are constructed as boundary conditions using the idea of “source function”. Models of the second category is relatively simple, which can better reflect the geological dynamic features of fractures or caves in reservoirs, and have good consistency with the actual production condition. 
     The third category is the discrete fracture-cave models with combinations of different fractures and caves established through combing the isopotential body models of tube flow (free flow) based on material balance and the percolation channel models of connecting isopotential body based on Darcy percolationt. Representative technical results: (1) For large-scale fractured vuggy reservoirs, Xiaolong Peng et al. (2008), Lijun Zhang et al. (2010), Baohua Chang et al. (2011), and Baojiang Duan et al. (2012) established a fractured vuggy models of psedo-steady flow between karst vuggy and karst vuggy through fractures, in which it is considered that the pressure change propagates rapidly, and the shape of fracture vuggy has no effect on the pressure change. (2) Yu Xiong et al. (2018) considered the fluid flow in the fractures as unsteady flow on the basis of the previous work, and constructed the corresponding fracture cavern model of the well drilled in large scale fractures. (3) Yonghui Wu et al. (2018) also divided the fracture cavern reservoirs into rock block systems (including matrix, micro fractures, and micro caves), fractures, karst cave systems. Fractures and caverns are embedded in the rock block system and are interconnected. The fluid flows in the rock block system is characterized by a triple porosity model. The fluid flows in the fracture system is Darcy percolation, which is characterized by Darcy percolation model. The fluid flow in the caverns is pseudo-steady flow, which is characterized by the isopotential body model. The models of the third category are easy to be constructed and solved, and the calculation speed of the model is fast, and the volume of the fractures and caverns can be calculated. (4) Wei Pang et al. (2019) based on the energy conservation equation (Bernoulli equation) of fluid flow in the wellbore and karst cave, gave the relationship between the bottom hole pressure and karst cave pressure, and on this basis, constructed the well test interpretation model with a well drilled in a large karst cave. However, in the process of simplifying the energy conservation equation. Wei Pang et al. (2019) ignored the flow velocity of the fluid in the cave, so the cave is essentially an isopotential body. This kind of isopotential model is easy to construct and solve. At the same time, the calculation speed of the model is not only fast, but also can calculate the parameters such as the volume of the fractured caves. 
     The fourth category is the Darcy-Navier-Stokes coupling model of tube flow (free flow)-percolation established based on Darcy percolation and free flow. Representative technical achievements: (1) Aiming at carbonate fracture and cavern reservoirs, Layton et al. (2003). Mu et al. (2007), Xueli Liu et al. (2007), Xiaolong Peng et al. (2007), Jun Yao et al. (2010), Yajun Li et al. (2011), Na Zhang et al. (2015), Akanni et al. (2017) treat the fluid flow in large-scale fractures and caverns as free flow. Based on the Beavers-Joseph boundary conditions, the fluid flow in fractures and matrix is described by using the Navier-Stokes equation as flow law. (2) Chaoqin Huang et al. (2010) established a macroscopic flow mathematical model of discrete fracture and cave network based on the Brinkam equation, which is the empirical equation proposed by Brinkman considering the terms of fluid viscous shear stress of Navier-Stokes equations on the basis of the Darcy formula in 1974. (3) Popov et al. (2009) proposed to use Stokes Brinkman equation to describe the flow in the fracture cavern media based on the fact, i.e. the caverns are often filled with different degrees. (4) Based on the idea and method of discrete fracture model (Baca, 1984), Dongyan et al. (2013) constructed the multi-stage fracturing horizontal well coupling flow model, the main idea of which is to use Darcy&#39;s law to describe the flow law of the fluid in the reservoir and fractures, and Hagen Poiscuille&#39;s law to describe the flow law of the fluid in the horizontal well bores. The model of the fourth category is currently the most accurate model describing the tube flow (free flow)-percolation coupling and is able to be used for simulation calculations of various complex fluid flow laws. 
     The fifth category is based on the Darcy-Weisbach (1845) formula. The corresponding equivalent permeability or equivalent percolation coefficient or converted percolation coefficient are defined by changing the formula into the form of “Darcy&#39;s law” to achieve the unification of percolation, laminar tube flow and turbulent tube flow, and then realize the flow simulation of tube flow-percolation coupling. Collins et al. (1991) and Shuhong Vu et al. (1999) built up the equivalent permeability model for laminar flow, turbulent flow and transition flow in horizontal wellbore, which simplified and reduced the coupling complexity between formation percolation and horizontal wellbore tube flow. In the field of groundwater exploitation, in order to overcome the difficult problem of coupling flow between different tube flows and formation water percolation in a wellbore-formation system, Chongxi Chen (1994) proposed the concept of converted (laminar) percolation coefficient for turbulent flow in wellbore (with 4 flow zones) for the first time. At the same time, the expressions of converted (laminar) percolation coefficients of four turbulent flow zones are given, which transformed the tube flow law into a linear law and solved the difficult problem in tube flow-percolation coupling. Based on the concept of converted (laminar flow) percolation coefficient, Chongxi Chen (1995) developed the groundwater flow model of karst pipe-fracture-pore triple porous media. Yanlin Zhao (2014) also established the nonlinear percolation-tube flow coupling model for water inrush in pressure-bearing karst caves based on the converted percolation coefficient. The results of Chongxi Chen (1994, 1995) and Yanlin Zhao et al. (2014) so far are the unique tube flow-percolation coupling model applied to the reservoirs on the basis of equivalent percolation coefficient. It should be noticed that the above concept of “equivalent permeability” is relative to the pressure gradient and mainly used in the field of oil and gas extraction engineering. The concepts of “equivalent percolation coefficient” and “converted percolation coefficient” have the exact same physical meaning, except that the names are different. These two concepts are relative to the hydraulic gradient and mainly used in the field of hydraulic geological engineering. The model of the fifth category can use a unified flow law to represent the flow states in different zones of percolation, laminar flow and turbulence flow, which makes the construction and solution of the model equations relatively simple. 
     The first category of models is simple to construct and solve, but the main problem is that the model is not suitable when there are large-scale cavities in the reservoir and the distribution of different media in the reservoir is not uniform. The modeling process of the second category of models is complex and costly, which is not conducive to the widespread application. The third category of models cannot be used to determine the geometric size of the karst caves, which is the poor adaptability to the long-strip or strip-type flow system, so that the flow capacity of the large-scale karst caverns cannot be obtained. The fourth category of models is mainly used in numerical forward modeling because of its complex process and large amount of computation. At present, most of the research of models of this category assume that the fluid is incompressible steady flow, but the fluid in the real oil, gas and water reservoir is compressible or slightly compressible unstable flow. In addition, it is very difficult to construct the model because the serious heterogeneities of the oil and gas fractured cave reservoirs exist, both percolation and tube flow exist, and the tube flow-percolation coupling boundary cannot be obtained accurately. The fifth category of models has long been mainly used to realize the simple coupling cases related to wellbore and formation based on equivalent permeability or equivalent percolation coefficient or converted percolation coefficient, and tube flow (free flow)-percolation coupling flow simulation within the reservoir is solely limited in the field of underground hydraulics with published finite application, No achievements have been published in the fields of numerical simulation of oil and gas reservoirs, transient pressure well test analysis, deliverability test analysis, multi-well interference test analysis, rate production data analysis, transient temperature well test analysis, and permanent downhole monitoring data analysis. In addition, the fifth category method is only limited to the Newtonian fluid percolation, laminar flow, and turbulent flow, but not used to consider the non-Newtonian characteristics of the fluid, non-isothermal flow, diffusion and adsorption desorption effects and so on. 
     At present, there is no uniform motion equation expression of describing flow law of underground oil, gas, and water. In the coupling problem of various complex reservoirs and various types of wellbores, it is necessary to construct different governing equations and coupling conditions based on various types of reservoir and wellbore conditions, which complicates the construction and solution of the models. 
     SUMMARY OF THE INVENTION 
     The main goal of this invention is to provide a flow simulation and transient well analysis method based on generalized tube flow and percolation coupling, which can be used to overcome various types of complicated tube flow and percolation coupling problems in underground reservoirs. 
     The technical scheme of the invention is as follows: 
     This invention provides a flow simulation and transient well analysis method based on generalized tube flow and percolation coupling, which is comprised of the following steps: 
     S1: Based on defining the generalized mobility, fluid flow laws in different subset of study area are characterized by using the generalized mobility models with the same form; 
     S2: On the basis of generalized mobility, the multi-component multi-phase flow governing equations are established by considering convection term, diffusion term, accumulation term, adsorption term, and source/sink term. Then a whole set of multi-component multi-phase flow simulation equations are created through combining energy conservation equation, auxiliary equations with saturation and capillary pressure, initial saturation equation, initial pressure equation, initial temperature equation, phase equilibrium equation, and boundary condition equations: The pressure, temperature and saturation of multi-phase fluid as well as the mole percentage of each component in each phase at any point in the study areas are obtained by solving the above mentioned multi-component multi-phase flow simulation equation; 
     S3: The corresponding application software are formed by using the established multi-component multi-phase flow simulation and analysis equations. 
     Furthermore, Step S1 is as follows: 
     S11: For any type of fluid motion equation, if it can be written as v=−λ∇p, λ is called generalized mobility, where v denotes the fluid flow velocity, ∇p denotes the pressure gradient, and the generalized mobility λ is a function of space position and time; 
     S12 The general mobility models with the same form are used to characterize the flow laws of fluid in the tube flow area of wellbores, pipes, fractures, vugs, holes, cavities, caves, fracture caves, karst caves and caverns, and the percolation area of porous media. 
     Furthermore, Step S2 is as follows: 
     S2: The generalized mobility of three-dimensional multi-phase flow 
     
       
         
           
             
               
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     where k=1, 2, . . . , n p , n p  denotes the total phase number, x denotes space position, t denotes time, 9 components of generalized mobility λ k,xx , λ k,xy , λ k,xz , λ k,yx , λ k,yy , λ k,yz , λ k,zx , λ k,zy , λ k,zz  are all functions of space position x and time t; 
     S22: Rewriting the three-dimensional multi-phase flow motion equation into a standard form v k =−λ k ∇p k  by applying three-dimensional multi-phase flow generalized mobility, wherc k=1, 2, . . . , n p , n p  denotes the total phase number, 
     
       
         
           
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     is Hamilton operator, v k  is fluid velocity of k-phase, p k  is pressure of k-phase; 
     S23: Multi-component multi-phase flow simulation equation 
     The multi-component multi-phase flow equations considering convection term, diffusion term, accumulation term, adsorption term, and source/sink term are written as: 
     Component governing equation: 
     
       
         
           
             
               
                 
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     Auxiliary equations with saturation and capillary pressure: 
     Saturation equation 
     
       
         
           
             
               
                 
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     Capillary pressure equation at α-β phase interface 
         p   cαβ ( S   w )= p   α   −p   β ,( x,t )∈Ω×(0, T ],α=1, . . . , n   p ;β=1, . . . , n   p  
 
     Phase equilibrium equation: 
     The phase equilibrium constant of component i in α and β phase: 
         K   iαβ   =C   iα   /C   iβ   ,α=, . . . ,n   p ;β=1, . . . , n   p   ;i= 1, . . . , n   c  
 
     Mole percentage normalization condition of components in each phase: 
     
       
         
           
             
               
                 
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     The total mole percent of component i: 
     
       
         
           
             
               
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                     C 
                     ik 
                   
                    
                 
               
             
             , 
             
               i 
               = 
               1 
             
             , 
             … 
              
             
                 
             
             , 
             
               n 
               c 
             
           
         
       
     
     Normalization conditions for ratio of moles of each phase to the whole system: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           n 
                           c 
                         
                       
                        
                     
                     = 
                     1 
                   
                   , 
                   
                     k 
                     = 
                     1 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     n 
                     p 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Boundary Condition Equation: 
     Boundary condition equation of pressure for each phase 
     
       
         
           
             
               
                 ( 
                 
                   
                     
                       c 
                       
                         k 
                         , 
                         1 
                       
                     
                      
                     
                       p 
                       k 
                     
                   
                   + 
                   
                     
                       c 
                       
                         k 
                         , 
                         2 
                       
                     
                      
                     
                       λ 
                       k 
                     
                      
                     
                       
                         ∂ 
                         
                           p 
                           k 
                         
                       
                       
                         ∂ 
                         
                           n 
                           
                             ∂ 
                             Ω 
                           
                         
                       
                     
                   
                 
                 ) 
               
               = 
               
                 
                   g 
                   k 
                 
                  
                 
                   ( 
                   
                     x 
                     , 
                     t 
                   
                   ) 
                 
               
             
             , 
             
               
                 ( 
                 
                   x 
                   , 
                   t 
                 
                 ) 
               
               ∈ 
               
                 
                   ∂ 
                   Ω 
                 
                 × 
                 
                   ( 
                   
                     0 
                     , 
                     
                       t 
                       max 
                     
                   
                   ] 
                 
               
             
             , 
             
               k 
               = 
               1 
             
             , 
             … 
              
             
                 
             
             , 
             
               n 
               p 
             
           
         
       
     
     Boundary condition equation of temperature 
     
       
         
           
             
               
                 ( 
                 
                   
                     
                       d 
                       1 
                     
                      
                     T 
                   
                   + 
                   
                     
                       d 
                       2 
                     
                      
                     κ 
                      
                     
                       
                         ∂ 
                         T 
                       
                       
                         ∂ 
                         
                           n 
                           δΩ 
                         
                       
                     
                   
                 
                 ) 
               
               = 
               
                 w 
                  
                 
                   ( 
                   
                     x 
                     , 
                     t 
                   
                   ) 
                 
               
             
             , 
             
               
                 ( 
                 
                   x 
                   , 
                   t 
                 
                 ) 
               
               ∈ 
               
                 
                   ∂ 
                   Ω 
                 
                 × 
                 
                   ( 
                   
                     0 
                     , 
                     
                       t 
                       max 
                     
                   
                   ] 
                 
               
             
           
         
       
     
     Initial saturation equation: 
     Initial saturation equation of each phase S k (x,0)=l k (x), x∈Ω,k=1, . . . ,n p    
     Initial pressure equation: 
     Initial pressure equation of each phase 
         p   k ( x, 0)=ƒ k ( x ), x∈Ω,k= 1, . . . , n   p  
 
     Initial temperature equation: 
         T ( x, 0)=τ( x ), x∈Ω 
 
     Variable symbols description in multi-component multi-phase flow simulation analysis equations: ϕ is porosity, which is the function of average pressure  p , %; n c  is the total component number, dimensionless; n p  is the total phase number, dimensionless; λ k  is the generalized mobility of k-phase, m 2 /(Pa·s); ρ k , ρ r  are densities of k-phase and rock, kg/m 3 ; π k  is enthalpy of k phase, J/kg; U k , U r  are internal energy of k-phase and rock, J/kg; V i  is adsorption amount of component i, dimensionless; Sk is the k-phase saturation; t is time, s; t max  is the maximum of time, s; p k  is the k-phase pressure, Pa; l k  is the k-phase initial saturation distribution function; q k  is the source/sink term of k-phase, kg/(m 3 ·s); ƒ k  is the k-phase initial pressure distribution function, Pa; K iαβ  is the phase equilibrium constant of component i in α and β phase, dimensionless; C ik  is the mole percent of component i in k-phase, dimensionless; D ik  is the diffusion coefficient of component i in k-phase, m 2 /s; Z i  is the total mole percent of component i, dimensionless;   is the ratio of k-phase to the whole system, dimensionless; g k  is the boundary functions of k-phase on reservoir boundary, dimensionless; w is boundary function of temperature on reservoir boundary, dimensionless; τ is the initial temperature distribution function of reservoirs, K; p cαβ  is the capillary pressure at α-β phase interface, Pa; Ω is reservoir space; ∂Ω is reservoir boundary including internal boundary and outer boundary; c k,1  is the pressure term coefficients of k-phase on reservoir boundary, 1/Pa; c k,2  is the k-phase coefficients of derivative terms along the outer normal direction on reservoir boundary condition, s/m; d 1  is the temperature term coefficient on reservoir boundary, 1/K; d 2  is temperature coefficients of derivative terms along the outer normal direction on reservoir boundary condition, s·m 2 /J; κ is coefficient of the thermal conductivity, J/(m·s·K); n ∂Ω  is the outer normal direction on reservoir boundary, m; ∇ is the Hamilton operator; ∂ is the partial derivative sign; 
     S24: The analytical solution algorithms of above multi-component multi-phase flow simulation equations include direct solving method, Laplace transformation, Fourier transformation, and orthogonal transformation method; Their numerical methods include finite difference method, finite volume method, boundary element method, and finite element method; After solving, the pressure, temperature, and saturation in multi-phase flow are obtained, including the pressure, temperature, saturation, and mole percent of each component in each phase at any time and at any position in the study area. 
     Furthermore, Step S3 is as follows: 
     S31: The application software described above includes 5 main parts: data pre-processing system, numerical simulation system, analytical analysis system, and analysis results output system, data input and output management system, etc. Its analysis process includes reservoir definition, setting of initial and boundary conditions, wellbore and fracture setting, numerical simulator selection or fluid type and composition setting, generalized mobility model definition, grid design of numerical simulation, wellbore storage model setting, flow period and regime definition, coordinate transformation, setting of models and their type curve analysis, parameter adjustment and history fitting, dynamic prediction; 
     S32: The application software described above can be used for single and multi-well flow simulation of complex multi-component multi-phase flow reservoirs, multi-well interference analysis, deliverability analysis (i.e. steady well test analysis), transient pressure analysis (i.e. unsteady pressure well test analysis), transient rate analysis (i.e. production data analysis or rate decline analysis), transient temperature analysis (i.e. unsteady temperature analysis), well test design, and permanent downhole monitoring data analysis. The complex multi-component multi-phase flow reservoirs mentioned above include all types of fluid reservoirs, reservoirs with fluid injection, underground gas storage, ground water reservoirs, geothermal reservoirs, etc. 
     Beneficial Effects of the Invention as the Following 
     This invention simplifies the application of complex problems by defining generalized mobility, and realizes a wider application. Compared with the definition of equivalent permeability or equivalent percolation coefficient or converted percolation coefficient, it has a great improvement in the depth and breadth of application. The concept of generalized mobility has a wider range than the traditional concept. Generalized mobility covers generalized permeability or equivalent permeability or equivalent percolation coefficient or converted percolation coefficient, and is able to adapt to more complex applications. This invention extends the application of generalized permeability or equivalent permeability or equivalent percolation coefficient or converted percolation coefficient, etc. In addition, generalized mobility can be used to further define the parameters (fluid flow coefficient or transmissibility, pressure transmitting coefficient, etc.) related to fluid flow capacity and reservoir conductivity, so that the application scope of these parameters can be expanded. 
     In solving the problems of single well and multi well flow simulation analysis of complex reservoir with multi-component and multi-phase flow, inter well interference analysis, productivity analysis, transient pressure analysis (i.e. unsteady pressure well test analysis), transient flow analysis (i.e. production data analysis or production decline analysis), transient temperature analysis, well test design and downhole permanent monitoring, the modeling methods of this invention with defining the generalized mobility is more efficient than the methods of the first to the fourth categories mentioned above. At the same time, this invention realizes more and more complex models and methods compared with the fifth category method described above, and extends the application scope of the fifth category method described above. 
     The above mentioned complex multi-component multi-phase flow reservoirs include all types of reservoirs with fluid injection, underground gas storage, groundwater reservoirs, geothermal reservoirs, etc. The fluid can be a single phase of multi-component oil/gas/water and their multi-phase combinations, or it can be Newtonian fluid and non-Newtonian fluid. The compressibility of the fluid and rock as well as the transient flow characteristics are considered. The invention can unify different linear and nonlinear fluid motion equations by applying generalized mobility, so that for oil and gas reservoirs unified governing equations are able to be constructed by using the same form of motion equations in different regions and different scales. Therefore, it is more convenient to discretize the models, which reduces the complexity of the coupling problem and makes the model solution simple and unified. 
     Meanwhile, the invention can be used not only for the modeling of numerical simulation of pure cave system and large-scale fracture system, multi-well interference analysis, deliverability test analysis, transient pressure analysis, transient rate analysis, transient temperature analysis, and permanent downhole monitoring data analysis, it can also be used for well testing model constructions of fluid injection reservoirs with dominant Seepage flow channel. For the situation that fluid flow obeys linear flow laws or the generalized mobility is a constant, it is not necessary to pay attention to whether both tube flow (or free flow) and percolation exist at the same time so that the method of this invention can be applied for situations of either one of tube flow (or free flow) and percolation exists or both exist. Therefore, the modelling method of this invention could adapt to more complex reservoirs than conventional Darcy percolation modelling method. 
     This invention plays an important role for solving the problem of the single and multi-well flow simulation of complex multi-component multi-phase flow reservoirs, multi-well interference analysis, deliverability analysis, transient pressure analysis, transient rate analysis, transient temperature analysis, well test design, and permanent downhole monitoring data analysis. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows 18 types of basic structural units of a reservoir, which is provided by this invention. A complex reservoir can be formed through nesting, stacking and splicing of a plurality of basic reservoir structural units. 
         FIG. 2  shows 32 types of basic reservoir bodies, which is provided by this invention. One basic reservoir structural unit can be selected from one of 32 types of basic reservoirs. A complex reservoir can be formed through nesting, stacking and splicing of a plurality of basic reservoir structural units. Every basic reservoir structural unit can be selected from one of 32 types of basic reservoirs. 
         FIG. 3  is a schematic diagram of a complex reservoir model for tube flow and percolation coupling, which is provided in embodiment 1 of the invention. The whole reservoir space is consisted of four reservoir bodies and three different wells (wells may also be considered as a part of the reservoirs). Four reservoir bodies include two low-permeability reservoirs, one karst cave reservoir, and one high permeability reservoir; three wells include one inclined well, one vertical well, and one multi-segment fractured horizontal well. This is merely a kind of special case of the example. 
         FIG. 4  is an auxiliary plot of the product of the oil phase generalized mobility and the normal vector at the internal discrete control point, which is provided in embodiment 1 of the invention. 
         FIG. 5  is an auxiliary plot of the product of the water phase generalized mobility and the normal vector at the internal discrete control point, which is provided in embodiment 1 of the invention. 
         FIG. 6  is an auxiliary plot of the product of the gas phase generalized mobility and the normal vector at the internal discrete control point, which is provided in embodiment 1 of the invention. 
         FIG. 7  is an auxiliary plot of the product of the oil phase generalized mobility and the normal vector at the boundary discrete control point, which is provided in embodiment 1 of the invention. 
         FIG. 8  is an auxiliary plot of the product of the water phase generalized mobility and the normal vector at the boundary discrete control point, which is provided in embodiment 1 of the invention. 
         FIG. 9  is an auxiliary plot of the product of the gas phase generalized mobility and the normal vector at the boundary discrete control point, which is provided in embodiment 1 of the invention. 
         FIG. 10  is a schematic diagram of black oil model of oil/gas/water three-phase flow through treating the wellbore as inner boundary for a partially open vertical well located in a homogeneous reservoir, which is provided in embodiment 2 of the invention. 
         FIG. 11  is a schematic diagram of black oil model of oil/gas/water three-phase flow through treating the wellbore as reservoir for a partially open vertical well located in a homogeneous reservoir, which is provided in embodiment 3 of the invention. 
         FIG. 12  is a schematic diagram of oil phase flow model for a partially open vertical well located in a homogeneous reservoir through treating the wellbore as inner boundary, which is provided in embodiment 4 of the invention. 
         FIG. 13  is a schematic diagram of oil phase flow model for a partially open vertical well located in a homogeneous reservoir through treating the wellbore as reservoir, which is provided in embodiment 5 of the invention. 
         FIG. 14  is a schematic diagram of oil phase flow model for a fully open vertical well located in a homogeneous radial 2-zone composite reservoir through treating the wellbore as inner boundary, which is provided in embodiment 6 of the invention. 
         FIG. 15  is a schematic diagram of oil phase flow model for a fully open vertical well located in a homogeneous radial 2-zone composite reservoir through treating the wellbore as reservoir, which is provided in embodiment 7 of the invention. 
         FIG. 16  is a schematic diagram of a tube-shaped reservoir model, which is provided in embodiment 8 of the invention. 
         FIG. 17  is the typical pressure draw-down type curves of a tube-shaped reservoir model, which is provided in embodiment 8 of the invention. The typical characteristics include: {circle around (1)} early-stage of pressure derivative curve is parallel to the pressure difference curve with a ½ slope (linear flow); {circle around (2)} late-stage pressure derivative curve has a 1 slope (pseudo-steady flow). 
         FIG. 18  is the pressure build-up type curve corresponding to  FIG. 17 . 
         FIG. 19  is the typical pressure draw-down type curves corresponding to  FIG. 17  with consideration of skin and wellbore storage effect. The early-stage pressure derivative curve overlaps with the pressure difference curves with a slope of 1. The pressure derivative curve then goes upward due to (positive) skin action. 
         FIG. 20  is a schematic diagram of a cylindrical reservoir model, which is provided in embodiment 9 of the invention. 
         FIG. 21  is the typical pressure draw-down type curves of a cylindrical reservoir model, which is provided in embodiment 9 of the invention. The typical characteristics include: {circle around (1)} early-stage of pressure derivative curve has a −½ slope (spherical flow); {circle around (2)} middle-stage pressure derivative curve has a 0 slope (radial flow); {circle around (3)} late-stage pressure derivative curve has a 1 slope (pseudo-steady flow). 
         FIG. 22  is the pressure build-up type curve corresponding to  FIG. 21 . 
         FIG. 23  is the typical pressure draw-down type curves corresponding to  FIG. 21  with consideration of skin and wellbore storage effect. The early-stage pressure derivative curve overlaps with the pressure difference curves with a slope of 1. The pressure derivative curve then goes upward due to (positive) skin action. 
         FIG. 24  is the typical pressure draw-down type curves of a cylindrical reservoir model, which is provided in embodiment 9 of the invention. The typical characteristics include: {circle around (1)} early-stage of pressure derivative curve has a −½ slope (spherical flow); {circle around (2)} middle-stage pressure derivative curve has a ½ slope (linear flow); {circle around (3)} late-stage pressure derivative curve has a 1 slope (pseudo-steady flow). 
         FIG. 25  is the pressure build-up type curve corresponding to  FIG. 24 . 
         FIG. 26  is the typical pressure draw-down type curves corresponding to  FIG. 24  with consideration of skin and wellbore storage effect. The early-stage pressure derivative curve overlaps with the pressure difference curves with a slope of 1. The pressure derivative curve then goes upward due to (positive) skin action. 
         FIG. 27  is a schematic diagram of a spherical reservoir model, which is provided in embodiment 10 of the invention. 
         FIG. 28  is the typical pressure draw-down type curves of a spherical reservoir model, which is provided in embodiment 10 of the invention. The typical characteristics include: (Dearly-stage of pressure derivative curve has a −½ slope (spherical flow); late-stage pressure derivative curve has a 1 slope (pseudo-steady flow). 
         FIG. 29  is the pressure build-up type curve corresponding to  FIG. 28 . 
         FIG. 30  is the typical pressure draw-down type curves corresponding to  FIG. 28  with consideration of skin and wellbore storage effect. The early-stage pressure derivative curve overlaps with the pressure difference curves with a slope of 1. The pressure derivative curve then goes upward due to (positive) skin action. 
         FIG. 31  is a schematic diagram of a composite model of hollow cylinder and cylindrical reservoir, which is provided in embodiment 11 of this invention. 
         FIG. 32  is the typical pressure draw-down type curves of a composite model of hollow cylinder and cylindrical reservoir, which is provided in embodiment 11 of the invention. The typical characteristics is that pressure derivative curve successively shows a −½ slope (spherical flow), 0 slope (radial flow of cylindrical reservoir), the junction of cylindrical reservoir and hollow cylindrical reservoir goes upward (transition regime), 0 slope (radial flow of hollow cylindrical reservoir), 1 slope (pseudo-steady flow). 
         FIG. 33  is the pressure build-up type curve corresponding to  FIG. 32 . 
         FIG. 34  is the typical pressure draw-down type curves corresponding to  FIG. 32  with consideration of skin and wellbore storage effect. The early-stage pressure derivative curve overlaps with the pressure difference curves with a slope of 1. The pressure derivative curve then goes upward due to (positive) skin action. 
         FIG. 35  is the typical pressure draw-down type curves of a hollow cylinder and cylindrical reservoir composite model, which is provided in embodiment 11 of the invention. The typical characteristics is that pressure derivative curve successively shows a −½ slope (spherical flow), 0 slope (radial flow of cylindrical reservoir), the junction of cylindrical reservoir and hollow cylindrical reservoir goes downward with a −½ slope (transient period), 0 slope (radial flow of hollow cylindrical reservoir), 1 slope (pseudo-steady flow). 
         FIG. 36  is the pressure build-up type curve corresponding to  FIG. 35 . 
         FIG. 37  is the typical pressure draw-down type curves corresponding to  FIG. 35  with consideration of skin and wellbore storage effect. The early-stage pressure derivative curve overlaps with the pressure difference curves with a slope of 1. The pressure derivative curve then goes upward due to (positive) skin action. 
         FIG. 38  is a schematic diagram of a composite model of tube-shaped and spherical reservoir, which is provided in embodiment 12 of the invention. 
         FIG. 39  is the typical pressure draw-down type curves of tube-shaped and spherical reservoir composite model, which is provided in embodiment 12 of the invention. The typical characteristics include: {circle around (1)} early-stage of pressure derivative curve is parallel to the pressure difference curve with a ½ slope (linear flow); {circle around (2)} middle-stage pressure derivative curve goes downward; {circle around (3)} late-stage pressure derivative curve has a 1 slope (pseudo-steady flow). 
         FIG. 40  is the pressure build-up type curve corresponding to  FIG. 39 . 
         FIG. 41  is the typical pressure draw-down type curves corresponding to  FIG. 39  with consideration of skin and wellbore storage effect. The early-stage pressure derivative curve overlaps with the pressure difference curves with a slope of 1. The pressure derivative curve then goes upward due to (positive) skin action. 
         FIG. 42  is the vertical well Blasingame type curves based on a cylindrical reservoir model, which is provided in embodiment 9 of the invention. 
         FIG. 43  is the vertical well Agarwal-Gardner type curves based on a cylindrical reservoir model, which is provided in embodiment 9 of the invention. 
         FIG. 44  is the vertical well NP1 type curves based on a cylindrical reservoir model, which is provided in embodiment 9 of the invention. 
         FIG. 45  is the vertical well Transient type curves based on a cylindrical reservoir model, which is provided in embodiment 9 of the invention. 
         FIG. 46  is the schematic diagram of a one-dimensional tube-shaped reservoir model, which is provided in embodiment 13 [high-speed non-Darcy flow regime I (0≤x≤X 1 )+Darcy flow regime II (X 1 ≤x≤X 2 )]. 
         FIG. 47  is the generalized mobility type curves of Darcy flow regime II of a one-dimensional tube-shaped reservoir model, which is provided in embodiment 13 [high-speed non-Darcy flow regime I (0≤x≤X 1 )+Darcy flow regime II (X 1 ≤x≤X 2 )]. 
         FIG. 48  is the generalized mobility type curves of Darcy flow regime II of a one-dimensional tube-shaped reservoir model, which is provided in embodiment 13 [β=0 in high-speed non-Darcy flow regime I becomes Darcy flow regime I (0≤x≤X 1 )+Darcy flow regime II(X 1 ≤x≤X 2 )]. 
         FIG. 49  is the comparison plot between high-speed non-Darcy flow and Darcy flow of a one-dimensional tube-shaped reservoir model, which is provided in embodiment 13 (it becomes Darcy flow when β=0 in high-speed non-Darcy flow). 
         FIG. 50  is a generalized mobility variation type curve of transient temperature analysis model with oil single-phase flow, which is provided by embodiment 18: The larger the generalized mobility, the lower the position of temperature difference and temperature difference derivative curve. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     The name of the invention is a flow simulation and transient well analysis method based on generalized tube flow and percolation coupling. 
     Through comparing the Hagen-Poiseuille (1839,1840) formula of laminar flow and tube flow with the Darcy formula of conventional percolation, it is found that there is no essential difference in mathematical expression form between the two formulas. They both are linear functions with positive proportion i.e. The average velocity of fluid is proportional to the pressure gradient. Fluid flow laws described by both laminar and tube flow formula and Darcy&#39;s conventional percolation formula is linear flow. However, Fluid flow laws, which is described by turbulent tube flow equation, Forchheimer (1901) binomial high-speed non-Darcy formula, power-law non-Newton formula, low-speed non-Darcy percolation equation, free flow Navier-Stokes equation, etc, is a nonlinear flow. In order to unify the linear flow and nonlinear flow and consider the anisotropy of the reservoir or others, this invention proposed that the fluid flow velocity of underground fluid flow motion equations are uniformly written as the product of the generalized mobility and the pressure gradient of the negative direction, which means 
         v   k =−λ k   ∇p   k  
 
     where 
     
       
         
           
             
               λ 
               k 
             
             = 
             
               [ 
               
                 
                   
                     
                       
                         λ 
                         
                           k 
                           , 
                           xx 
                         
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   
                     
                         
                     
                   
                   
                     
                       
                         λ 
                         
                           k 
                           , 
                           xy 
                         
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         λ 
                         
                           k 
                           , 
                           xz 
                         
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       
                         λ 
                         
                           k 
                           , 
                           yx 
                         
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   
                     
                         
                     
                   
                   
                     
                       
                         λ 
                         
                           k 
                           , 
                           yy 
                         
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         λ 
                         
                           k 
                           , 
                           yz 
                         
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       
                         λ 
                         
                           k 
                           , 
                           zx 
                         
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   
                     
                         
                     
                   
                   
                     
                       
                         λ 
                         
                           k 
                           , 
                           zy 
                         
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         λ 
                         
                           k 
                           , 
                           zz 
                         
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     is the generalized mobility of three-dimensional multi-phase flow [unit: m2/(Pa·s)]. where k=1, 2, . . . , n p , n p  denotes the total phase number, x denotes space position, t denotes time, 9 components of generalized mobility λ k,xx , λ k,xy , λ k,xz , λ k,yx , λ k,yy , λ k,yz , λ k,zx , λ k,zy , λ k,zz  are all functions of space position x and time t. 
     
       
         
           
             ∇ 
             
               = 
               
                 
                   ( 
                   
                     
                       ∂ 
                       
                         ∂ 
                         x 
                       
                     
                     , 
                     
                       ∂ 
                       
                         ∂ 
                         y 
                       
                     
                     , 
                     
                       ∂ 
                       
                         ∂ 
                         z 
                       
                     
                   
                   ) 
                 
                 T 
               
             
           
         
       
     
     is Hamilton operator (unit: 1/m), v k  is fluid velocity (m/s), p k  is pressure (Pa). 
     Because the generalized mobility defined in this invention itself covers the concepts of generalized permeability or equivalent permeability, equivalent percolation coefficient or converted percolation coefficient, when solving some relatively simple practical problems, the use of these definitions can also achieve the similar effect of generalized mobility. If the parameters (such as fluid flow coefficient or transmissibility, pressure transmitting coefficient, etc.) related to fluid flow capacity and reservoir conductivity are defined by using the definition method of this invention, the application scope of the relevant parameters can be extended. 
     The invention can be applied to fluid types including multi-component oil/gas/water single-phase flow and multi-component multi-phase flow. The fluid types can be Newton and non-Newtonian. The flowing laws can be linear, non-linear, or locally linear and locally nonlinear. 
     The further detailed descriptions of this invention are given in following by combining with drawings. The following embodiments are described only for the purpose of illustration and not for any limitation to the scope of the present disclosure. 
     A flow simulation and transient well analysis method based on generalized tube flow and percolation coupling includes: 
     S1: Based on defining the generalized mobility, fluid flow laws in different subset of study area are characterized by using the generalized mobility models with the same form; 
     S2: On the basis of generalized mobility, the multi-component multi-phase flow governing equations are established by considering convection term, diffusion term, accumulation term, adsorption term, and source/sink term. Then a whole set of multi-component multi-phase flow simulation equations are created through combining energy conservation equation, auxiliary equations with saturation and capillary pressure, initial saturation equation, initial pressure equation, initial temperature equation, phase equilibrium equation, and boundary condition equations; The pressure, temperature and saturation of multi-phase fluid as well as the mole percentage of each component in each phase at any point in the study areas are obtained by solving the above multi-component multi-phase flow simulation equation; 
     S3: The corresponding application software are formed by using the established multi-component multi-phase flow simulation and analysis equations. 
     wherein S1 (Based on defining the generalized mobility, fluid flow laws in different subset of study area are characterized by using the same formed generalized mobility models) includes: 
     (1) For any type of fluid motion equation, if it can be written as v=−λ∇p, λ is called generalized mobility, where v denotes the fluid flow velocity, ∇p denotes the pressure gradient, and the generalized mobility λ is a function of space position and time.
 
(2) The general mobility models with the same form are used to characterize the flow laws of fluid in the tube flow area of wellbores, pipes, fractures, vugs, holes, cavities, caves, fracture caves, karst caves and caverns, and the percolation area of porous media.
 
     wherein S2 (On the basis of generalized mobility, the multi-component multi-phase flow governing equations are established by considering convection term, diffusion term, accumulation term, adsorption term, and source/sink term. Then a whole set of multi-component multi-phase flow simulation equations are created through combining energy conservation equation, auxiliary equations with saturation and capillary pressure, initial saturation equation, initial pressure equation, initial temperature equation, phase equilibrium equation, and boundary condition equations; The pressure, temperature and saturation of multi-phase fluid as well as the mole percentage of each component in each phase at any point in the study areas are obtained by solving the above multi-component multi-phase flow simulation equation) includes: 
     (1) The generalized mobility of three-dimensional multi-phase flow 
     
       
         
           
             
               
                 λ 
                 k 
               
               = 
               
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                           λ 
                           
                             k 
                             , 
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                          
                         
                           ( 
                           
                             x 
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                           ) 
                         
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         
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                          
                         
                           ( 
                           
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                           λ 
                           
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                           ( 
                           
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                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           λ 
                           
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                          
                         
                           ( 
                           
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                           ) 
                         
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         
                           λ 
                           
                             k 
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                          
                         
                           ( 
                           
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                           ( 
                           
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                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           λ 
                           
                             k 
                             , 
                             zx 
                           
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         
                           λ 
                           
                             k 
                             , 
                             zy 
                           
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           λ 
                           
                             k 
                             , 
                             zz 
                           
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
           
         
       
     
     where k=1, 2, . . . , n p , n p  denotes the total phase number, x denotes space position, t denotes time, 9 components of generalized mobility λ k,xx , λ k,xy , λ k,xz , λ k,yx , λ k,yy , λ k,yz , λ k,zx , λ k,zy , λ k,zz  are all functions of space position x and time t. 
     From the generalized mobility of three-dimensional multi-phase flow, it is easy to obtain the generalized mobility of two-dimensional multi-phase flow 
     
       
         
           
             
               
                 λ 
                 k 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           λ 
                           
                             k 
                             , 
                             xx 
                           
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           λ 
                           
                             k 
                             , 
                             xy 
                           
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           λ 
                           
                             k 
                             , 
                             yx 
                           
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           λ 
                           
                             k 
                             , 
                             yy 
                           
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
           
         
       
     
     where k=1, 2, . . . , n p , n p  denotes the total phase number, x denotes space position, t denotes time, 4 components of generalized mobility λ k,xx , λ k,xy , λ k,yx , λ k,yy , are all functions of space position x and time t. 
     From the generalized mobility of three-dimensional multi-phase flow, it is easy to obtain the generalized mobility of one-dimensional multi-phase flow λ k =λ k,xx (x,t), where k=1, 2, . . . , n p , n p  denotes the total phase number, x denotes space position, t denotes time. 
     Rewriting the three-dimensional multi-phase flow motion equation into a standard form v k =−λ k ∇p k  by applying three-dimensional multi-phase flow generalized mobility, where k=1, 2, . . . , n p , n p  denotes the total phase number, 
     
       
         
           
             ∇ 
             
               = 
               
                 
                   ( 
                   
                     
                       ∂ 
                       
                         ∂ 
                         x 
                       
                     
                     , 
                     
                       ∂ 
                       
                         ∂ 
                         y 
                       
                     
                     , 
                     
                       ∂ 
                       
                         ∂ 
                         z 
                       
                     
                   
                   ) 
                 
                 T 
               
             
           
         
       
     
     is Hamilton operator, v k  is fluid velocity of k-phase, p k  is pressure of k-phase. 
     Rewriting the two-dimensional multi-phase flow motion equation into a standard form v k =−λ k ∇p k  by applying two-dimensional multi-phase flow generalized mobility, where k=1, 2, . . . , n p , n p  denotes the total phase number. 
     
       
         
           
             ∇ 
             
               = 
               
                 
                   ( 
                   
                     
                       ∂ 
                       
                         ∂ 
                         x 
                       
                     
                     , 
                     
                       ∂ 
                       
                         ∂ 
                         y 
                       
                     
                   
                   ) 
                 
                 T 
               
             
           
         
       
     
     is Hamilton operator, v k  is fluid velocity of k-phase, p k  is pressure of k-phase. 
     Rewriting the one-dimensional multi-phase flow motion equation into a standard form v k =λ k ∇p k  by applying one-dimensional multi-phase flow generalized mobility, where k=1, 2, . . . , n p , n p  denotes the total phase number, 
     
       
         
           
             ∇ 
             
               = 
               
                 ∂ 
                 
                   ∂ 
                   x 
                 
               
             
           
         
       
     
     is Hamilton operator, v k  is fluid velocity of k-phase, p k  is pressure of k-phase.
 
(3) Multi-component multi-phase flow simulation equation
 
     The multi-component multi-phase flow equations considering convection term, diffusion term, accumulation term, adsorption term, and source/sink term are written as: 
     Component governing equation: 
     
       
         
           
             
               
                 
                   [ 
                   
                     ∇ 
                     
                       · 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           
                             n 
                             p 
                           
                         
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             
                               ρ 
                               k 
                             
                              
                             
                               C 
                               ik 
                             
                              
                             
                               λ 
                               k 
                             
                              
                             
                               ∇ 
                               
                                 p 
                                 k 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 + 
                 
                   [ 
                   
                     ∇ 
                     
                       · 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           
                             n 
                             p 
                           
                         
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             
                               φρ 
                               k 
                             
                              
                             
                               S 
                               k 
                             
                              
                             
                               D 
                               ik 
                             
                              
                             
                               ∇ 
                               
                                 C 
                                 ik 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
               
               = 
               
                 
 
               
                
               
                 
                   
                     ∂ 
                     
                       ∂ 
                       t 
                     
                   
                    
                   
                     [ 
                     
                       φ 
                        
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           
                             n 
                             p 
                           
                         
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             
                               ρ 
                               k 
                             
                              
                             
                               S 
                               k 
                             
                              
                             
                               C 
                               ik 
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
                 + 
                 
                   
                     ∂ 
                     
                       ∂ 
                       t 
                     
                   
                    
                   
                     [ 
                     
                       
                         ( 
                         
                           1 
                           - 
                           φ 
                         
                         ) 
                       
                        
                       
                         ρ 
                         r 
                       
                        
                       
                         V 
                         i 
                       
                     
                     ] 
                   
                 
                 + 
                 
                   
 
                 
                  
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       1 
                     
                     
                       n 
                       p 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       C 
                       ik 
                     
                      
                     
                       q 
                       k 
                     
                   
                 
               
             
             , 
             
               
                 ( 
                 
                   x 
                   , 
                   t 
                 
                 ) 
               
               ∈ 
               
                 Ω 
                 × 
                 
                   ( 
                   
                     0 
                     , 
                     
                       t 
                       max 
                     
                   
                   ] 
                 
               
             
             , 
             
               i 
               = 
               1 
             
             , 
             2 
             , 
             ... 
             
                 
             
             , 
             
                 
             
              
             
               n 
               c 
             
           
         
       
     
     Energy conservation equation: 
     
       
         
           
             
               
                 
                   [ 
                   
                     ∇ 
                     
                       · 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           
                             n 
                             p 
                           
                         
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             
                                
                               k 
                             
                              
                             
                               ρ 
                               k 
                             
                              
                             
                               λ 
                               k 
                             
                              
                             
                               ∇ 
                               
                                 p 
                                 k 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 + 
                 
                   [ 
                   
                     
                       ∇ 
                       
                         · 
                         κ 
                       
                     
                      
                     
                       ∇ 
                       T 
                     
                   
                   ] 
                 
               
               = 
               
                 
 
               
                
               
                 
                   
                     ∂ 
                     
                       ∂ 
                       t 
                     
                   
                    
                   
                     [ 
                     
                       φ 
                        
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           
                             n 
                             p 
                           
                         
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             
                               ρ 
                               k 
                             
                              
                             
                               S 
                               k 
                             
                              
                             
                               U 
                               k 
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
                 + 
                 
                   
                     ∂ 
                     
                       ∂ 
                       t 
                     
                   
                    
                   
                     [ 
                     
                       
                         ( 
                         
                           1 
                           - 
                           φ 
                         
                         ) 
                       
                        
                       
                         ρ 
                         r 
                       
                        
                       
                         U 
                         r 
                       
                     
                     ] 
                   
                 
                 + 
                 
                   
 
                 
                  
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       1 
                     
                     
                       n 
                       p 
                     
                   
                    
                   
                     
                        
                       k 
                     
                      
                     
                       q 
                       k 
                     
                   
                 
               
             
             , 
             
               
                 ( 
                 
                   x 
                   , 
                   t 
                 
                 ) 
               
               ∈ 
               
                 Ω 
                 × 
                 
                   ( 
                   
                     0 
                     , 
                     
                       t 
                       max 
                     
                   
                   ] 
                 
               
             
           
         
       
     
     Auxiliary equations with saturation and capillary pressure: 
     Saturation equation 
     
       
         
           
             
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   
                     n 
                     p 
                   
                 
                  
                 
                   S 
                   k 
                 
               
               = 
               1 
             
             , 
             
               
                 ( 
                 
                   x 
                   , 
                   t 
                 
                 ) 
               
               ∈ 
               
                 Ω 
                 × 
                 
                   ( 
                   
                     0 
                     , 
                     
                       t 
                       max 
                     
                   
                   ] 
                 
               
             
           
         
       
     
     Capillary pressure equation at α-β phase interface 
         p   cαβ ( S   w )= p   α - p   β ,( x,t )∈Ω×(0, T ],α=1, . . . , n   p ;β 1   , . . . ,n   p  
 
     Equilibrium equation: 
     The equilibrium constant of component i in α and β phase: 
         K   iαβ   =C   iα   /C   iβ ,α=1, . . . , n   p ;β=1, . . . , n   p   ;i= 1, . . . , n   c  
 
     Mole percentage normalization condition of components in each phase: 
     
       
         
           
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   
                     n 
                     c 
                   
                 
                  
                 
                     
                 
                  
                 
                   C 
                   ik 
                 
               
               = 
               1 
             
             , 
             
               k 
               = 
               1 
             
             , 
             ... 
             
                 
             
             , 
             
               n 
               p 
             
           
         
       
     
     The total mole percent of component i: 
     
       
         
           
             
               
                 Z 
                 i 
               
               = 
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   
                     n 
                     p 
                   
                 
                  
                 
                     
                 
                  
                 
                   
                     C 
                     ik 
                   
                    
                 
               
             
             , 
             
               i 
               = 
               1 
             
             , 
             ... 
             
                 
             
             , 
             
               n 
               c 
             
           
         
       
     
     Normalizing conditions for ratio of moles of each phase to the whole system: 
     
       
         
           
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   
                     n 
                     c 
                   
                 
                  
                 
                     
                 
                  
               
               = 
               1 
             
             , 
             
               k 
               = 
               1 
             
             , 
             ... 
             
                 
             
             , 
             
               n 
               p 
             
           
         
       
     
     Boundary condition equation: 
     Boundary condition equation of pressure for each phase 
     
       
         
           
             
               
                 ( 
                 
                   
                     
                       c 
                       
                         k 
                         , 
                         1 
                       
                     
                      
                     
                       p 
                       k 
                     
                   
                   + 
                   
                     
                       c 
                       
                         k 
                         , 
                         2 
                       
                     
                      
                     
                       λ 
                       k 
                     
                      
                     
                       
                         ∂ 
                         
                           p 
                           k 
                         
                       
                       
                         ∂ 
                         
                           n 
                           
                             ∂ 
                             Ω 
                           
                         
                       
                     
                   
                 
                 ) 
               
               = 
               
                 
                   g 
                   k 
                 
                  
                 
                   ( 
                   
                     x 
                     , 
                     t 
                   
                   ) 
                 
               
             
             , 
             
               
                 ( 
                 
                   x 
                   , 
                   t 
                 
                 ) 
               
               ∈ 
               
                 
                   ∂ 
                   Ω 
                 
                 × 
                 
                   ( 
                   
                     0 
                     , 
                     
                       t 
                       max 
                     
                   
                   ] 
                 
               
             
             , 
             
               k 
               = 
               1 
             
             , 
             ... 
             
                 
             
             , 
             
               
                 
                   n 
                   p 
                 
                  
                 
                   
 
                 
                 ( 
                 
                   
                     
                       d 
                       1 
                     
                      
                     T 
                   
                   + 
                   
                     
                       d 
                       2 
                     
                      
                     κ 
                      
                     
                       
                         ∂ 
                         T 
                       
                       
                         ∂ 
                         
                           n 
                           
                             ∂ 
                             Ω 
                           
                         
                       
                     
                   
                 
                 ) 
               
               = 
               
                 w 
                  
                 
                   ( 
                   
                     x 
                     , 
                     t 
                   
                   ) 
                 
               
             
             , 
             
               
                 ( 
                 
                   x 
                   , 
                   t 
                 
                 ) 
               
               ∈ 
               
                 
                   ∂ 
                   Ω 
                 
                 × 
                 
                   ( 
                   
                     0 
                     , 
                     
                       t 
                       max 
                     
                   
                   ] 
                 
               
             
           
         
       
     
     Initial saturation equation: 
     Initial saturation equation of each phase 
         S   k ( x, 0)= l   k ( x ), x∈Ω,k= 1, . . . , n   p    
     Initial pressure equation: 
     Initial pressure equation of each phase: 
         p   k ( x, 0)=ƒ k ( x ), x∈Ω,k= 1, . . . , n   p  
 
     Initial temperature equation: 
         T ( x, 0)=τ( x ), x∈Ω 
 
     Variable symbols description in multi-component multi-phase flow simulation analysis equations: ϕ is porosity, which is the function of average pressure  p , %; n c  is the total component number, dimensionless; n p  is the total phase number, dimensionless; λ k  is the generalized mobility of k-phase, m 2 /(Pa·s); ρ k , ρ r  are densities of k-phase and rock, kg/m 3 ; π k  is enthalpy of k phase, J/kg; U k , U r  are internal energy of k-phase and rock, J/kg; Vi is adsorption amount of component i, dimensionless; Sk is the k-phase saturation; t is time, s; t max  is the maximum of time, s; p k  is the k-phase pressure, Pa; l k  is the k-phase initial saturation distribution function q k  is the source/sink term of k-phase, kg/(m 3 ·s); ƒ k  is the k-phase initial pressure distribution function, Pa; K iαβ  is the phase equilibrium constant of component i in α and β phase, dimensionless; C ik  is the mole percent of component i in k-phase, dimensionless; D ik  is the diffusion coefficient of component i in k-phase, m 2 /s; Z i  is the total mole percent of component i, dimensionless;   is the ratio of k-phase to the whole system, dimensionless; g k  is the boundary functions of k-phase on reservoir boundary, dimensionless; w is boundary function of temperature on reservoir boundary, dimensionless; τ is the initial temperature distribution function of reservoirs, K; pee is the capillary pressure at α-β phase interface, Pa; Ω is reservoir space; ∂Ω is reservoir boundary including internal boundary and outer boundary; c k,1  is the pressure term coefficients of k-phase on reservoir boundary, 1/Pa c k,2  is the k-phase coefficients of derivative terms along the outer normal direction on reservoir boundary condition, s/m; d 1  is the temperature term coefficient on reservoir boundary, 1/K; d 2  is temperature coefficients of derivative terms along the outer normal direction on reservoir boundary condition, s·m 2 /J; κ is coefficient of the thermal conductivity, J/(m·s·K); n ∂Ω  is the outer normal direction on reservoir boundary, m; ∇ is the Hamilton operator; ∂ is the partial derivative sign; 
     (4) The analytical solution of above multi-component multi-phase flow simulation equations include direct solving method, Laplace transformation, Fourier transformation, and orthogonal transformation method; The numerical methods include finite difference method, finite volume method, boundary element method, and finite element method; After solving, the pressure, temperature, and saturation in multi-phase flow are obtained, including the pressure, temperature, saturation, and mole percent of each component in each phase at any time and at any position in the study area. 
     Wherein S3 (The corresponding application software are developed on the basis of multi-component multi-phase flow simulation analysis equations) includes: 
     (1) The application software described above includes 5 main parts: data pre-processing system, numerical simulation system, analytical analysis system, and analysis results output system, data input and output management system, etc. The key technologies include reservoir definition, setting of initial and boundary conditions, wellbore and fracture setting, numerical simulator selection or fluid type and composition setting, generalized mobility model definition (including defining generalized permeability or equivalent permeability, etc., in simplified cases), grid design of numerical simulation, wellbore storage model setting, flow period and regime definition, coordinate transformation, setting of models and their type curve analysis, parameter adjustment and history fitting, dynamic prediction. The details are as follows: 
     (1.1) Reservoir Definition 
     The basic reservoir structural unit is shown in  FIG. 1 . The reservoir is composed of a single or several basic reservoir structural units, which can be nested, stacked and spliced among each other. 
     Reservoir boundaries include constant pressure boundary, variable pressure boundary, sealed boundary, semi-permeable boundary, infinite acting boundary and so on. 
     The reservoir types are as shown in  FIG. 2 . The reservoir types of basic reservoir structural unit include homogeneous, dual porous media, triple porous media, fractal media, cavities caves, caverns, fractures and so on. 
     (1.2) Setting of Initial and Boundary Conditions 
     Initialization of pressure, temperature, saturation and internal and external boundary conditions. The Inner boundary models include effective wellbore diameter model, source function model and so on. 
     (1.3) Wellbore and Fracture Setting 
     Well types include vertical well, horizontal well, inclined well, fractured well, multibranch well and so on. 
     The wellbore and fracture types include uniform flux, infinite conductivity, and finite conductivity and so on. 
     The wellbore and fracture opening degree includes fully open and partially open. 
     (1.4) Numerical Simulator Selection or Fluid Type and Composition Setting 
     The numerical simulator selection includes black oil model, compositional model and thermal recovery model. The fluid phases (including composition) include oil single-phase, water single-phase, gas single-phase, oil-water two-phase, oil-gas two-phase, gas-water two-phase, oil-gas-water three phase, condensate gas, carbon dioxide, foam fluid, polymer and so on. 
     (1.5) Generalized Mobility Model Definition (Including Defining Generalized Permeability or Equivalent Permeability, Etc., in Simplified Cases) 
     The generalized mobility models include: {circle around (1)} Newton fluid+Darcy flow, {circle around (2)} Newton fluid+Laminar tube flow, {circle around (3)} Newton fluid+High speed Non-Darcy flow, {circle around (4)} Newton fluid+turbulent tube flow, {circle around (5)} Newton fluid+pressure sensitive flow, {circle around (6)} Non-Newton fluid+Darcy flow, {circle around (7)} Non-Newton fluid+Laminar tube flow, {circle around (8)} Non-Newton fluid+pressure sensitive flow, {circle around (9)} Non-Newton fluid+turbulent tube flow, {circle around (10)} Newton fluid+low speed non-Darcy flow, {circle around (11)} Newton fluid+slippage effect, etc. 
     EXAMPLES 
     {circle around (1)} Newton fluid+Darcy flow 
     On the basis of Darcy&#39;s percolation formula (1856), λ is written as: 
     
       
         
           
             λ 
             = 
             
               K 
               μ 
             
           
         
       
     
     Where μ is the viscosity of the fluid, K is the permeability, 
     For the form of considering the influence of gravity, λ is written as: 
     
       
         
           
             λ 
             = 
             
               
                 K 
                 μ 
               
                
               
                 ( 
                 
                   1 
                   - 
                   
                     
                       ρ 
                        
                       g 
                        
                       cos 
                        
                       α 
                     
                     
                        
                       
                         
                           ∂ 
                           p 
                         
                         
                           ∂ 
                           l 
                         
                       
                        
                     
                   
                 
                 ) 
               
             
           
         
       
     
     Where μ is the viscosity of the fluid, K is the permeability, ρ is the density of the fluid, g is the acceleration of gravity, p is the pressure, l is the distance, α is the angle between pressure gradient direction and gravity direction. 
     {circle around (2)} Newton fluid+Laminar tube flow 
     On the basis of Hagen-poiseuille&#39;s formula (1839,1840), λ is written as: 
     
       
         
           
             λ 
             = 
             
               
                 d 
                 2 
               
               
                 3 
                  
                 2 
                  
                 μ 
               
             
           
         
       
     
     Where μ is the viscosity of the fluid, d is the hydraulic diameter of tubes. 
     {circle around (3)} Newton fluid+High speed Non-Darcy flow 
     On the basis of Forchheimer binomial high-speed Non-darcy formula (1901), λ is written as: 
     
       
         
           
             λ 
             = 
             
               1 
               
                 
                   μ 
                   K 
                 
                 + 
                 
                   βρ 
                    
                   
                      
                     v 
                      
                   
                 
               
             
           
         
       
     
     Where μ is the viscosity of the fluid, K is the permeability, β is the high speed Non-Darcy factor, ρ is the density of the fluid, v is the velocity of the fluid. 
     {circle around (4)} Newton fluid+turbulent tube flow 
     On the basis of Darcy-Weisbach formula (1845) and Colebrook on-way resistance formula (1938), λ is written as: 
     
       
         
           
             λ 
             = 
             
               0.0232 
                
               
                 d 
                 
                   ρ 
                    
                   
                       
                   
                    
                   v 
                 
               
                
               
                 
                   ( 
                   
                     
                       
                         
                           
                             d 
                              
                             
                                 
                             
                              
                             ρ 
                              
                             
                                 
                             
                              
                             v 
                           
                            
                           
                               
                           
                         
                         d 
                       
                        
                       
                         ɛ 
                         d 
                       
                     
                     - 
                     
                       18.574 
                        
                       
                         W 
                          
                         
                           ( 
                           
                             0.199 
                              
                             
                               
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 ρ 
                                  
                                 
                                     
                                 
                                  
                                 v 
                               
                               μ 
                             
                              
                             
                               e 
                               
                                 0.0538 
                                  
                                 
                                   
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     ρ 
                                      
                                     
                                         
                                     
                                      
                                     v 
                                   
                                   μ 
                                 
                                  
                                 
                                   ɛ 
                                   d 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
                 2 
               
             
           
         
       
     
     Where μ is the viscosity of the fluid, d is the hydraulic diameter of tubelines, ρ is the density of the fluid, ε/d is the relative roughness, v is the velocity, W(g) is the Lambert W function or product logarithm function. 
     {circle around (5)} Newton fluid+pressure sensitive flow 
     On the basis of permeability modulus (pressure sensitivity) formula, λ is written as 
     
       
         
           
             λ 
             = 
             
               
                 
                   K 
                   i 
                 
                  
                 
                   e 
                   
                     γ 
                      
                     
                       ( 
                       
                         p 
                         - 
                         
                           p 
                           i 
                         
                       
                       ) 
                     
                   
                 
               
               μ 
             
           
         
       
     
     Where μ is the viscosity of the fluid, K i  is the initial permeability of reservoir, γ is the permeability modulus, P is the pressure, p i  is the initial reservoir pressure. 
     {circle around (6)} Non-Newton fluid+Darcy flow 
     On the basis of the Darcy&#39;s percolation formula, Qstwald-DeWaele power-law fluid viscosity formula (1923,1925) and Hirasaki-Pope shear rate formula (1974), λ is written as: 
     
       
         
           
             λ 
             = 
             
               
                 ( 
                 
                   k 
                   / 
                   
                     μ 
                     eff 
                   
                 
                 ) 
               
               · 
               
                 
                    
                   v 
                    
                 
                 
                   1 
                   - 
                   n 
                 
               
             
           
         
       
       
         
           
             
               μ 
               eff 
             
             = 
             
               
                 H 
                 
                   1 
                    
                   2 
                 
               
                
               
                 
                   ( 
                   
                     9 
                     + 
                     
                       3 
                       n 
                     
                   
                   ) 
                 
                 n 
               
                
               
                 
                   ( 
                   
                     1 
                      
                     5 
                      
                     0 
                      
                     K 
                      
                     φ 
                   
                   ) 
                 
                 
                   
                     1 
                     - 
                     n 
                   
                   2 
                 
               
             
           
         
       
     
     Where μ eff  is the effective viscosity, K is the permeability, n is the power-law index, v is the fluid flow velocity, H is the consistency coefficient, ϕ is the porosity. 
     {circle around (7)} Non-Newton fluid+Laminar tube flow 
     On the basis of Hagen-poiseuille laminar tube flow formula (1839,1840), Qstwald-DeWaele power-law fluid viscosity formula (1923,1925) and Hirasaki-Pope shear rate formula (1974), λ is written as: 
     
       
         
           
             λ 
             = 
             
               
                 d 
                 2 
               
               
                 32 
                  
                 
                   μ 
                   eff 
                 
                  
                 
                   v 
                   
                     n 
                     - 
                     1 
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               μ 
               
                 e 
                  
                 f 
                  
                 f 
               
             
             = 
             
               
                 H 
                 12 
               
                
               
                 
                   ( 
                   
                     9 
                     + 
                     
                       3 
                       n 
                     
                   
                   ) 
                 
                 n 
               
                
               
                 
                   ( 
                   
                     
                       75 
                        
                       
                         d 
                         2 
                       
                     
                     
                       1 
                        
                       6 
                     
                   
                   ) 
                 
                 
                   
                     1 
                     - 
                     n 
                   
                   2 
                 
               
             
           
         
       
     
     Where μ eff  is the effective viscosity, d is the hydraulic diameter of tubes, n is the power-law index, v is the fluid flow velocity, H is the consistency coefficient. 
     {circle around (8)} Non-Newton fluid+pressure sensitive flow 
     On the basis of the permeability modulus(stress sensitivity) formula, Qstwald-DeWaele power-law fluid viscosity formula (1923,1925) and Hirasaki-Pope shear rate formula (1974), λ is written as: 
     
       
         
           
             λ 
             = 
             
               
                 
                   K 
                   i 
                 
                  
                 
                   e 
                   
                     γ 
                      
                     
                       ( 
                       
                         p 
                         - 
                         
                           p 
                           i 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   μ 
                   eff 
                 
                  
                 
                   v 
                   
                     n 
                     - 
                     1 
                   
                 
               
             
           
         
       
       
         
           
             
               μ 
               eff 
             
             = 
             
               
                 H 
                 
                   1 
                    
                   2 
                 
               
                
               
                 
                   ( 
                   
                     9 
                     + 
                     
                       3 
                       n 
                     
                   
                   ) 
                 
                 n 
               
                
               
                 
                   ( 
                   
                     1 
                      
                     5 
                      
                     0 
                      
                     K 
                      
                     φ 
                   
                   ) 
                 
                 
                   
                     1 
                     - 
                     n 
                   
                   2 
                 
               
             
           
         
       
     
     Where μ eff  is the effective viscosity, K i  is the initial permeability of reservoir, γ is the permeability modulus, p is the pressure, p i  is the initial reservoir pressure, v is the fluid flow velocity, n is the power-law index, H is the consistency coefficient, K is the permeability, ϕ is the porosity. 
     {circle around (9)} Non-Newton fluid+turbulent tube flow 
     On the basis of Darcy-Weisbach formula (1845), Colebrook resistance formula along the way (1938), Qstwald-DeWaele power-law fluid viscosity formula (1923,1925), and Hirasak-Pope shear rate formula (1974), λ is written as: 
     
       
         
           
             λ 
             = 
             
               0.0232 
                
               
                 d 
                 
                   ρ 
                    
                   
                       
                   
                    
                   v 
                 
               
                
               
                 
                   ( 
                   
                     
                       
                         
                           d 
                            
                           ρ 
                         
                         
                           
                             μ 
                             eff 
                           
                            
                           
                             v 
                             
                               n 
                               - 
                               2 
                             
                           
                         
                       
                        
                       
                         ɛ 
                         d 
                       
                     
                     - 
                     
                       18.574 
                        
                       
                         W 
                          
                         
                           ( 
                           
                             0.199 
                              
                             
                                 
                             
                              
                             
                               
                                 d 
                                  
                                 p 
                               
                               
                                 
                                   μ 
                                   eff 
                                 
                                  
                                 
                                   v 
                                   
                                     n 
                                     - 
                                     2 
                                   
                                 
                               
                             
                              
                             
                               e 
                               
                                 0.0538 
                                  
                                 
                                   
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     ρ 
                                   
                                   
                                     
                                       μ 
                                       eff 
                                     
                                      
                                     
                                       v 
                                       
                                         n 
                                         - 
                                         2 
                                       
                                     
                                   
                                 
                                  
                                 
                                   ɛ 
                                   d 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
                 2 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 μ 
                 eff 
               
               = 
               
                 
                   H 
                   12 
                 
                  
                 
                   
                     ( 
                     
                       9 
                       + 
                       
                         3 
                         n 
                       
                     
                     ) 
                   
                   n 
                 
                  
                 
                   
                     ( 
                     
                       
                         7 
                          
                         5 
                          
                         
                           d 
                           2 
                         
                       
                       
                         1 
                          
                         6 
                       
                     
                     ) 
                   
                   
                     
                       1 
                       - 
                       n 
                     
                     2 
                   
                 
               
             
           
         
       
     
     Where μ eff  a is the effective viscosity, d is the hydraulic diameter of tubes, ρ is the density of the fluid, 
     
       
         
           
             ɛ 
             d 
           
         
       
     
     is the relative roughness, v is the velocity of the fluid, n is the power-law index, H is the consistency coefficient, W(g) is the Lambert W function or product logarithm function.
 
{circle around (10)} Newton fluid+low speed non-Darcy flow or Bingham (1919) Non-Newton fluid+Darcy percolation flow
 
     λ is written as: 
     
       
         
           
             λ 
             = 
             
               
                 K 
                 μ 
               
                
               
                 ( 
                 
                   1 
                   - 
                   
                     G 
                     
                        
                       
                         
                           ∂ 
                           p 
                         
                         
                           ∂ 
                           l 
                         
                       
                        
                     
                   
                 
                 ) 
               
             
           
         
       
     
     Where μ is the viscosity of the fluid, K is the permeability, G is the low-speed non-darcy factor or pseudo-threshold pressure gradient, p is the pressure, l is the distance. 
     {circle around (11)} Newton fluid+slippage effect 
     λ is written as: 
     
       
         
           
             λ 
             = 
             
               
                 k 
                  
                 
                   ( 
                   
                     1 
                     + 
                     
                       b 
                       / 
                       p 
                     
                   
                   ) 
                 
               
               μ 
             
           
         
       
     
     Where μ is the viscosity of the fluid, K is the permeability, b is the Klinkenberg factor, p is the pressure. 
     For non-Darcy percolation in the above formula, besides Forchheimer (1901) binomial high-speed non-Darcy formula and permeability modulus formula (stress sensitivity), there are also formulas created as a replacement by Irmay (1968), Izbash (1971), Swartzendruber (1962). Huang Tingzhang (1997), Halex (1979) and so on. The Colebrook (1939)&#39;s along-way resistance formula above can be replaced by other formulas such as Blasius (1913). Nikuradse (1933), Churchill (1977). The Qstwald-DeWaele (1923,1925)&#39;s power-law fluid viscosity formula above can be replaced by other formulas such as Bingham (1919), Cross (1979). Carreau (1979). Meter (1964), Sisko (1958); The Hirasaki-Pope (1974) &#39;s shear rate formula can be replaced by other formulas such as Gogarty (1967), W. Littmann (1988), Camillen (1987), Rabinowitsch (1929), Jennings (1971). In addition, for low-density gas flow cases, the generalized mobility expression can be obtained by correcting the non-Darcy effect with the Klinkenberg equation (1941). 
     (1.6) Grid Design of Numerical Simulation 
     The design of grid types (orthogonal, corner, hybrid, etc.), grid directions (considering sediment source direction, flow direction, well location, etc.), grid boundaries, grid layers, grid sizes, grid density, etc. 
     (1.7) Wellbore Storage Model Setting 
     Wellbore storage models include constant wellbore storage model, changing wellbore storage model [Fair model (1981), Hegeman model (1993), leaky packer model and near wellbore fracture storage model (Spivey, 1999. 
     (1.8) Flow Period and Regime Definition 
     The flow period definition includes pressure build-up, pressure falloff, pressure drawdown, injection test, and slug flow. The flow regime definition includes linear flow, bilinear flow, radial flow and spherical flow. 
     (1.9) Coordinate Transformation (Flow Rate History Influence Treatment) 
     The coordinate transformation methods include equivalent constant flow rate history Horner method, equivalent constant flow rate history Agarwal method, variable flow rate history Horner method, variable flow rate history Agarwal method, deconvolution method and so on. 
     (1.10) Setting of Models and their Type Curve Analysis 
     The type curves of model analysis include various model type curves for pressure, flow and temperature analysis. The pressure analysis p type curves include full view pressure history plot, linear plot, semi-logarithmic plot, Gringarten-Bourdet log-log plot, linear flow plot, bilinear flow plot, (hemi)spherical flow plot, PPD plot, SLPD plot, early time plot, ζ function plot, curve comparison of previous single well or well group and so on. 
     The type curves of production analysis include blasingame plate, Agarwal Gardner plate, NPI plate and transient plate. The Type curves of temperature analysis include double logarithm diagram of temperature change and its derivative. 
     See Table 1 for the combination of tube flow (free flow)—percolation coupling model methods available in the numerical simulation and type curve analysis. 
     (1.11) Parameter Adjustment and History Matching 
     The adjustable parameters include permeability, saturation and porosity. History matching indexes include pressure index, production index, temperature index, fluid property index and so on. 
     (1.12) Dynamic Prediction 
     Dynamic prediction includes various development indexes of the prediction scheme for economic evaluation. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 The tube flow percolation coupling models combined in the software (Scope cases of Claims) 
               
            
           
           
               
               
            
               
                 Different combinations of models 
                 Application field 
               
            
           
           
               
               
               
               
               
            
               
                 Well and reservoir coupling, 
                 Combination 
                 Transient well 
                 Numerical 
                 Numerical simulation 
               
               
                 coupling between reservoirs, 
                 of fluid types 
                 analysis (pressure, 
                 simulation of oil 
                 of underground 
               
               
                 boundary coupling, etc 
                 and motion laws 
                 rate and temperature) 
                 and gas reservoir 
                 water reservoir 
               
               
                   
               
               
                 Coupling of single/dual 
                 Remark 1 
                 This invention 
                 Ones published 
                 Ones published by 
               
               
                 medium homogeneous 
                   
                   
                 by others limited 
                 others limited to use 
               
               
                 reservoir and vertical or 
                   
                   
                 to use of 
                 of equivalent 
               
               
                 horizontal well 
                   
                   
                 equivalent 
                 percolation coefficient 
               
               
                   
                   
                   
                 permeability 
               
               
                   
                 Remark 2 
                 This invention 
                 This invention 
                 This invention 
               
               
                 Coupling of triple media 
                 Remark 1 
                 This invention 
                 This invention 
                 Ones published by 
               
               
                 hreservoir and vertical 
                   
                   
                   
                 others limited to 
               
               
                 or horizontal well 
                   
                   
                   
                 use of equivalent 
               
               
                   
                   
                   
                   
                 percolation coefficient 
               
               
                   
                 Remark 2 
                 This invention 
                 This invention 
                 This invention 
               
               
                 Coupling of different 
                 Remark 3 
                 This invention 
                 This invention 
                 This invention 
               
               
                 reservoirs and different 
               
               
                 well types (the above 
               
               
                 cases are not included) 
               
               
                 Coupling of different 
                 Remark 3 
                 This invention 
                 This invention 
                 This invention 
               
               
                 reservoirs 
               
               
                 Coupling of different 
                 Remark 3 
                 This invention 
                 This invention 
                 This invention 
               
               
                 reservoirs and different 
               
               
                 types of boundaries 
               
            
           
           
               
               
            
               
                 Remark 1 
                 Three cases of fluid type and motion law combinations: 
               
               
                   
                 {circle around (1)}Newtonian fluid + Darcy flow; 
               
               
                   
                 {circle around (2)}Newtonian fluid + laminar tube flow; 
               
               
                   
                 {circle around (4)}Newtonian fluid + turbulent tube flow; 
               
               
                 Remark 2 
                 Seven cases of fluid type and motion law combinations: 
               
               
                   
                 {circle around (3)}Newtonian fluid + high speed non-Darcy flow; 
               
               
                   
                 {circle around (5)}Newtonian fluid + stress-sensitive percolation; 
               
               
                   
                 {circle around (6)}non-Newionian fluid + Darcy flow; 
               
               
                   
                 {circle around (7)}non-Newtonian fluid + laminar tube flow; 
               
               
                   
                 {circle around (8)}non-Newtonian fluid + stress-sensitive percolation; 
               
               
                   
                 {circle around (9)}non-Newtonian fluid + turbulent tube flow; 
               
               
                   
                 {circle around (10)}Newtonian fluid + low speed non-Darcy flow 
               
               
                   
                 (or Binghamnon-Newtonian fluid + Darcy flow). 
               
               
                 Remark 3 
                 Ten cases of fluid type and motion law combinations: 
               
               
                   
                 {circle around (1)}Newtonian fluid + Darcy flow; 
               
               
                   
                 {circle around (2)}Newtonian fluid + laminar tube flow; 
               
               
                   
                 {circle around (3)}Newtonian fluid + high speed non-Darcy flow; 
               
               
                   
                 {circle around (4)} Newtonian fluid + turbulent tube flow; 
               
               
                   
                 {circle around (5)} Newtonian fluid + stress-sensitive percolation; 
               
               
                   
                 {circle around (6)}Non-Newtonian fluid + Darcy flow; 
               
               
                   
                 {circle around (7)}Non-Newtonian fluid + laminar tube flow; 
               
               
                   
                 {circle around (8)}Non-Newtonian fluid + stress-sensitive percolation; 
               
               
                   
                 {circle around (9)}Non-Newtonian fluid + turbulent tube flow; 
               
               
                   
                 {circle around (10)}Newtonian fluid + low speed non-Darcy flow 
               
               
                   
                 (or Bingham Non-Newtonian fluid + Darcy flow). 
               
            
           
           
               
               
            
               
                 Remark 4 
                 (1)“Published by others” are limited to the use of 
               
               
                   
                 equivalent permeability and equivalent percolation coefficient (or 
               
               
                   
                 converted percolation coefficient) in the above numerical 
               
               
                   
                 simulation models, where the use of generalized mobility (or 
               
               
                   
                 equivalent mobility) belongs to this invention; 
               
               
                   
                 (2) This invention includes characteristic methods for tube 
               
               
                   
                 flow (free flow) or percolation cases realized by the method of 
               
               
                   
                 this invention; 
               
               
                   
                 (3) The permanent downhole monitoring data analysis is 
               
               
                   
                 mainly based on models of “transient well analysis” (pressure 
               
               
                   
                 or temperature) or/and “transient rate production data analysis”. 
               
               
                   
                 (4) This invention includes above models that can be applied 
               
               
                   
                 for geothermal reservoirs. 
               
               
                 Remark 5 
                 The reservoirs refer to a reservoir of oil, gas or water. 
               
               
                   
                 Reservoirs contains tubes, porous media, fracture media, caves, 
               
               
                   
                 or their combinations, connected combinations or overlapping. 
               
               
                   
               
            
           
         
       
     
     (2) The software developed can be used for single and multi-well flow simulation of complex multi-component multi-component multi-phase flow reservoirs, multi-well interference analysis, deliverability analysis (i.e. steady well test analysis), transient pressure analysis (i.e. unsteady pressure well test analysis), transient rate analysis (i.e. production data analysis or rate decline analysis), transient temperature analysis (i.e. unsteady temperature analysis), well test design, and permanent downhole monitoring data analysis (including pressure, temperature, flow rate, and so on). The above complex multi-component multi-phase flow reservoirs refer to all types of reservoirs, reservoirs with fluid injection, underground gas storage, ground water reservoirs, geothermal reservoirs, etc. 
     Embodiment 1 is a general example that can be used to solve tube flow-percolation coupling problems and multi-phase flow problems for various complex reservoirs, embodiments 2 to 7 are simplified on the basis of embodiment 1 to reflect that the wellbore can be treated either as an inner boundary or as a reservoir, the specific solution methods of which can be solved according to the numerical methods given in embodiment 1. However, some embodiments are simple and can be solved by analytical methods. For example, embodiments 4 and 6 can be solved by Laplace integral transformation method and give analytical solutions in Laplace space. embodiments 8 to 12 are ones given where the generalized mobility is a constant and they have analytical solutions in Laplace space. embodiment 13 gives a situation where the generalized mobility is a non-constant. embodiment 14 is an example of multi-component condensate gas flow simulation analysis for complex reservoirs. Embodiment 15 is an example of multi-component carbon dioxide drive flow simulation analysis in complex reservoirs. Embodiment 16 is an example of multi-component polymer drive flow simulation analysis in complex reservoirs. Embodiment 17 is an example of multi-component three-phase flow simulation analysis considering non-isothermal process in complex reservoirs. Embodiment 18 is an example of transient temperature analysis of oil single-phase flow. 
     It should be noted that the generalized mobility is a constant (matrix) when only the fluid flow is Darcy flow or laminar tube flow. In general, the generalized mobility is a function of space position and time. In particular, the generalized mobility may be a function of variables such as fluid velocity, pressure gradient, and so on. If the generalized mobility can be expressed as the product of a constant matrix and a scalar function, the non-linear flow problem can be transformed into linear flow through defining pseudo-pressure and pseudo-time functions, which means to transform generalized mobility into a constant matrix; Otherwise, the problem cannot be transformed into a linear flow problem. 
     Let a function ƒ(t) be defined on [0,∞), then ƒ(t) is a real-valued function or a complex-valued function of a real variable t. The function 
     
       
         
           
             
               
                 f 
                 _ 
               
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 0 
                 ∞ 
               
                
               
                 
                   f 
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                  
                 
                   e 
                   
                     - 
                     zt 
                   
                 
                  
                 dt 
               
             
           
         
       
     
     determined by the Laplace integral 
     
       
         
           
             
               ∫ 
               0 
               ∞ 
             
              
             
               
                 f 
                  
                 
                   ( 
                   t 
                   ) 
                 
               
                
               
                 e 
                 
                   - 
                   zt 
                 
               
                
               dt 
             
           
         
       
     
     is called the Laplace transformation of the function ƒ(t). By applying the integral transformation above, the well testing models in embodiments 4, 6 and 8 to 12 can be transformed into homogeneous equation systems in Laplace space, which are used to obtain bottom hole pressure function solutions in Laplace space. 
     Let  p   wD (z) be dimensionless bottom hole pressure solution function in Laplace space, then the real space dimensionless bottom hole pressure solution p wD (t D ) can be obtained by Stehfest numerical inversion technique 
     
       
         
           
             
               
                 p 
                 wD 
               
                
               
                 ( 
                 
                   t 
                   D 
                 
                 ) 
               
             
             = 
             
               
                 
                   log 
                    
                   
                       
                   
                    
                   2 
                 
                 
                   t 
                   D 
                 
               
                
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   N 
                 
                  
                 
                   
                     ( 
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           
                             N 
                             2 
                           
                           + 
                           k 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             
                               [ 
                               
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                 / 
                                 2 
                               
                               ] 
                             
                           
                           
                             min 
                              
                             
                               ( 
                               
                                 k 
                                 , 
                                 
                                   N 
                                   / 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                          
                         
                           
                             
                               
                                 n 
                                 
                                   N 
                                   / 
                                   2 
                                 
                               
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   n 
                                 
                                 ) 
                               
                             
                             ! 
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     N 
                                     / 
                                     2 
                                   
                                   - 
                                   n 
                                 
                                 ) 
                               
                               ! 
                             
                              
                             
                               n 
                               ! 
                             
                              
                             
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               ! 
                             
                              
                             
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   n 
                                 
                                 ) 
                               
                               ! 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   - 
                                   k 
                                 
                                 ) 
                               
                               ! 
                             
                           
                         
                       
                     
                     ) 
                   
                    
                   
                     
                       
                         p 
                         _ 
                       
                       wD 
                     
                      
                     
                       ( 
                       
                         k 
                          
                         
                           
                             log 
                              
                             
                                 
                             
                              
                             2 
                           
                           
                             t 
                             D 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     where N is an even number, generally the value is between 8 and 16. 
     Other numerical inversion techniques can also be used here in addition to Stehfest numerical inversion technology. 
     Embodiment of transient rate analysis (i.e., production data analysis) is shown in  FIGS. 41 to 45 . These figures are realized on the basis of a cylindrical reservoir model provided in embodiment 9 of this invention, which are vertical well Blasingame type curve, vertical well Agarwal-Gardner type curve, vertical well NPI type curve, and vertical well Transient type curve. Similarly, various kinds of analysis type curves of other analysis models can also be created. The basic principles of drawing type curves are detailed in the book “Modern production decline analysis methods for oil and gas wells and their application” written by Sun Hedong (2013). 
     For some layered reservoirs, the problem can be further simplified by defining the generalized flow coefficient or the equivalent transmissibility (Kh/μ) to facilitate the analysis and solution of some problems. 
     In addition, based on the above principle method, It can be similar to follow the traditional analytical model methods to establish the corresponding deliverability well test analysis, multi-well interference well test analysis, well test analysis method for wells in geothermal reservoirs and permanent downhole monitoring data analysis. 
     For example, with regard to the establishment of single-phase flow deliverability test analysis model method for steady-flow vertical wells in unsaturated layered reservoirs, the formula of production index is written as: follows: 
     
       
         
           
             J 
             = 
             
               
                 q 
                 
                   
                     P 
                     e 
                   
                   - 
                   
                     P 
                     wf 
                   
                 
               
               - 
               
                 
                   2 
                    
                   πλh 
                 
                 
                   B 
                    
                   
                     [ 
                     
                       
                         ln 
                          
                         
                           ( 
                           
                             
                               r 
                               e 
                             
                             / 
                             
                               r 
                               w 
                             
                           
                           ) 
                         
                       
                       + 
                       S 
                     
                     ] 
                   
                 
               
             
           
         
       
     
     B—Volume factor of the fluid, m 3 /m 3 ; h—Reservoir thickness, m; p e -reservoir outer boundary pressure (average formation pressure of reservoir can be used in practical application), Pa; p wf -bottom hole flowing pressure, Pa; q—flow rate, m 3 /s; r e -outer radius of reservoir, m; r w -wellbor radius, m S-skin factor; λ-generalized mobility of the reservoir fluid, m 2 /Pa·s. 
     Embodiment 1 
     This embodiment provides a generalized oil/gas/water three-phase flow simulation analysis method for complex reservoirs.  FIG. 3  is the physical model, which only denotes a special case of this embodiment. 
     Based on the mass conservation principle, the general equation of oil/gas/water three-phase flow can be established (Note: the volume factor is equal to the fluid density divided by the surface reference density and the surface reference density is a constant. Therefore, the two sides of the governing equation in the following examples can be converted into the fluid density by multiplying the surface reference density of each phase, respectively): 
     Oil phase governing equation considering source/sink term: 
     
       
         
           
             
               
                 
                   
                     
                       ∇ 
                       
                         · 
                         
                           [ 
                           
                             
                               1 
                               
                                 B 
                                 o 
                               
                             
                              
                             
                               λ 
                               o 
                             
                              
                             
                               ∇ 
                               
                                 p 
                                 o 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           ( 
                           
                             
                               1 
                               
                                 B 
                                 o 
                               
                             
                              
                             
                               
                                 φ 
                                  
                                 S 
                               
                               o 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           q 
                           o 
                         
                         
                           ρ 
                           osc 
                         
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     Water phase governing equation considering source/sink term: 
     
       
         
           
             
               
                 
                   
                     
                       ∇ 
                       
                         · 
                         
                           [ 
                           
                             
                               1 
                               
                                 B 
                                 w 
                               
                             
                              
                             
                               λ 
                               w 
                             
                              
                             
                               ∇ 
                               
                                 p 
                                 w 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           ( 
                           
                             
                               1 
                               
                                 B 
                                 w 
                               
                             
                              
                             
                               
                                 φ 
                                  
                                 S 
                               
                               w 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           q 
                           w 
                         
                         
                           ρ 
                           osc 
                         
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Gas phase governing equation considering source/sink term: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∇ 
                         
                           · 
                           
                             [ 
                             
                               
                                 1 
                                 
                                   B 
                                   g 
                                 
                               
                                
                               
                                 λ 
                                 g 
                               
                                
                               
                                 ∇ 
                                 g 
                               
                             
                             ] 
                           
                         
                       
                       + 
                       
                         ∇ 
                         
                           · 
                           
                             [ 
                             
                               
                                 
                                   R 
                                   g 
                                 
                                 
                                   B 
                                   o 
                                 
                               
                                
                               
                                 λ 
                                 o 
                               
                                
                               
                                 ∇ 
                                 
                                   p 
                                   o 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 1 
                                 
                                   B 
                                   g 
                                 
                               
                                
                               φ 
                             
                             , 
                             
                               S 
                               g 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           q 
                           g 
                         
                         
                           ρ 
                           gsc 
                         
                       
                       + 
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 R 
                                 s 
                               
                               
                                 B 
                                 o 
                               
                             
                              
                             
                               
                                 φ 
                                  
                                 S 
                               
                               o 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Auxiliary equations with saturation and capillary pressure: 
         S   o   +S   w   +S   g =1,( x,t )∈Ω×(0, t   max ]  (4)
 
         p   cow ( S   w )= p   o   −p   w ,( x,t )∈Ω×(0, t   max ]  (5)
 
         p   cgo ( S   w )= p   g   −p   o ,( x,t )∈Ω×(0, t   max ]  (6)
 
     Boundary condition equations: 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               o 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             o 
                           
                         
                         + 
                         
                           
                             c 
                             
                               o 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             o 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 o 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         o 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               w 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             w 
                           
                         
                         + 
                         
                           
                             c 
                             
                               w 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             w 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 w 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         w 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               g 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             g 
                           
                         
                         + 
                         
                           
                             c 
                             
                               g 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             g 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 g 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         g 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Initial condition equations: 
         S   k ( x, 0)= l   k ( x ), x∈Ω   (10)
 
         p   k ( x, 0)=ƒ k ( x ), x∈Ω   (11)
 
       Where 
         S   k   =S   k ( x,t ), k=o,w,g   (12)
 
         p   k   =p   k ( x,t ), k=o,w,g   (13)
 
         q   k   =q   k ( x,t ), k=o,w,g   (14)
 
       λ k =λ k ( x,t ), k=o,w,g   (15)
 
     ϕ is porosity, which is the function of average pressure  p , %; B w , B o , B g  are volume factors of water/oil/gas, water volume factor is a function of water-phase pressure p w , oil volume factor is a function of oil-phase pressure p o  and bubble point pressure p b , gas volume factor is a function of gas-phase pressure p g , dimensionless; λ w , λ o , λ g  are generalized mobility for water/oil/gas, separately, m 2 /(Pa·s); ρ w , ρ o , ρ g  are densities of water/oil/gas, kg/m 3 ; ρ o,o  is oil component density in oil-phase, kg/m 3 ; ρ g,o  is gas component density in oil-phase, kg/m 3 ; ρ wsc , ρ osc , ρ gsc  are surface reference densities of water/oil/gas, kg/m 3 ; R s  is the dissolved gas-oil ratio, dimensionless; S w , S o , S g  are saturation of water/oil/gas; t is time, s; t max  is the total time, s; p w , p o , p g  are pressures of water/oil/gas, Pa; l w , l o , l g  are distribution functions of water/oil/gas initial saturation; q w , q o , q g  are source/sink terms of water/oil/gas, kg/(m 3 ·s); ƒ w , ƒ o , ƒ g  are distribution functions of water/oil/gas initial pressure, Pa; g w , g o , g g  are boundary functions of water/oil/gas on reservoir boundary, dimensionless; p cow , p cgo  are capillary pressure of oil-water/gas-oil contact, Pa; Ω is reservoir space; ∂Ω is reservoir boundary including internal boundary and outer boundary; c w1 , c o,1 , c g,1  are pressure term coefficients of water/oil/gas on reservoir boundary, 1/Pa; c w,2 , c o,2 , c g,2  are respectively directional derivative coefficient of water, oil and gas along the normal direction outside the boundary on reservoir boundary condition, s/m; n ∂Ω  is The direction of outer normal of reservoir boundary, m; ∇ is the Hamilton operator; ∂ is the partial derivative sign. 
     Problems can be optimized and discretely solved by applying the finite volume method. The following example of tetrahedral element is given to illustrate the process. 
     Firstly the region Ω is divided into a series of tetrahedrons Ω i , which do not overlap each other. The control point of tetrahedrons Ω i  is marked as x i  and the boundary of tetrahedrons Ω i  is marked as ∂Ω i . If the number of tetrahedrons that have a common surface with tetrahedrons Ω i  is 4, then x i  is called the internal control point, and the set of all internal control points is Ω I . If the number of tetrahedrons that have a common surface with tetrahedrons is less than Ω i  then x i  is called the boundary control point, and the set of all boundary control points is called Ω B . 
     Oil phase governing equation considering source/sink term is discretized as 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           σ 
                           ⋐ 
                           
                             ∂ 
                             
                               Ω 
                               η 
                             
                           
                         
                       
                        
                       
                         
                           u 
                           
                             o 
                             , 
                             
                               i 
                               2 
                             
                           
                         
                          
                         
                           a 
                           
                             o 
                             , 
                             
                               i 
                               2 
                             
                           
                         
                          
                         
                           
                             p 
                             o 
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 
                                   i 
                                   2 
                                 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     - 
                     
                       
                         u 
                         
                           o 
                           , 
                           
                             i 
                             1 
                           
                         
                       
                        
                       
                         a 
                         
                           o 
                           , 
                           
                             i 
                             1 
                           
                         
                       
                        
                       
                         
                           p 
                           o 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                             , 
                             
                               i 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         e 
                         
                           o 
                           , 
                           2 
                         
                       
                        
                       
                         
                           S 
                           o 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                             , 
                             
                               i 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         e 
                         
                           o 
                           , 
                           1 
                         
                       
                        
                       
                         
                           S 
                           o 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         V 
                         
                           i 
                           1 
                         
                       
                        
                       
                         
                           
                             q 
                             o 
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 
                                   i 
                                   1 
                                 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         
                           ρ 
                           osc 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Where x i     1   ∈Ω I , x i     2   ∈Ω I  UΩ B , and 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       e 
                       
                         o 
                         , 
                         1 
                       
                     
                     = 
                     
                       
                         
                           V 
                           
                             i 
                             1 
                           
                         
                          
                         
                           φ 
                            
                           
                             [ 
                             
                               
                                 p 
                                 _ 
                               
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     
                                       i 
                                       1 
                                     
                                   
                                   , 
                                   
                                     t 
                                     n 
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                       
                         
                           
                             B 
                             o 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   p 
                                   o 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     , 
                                     
                                       t 
                                       n 
                                     
                                   
                                   ) 
                                 
                               
                               , 
                               
                                 p 
                                 b 
                               
                             
                             ] 
                           
                         
                          
                         
                           ( 
                           
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       e 
                       
                         o 
                         , 
                         2 
                       
                     
                     = 
                     
                       
                         
                           V 
                           
                             i 
                             1 
                           
                         
                          
                         
                           φ 
                            
                           
                             [ 
                             
                               
                                 p 
                                 _ 
                               
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     
                                       i 
                                       1 
                                     
                                   
                                   , 
                                   
                                     t 
                                     
                                       n 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                       
                         
                           
                             B 
                             o 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   p 
                                   o 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     , 
                                     
                                       t 
                                       
                                         n 
                                         + 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               , 
                               
                                 p 
                                 b 
                               
                             
                             ] 
                           
                         
                          
                         
                           ( 
                           
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       a 
                       
                         o 
                         , 
                         
                           i 
                           1 
                         
                       
                     
                     = 
                     
                       
                         S 
                         
                           
                             i 
                             1 
                           
                           , 
                           σ 
                         
                       
                        
                       
                         ( 
                         
                           
                             α 
                             
                               o 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                           + 
                           
                             β 
                             
                               o 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                           + 
                           
                             γ 
                             
                               o 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       α 
                       
                         o 
                         , 
                         
                           i 
                           2 
                         
                       
                     
                     = 
                     
                       
                         S 
                         
                           
                             i 
                             2 
                           
                           , 
                           σ 
                         
                       
                        
                       
                         ( 
                         
                           
                             α 
                             
                               o 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                           + 
                           
                             β 
                             
                               o 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                           + 
                           
                             γ 
                             
                               o 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     b 
                     
                       o 
                       , 
                       
                         i 
                         1 
                       
                     
                   
                   = 
                   
                     
                       S 
                       
                         
                           i 
                           1 
                         
                         , 
                         σ 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             α 
                             
                               o 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             β 
                             
                               o 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     2 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             γ 
                             
                               o 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     4 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   
                     b 
                     
                       o 
                       , 
                       
                         i 
                         2 
                       
                     
                   
                   = 
                   
                     
                       S 
                       
                         
                           i 
                           2 
                         
                         , 
                         σ 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             α 
                             
                               o 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     2 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             β 
                             
                               o 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     3 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             γ 
                             
                               o 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     5 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       u 
                       
                         o 
                         , 
                         
                           i 
                           1 
                         
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 1 
                                 / 
                                 2 
                               
                               , 
                             
                           
                           
                             
                               
                                 
                                   b 
                                   
                                     o 
                                     , 
                                     
                                       i 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   b 
                                   
                                     o 
                                     , 
                                     
                                       i 
                                       2 
                                     
                                   
                                 
                               
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   b 
                                   
                                     o 
                                     , 
                                     
                                       i 
                                       2 
                                     
                                   
                                 
                                 / 
                                 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                     
                                     + 
                                     
                                       b 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           2 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               , 
                             
                           
                           
                             
                               
                                 
                                   b 
                                   
                                     o 
                                     , 
                                     
                                       i 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   b 
                                   
                                     o 
                                     , 
                                     
                                       i 
                                       2 
                                     
                                   
                                 
                               
                               ≠ 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       u 
                       
                         o 
                         , 
                         
                           i 
                           2 
                         
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 1 
                                 / 
                                 2 
                               
                               , 
                             
                           
                           
                             
                               
                                 
                                   b 
                                   
                                     o 
                                     , 
                                     
                                       i 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   b 
                                   
                                     o 
                                     , 
                                     
                                       i 
                                       2 
                                     
                                   
                                 
                               
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   b 
                                   
                                     o 
                                     , 
                                     
                                       i 
                                       1 
                                     
                                   
                                 
                                 / 
                                 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                     
                                     + 
                                     
                                       b 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           2 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               , 
                             
                           
                           
                             
                               
                                 
                                   b 
                                   
                                     o 
                                     , 
                                     
                                       i 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   b 
                                   
                                     o 
                                     , 
                                     
                                       i 
                                       2 
                                     
                                   
                                 
                               
                               ≠ 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     The vertex pressures in the formulas [(21) and (22)] above are treated by the inverse distance weighting method, i. e, 
     
       
         
           
             
               
                 
                   
                     
                       
                         p 
                         o 
                       
                        
                       
                         ( 
                         
                           
                             d 
                             j 
                           
                           , 
                           
                             t 
                             
                               n 
                               + 
                               1 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           ∈ 
                           
                             D 
                             j 
                           
                         
                       
                        
                       
                         
                           
                             || 
                             
                               
                                 d 
                                 j 
                               
                               - 
                               
                                 x 
                                 i 
                               
                             
                              
                             
                               || 
                               2 
                               l 
                             
                           
                           
                             
                               ∑ 
                               
                                 i 
                                 ∈ 
                                 
                                   D 
                                   j 
                                 
                               
                             
                              
                             
                               || 
                               
                                 
                                   d 
                                   j 
                                 
                                 - 
                                 
                                   x 
                                   i 
                                 
                               
                                
                               
                                 || 
                                 2 
                                 l 
                               
                             
                           
                         
                          
                         
                           
                             p 
                             o 
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 i 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     j 
                     = 
                     
                       j 
                       1 
                     
                   
                   , 
                   
                     j 
                     2 
                   
                   , 
                   
                     j 
                     3 
                   
                   , 
                   
                     j 
                     4 
                   
                   , 
                   
                     j 
                     5 
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     D j  denotes the number set of all discrete control body central points from the vertex d j , l is a real number less than 0. 
     The coefficients α o,i     1   , β o,i     1   , γ o,i     1    in the formulas [(19) and (21) above are determined by the following formula 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               λ 
                               o 
                             
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 
                                   i 
                                   1 
                                 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           n 
                           
                             
                               i 
                               1 
                             
                             , 
                             σ 
                           
                         
                       
                       
                         
                           B 
                           o 
                         
                          
                         
                           [ 
                           
                             
                               
                                 p 
                                 o 
                               
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     
                                       i 
                                       1 
                                     
                                   
                                   , 
                                   
                                     t 
                                     
                                       n 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             , 
                             
                               p 
                               b 
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         
                           α 
                           
                             o 
                             , 
                             
                               i 
                               1 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 1 
                               
                             
                             - 
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           β 
                           
                             o 
                             , 
                             
                               i 
                               1 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 2 
                               
                             
                             - 
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           γ 
                           
                             o 
                             , 
                             
                               i 
                               1 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 4 
                               
                             
                             - 
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       x 
                       
                         i 
                         1 
                       
                     
                     ∈ 
                     
                       Ω 
                       i 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     As shown in  FIG. 4 , the subscripts of vertices coordinates j 1 , j 2 , j 4 , are intersected by a ray in the direction of 
     
       
         
           
             
               
                 
                   
                     λ 
                     o 
                   
                   T 
                 
                  
                 
                   ( 
                   
                     
                       x 
                       
                         i 
                         1 
                       
                     
                     , 
                     
                       t 
                       
                         n 
                         + 
                         1 
                       
                     
                   
                   ) 
                 
               
                
               
                 n 
                 
                   
                     i 
                     1 
                   
                   , 
                   σ 
                 
               
             
             
               
                 B 
                 o 
               
                
               
                 [ 
                 
                   
                     
                       p 
                       o 
                     
                      
                     
                       ( 
                       
                         
                           x 
                           
                             i 
                             1 
                           
                         
                         , 
                         
                           t 
                           
                             n 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                     p 
                     b 
                   
                 
                 ] 
               
             
           
         
       
     
     through the starting point x i     1    with a sub-surface of the discrete control body Ω i     1   . The three vertices coordinates of the sub-surface are d j     1   , d j     2   , d j     4   , as shown in  FIG. 4 . 
     The coefficients α o,i     2   , β o,i     2   , γ o,i     2    in the formulas Eq,20 and Eq.22 above are determined by the following formula 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             λ 
                             o 
                           
                           T 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               
                                 i 
                                 2 
                               
                             
                             , 
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         n 
                         
                           
                             i 
                             2 
                           
                           , 
                           σ 
                         
                       
                     
                     
                       
                         B 
                         o 
                       
                        
                       
                         [ 
                         
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   x 
                                   
                                     i 
                                     2 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           , 
                           
                             p 
                             b 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         α 
                         
                           o 
                           , 
                           
                             i 
                             2 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               2 
                             
                           
                           - 
                           
                             x 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         β 
                         
                           o 
                           , 
                           
                             i 
                             2 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               3 
                             
                           
                           - 
                           
                             x 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         γ 
                         
                           o 
                           , 
                           
                             i 
                             2 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               5 
                             
                           
                           - 
                           
                             x 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     The subscripts of vertices coordinates j 2 , j 3 , j 5  are intersected by a ray in the direction of 
     
       
         
           
             
               
                 
                   
                     λ 
                     o 
                   
                   T 
                 
                  
                 
                   ( 
                   
                     
                       x 
                       
                         i 
                         2 
                       
                     
                     , 
                     
                       t 
                       
                         n 
                         + 
                         1 
                       
                     
                   
                   ) 
                 
               
                
               
                 n 
                 
                   
                     i 
                     2 
                   
                   , 
                   σ 
                 
               
             
             
               
                 B 
                 o 
               
                
               
                 [ 
                 
                   
                     
                       p 
                       o 
                     
                      
                     
                       ( 
                       
                         
                           x 
                           
                             i 
                             2 
                           
                         
                         , 
                         
                           t 
                           
                             n 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                     p 
                     b 
                   
                 
                 ] 
               
             
           
         
       
     
     through the starting point x i     2    with a sub-surface of the discrete control body χ i     2   . The three vertices coordinates of the sub-surface are d j     2   , d j     3   , d j     5   , as shown in  FIG. 4 . 
     Water phase governing equation considering source/sink term is discretized as 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           σ 
                           ⋐ 
                           
                             ∂ 
                             
                               Ω 
                               η 
                             
                           
                         
                       
                        
                       
                         
                           u 
                           
                             w 
                             , 
                             
                               i 
                               2 
                             
                           
                         
                          
                         
                           a 
                           
                             w 
                             , 
                             
                               i 
                               2 
                             
                           
                         
                          
                         
                           
                             p 
                             w 
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 
                                   i 
                                   2 
                                 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     - 
                     
                       
                         u 
                         
                           w 
                           , 
                           
                             i 
                             1 
                           
                         
                       
                        
                       
                         a 
                         
                           w 
                           , 
                           
                             i 
                             1 
                           
                         
                       
                        
                       
                         
                           p 
                           w 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                             , 
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         e 
                         
                           w 
                           , 
                           2 
                         
                       
                        
                       
                         
                           S 
                           w 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                             , 
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         e 
                         
                           w 
                           , 
                           1 
                         
                       
                        
                       
                         
                           S 
                           w 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         V 
                         
                           i 
                           1 
                         
                       
                        
                       
                         
                           
                             q 
                             w 
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 
                                   i 
                                   1 
                                 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         
                           ρ 
                           wsc 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     Where x i     1   ∈Ω I , 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       e 
                       
                         w 
                         , 
                         1 
                       
                     
                     = 
                     
                       
                         
                           V 
                           
                             i 
                             1 
                           
                         
                          
                         
                           φ 
                            
                           
                             [ 
                             
                               
                                 p 
                                 _ 
                               
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     
                                       i 
                                       1 
                                     
                                   
                                   , 
                                   
                                     t 
                                     n 
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                       
                         
                           
                             B 
                             w 
                           
                            
                           
                             [ 
                             
                               
                                 p 
                                 w 
                               
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     
                                       i 
                                       1 
                                     
                                   
                                   , 
                                   
                                     t 
                                     n 
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                          
                         
                           ( 
                           
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       e 
                       
                         w 
                         , 
                         2 
                       
                     
                     = 
                     
                       
                         
                           V 
                           
                             i 
                             1 
                           
                         
                          
                         
                           φ 
                            
                           
                             [ 
                             
                               
                                 p 
                                 _ 
                               
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     
                                       i 
                                       1 
                                     
                                   
                                   , 
                                   
                                     t 
                                     
                                       n 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                       
                         
                           
                             B 
                             w 
                           
                            
                           
                             [ 
                             
                               
                                 p 
                                 w 
                               
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     
                                       i 
                                       1 
                                     
                                   
                                   , 
                                   
                                     t 
                                     
                                       n 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                          
                         
                           ( 
                           
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       a 
                       
                         w 
                         , 
                         
                           i 
                           1 
                         
                       
                     
                     = 
                     
                       
                         S 
                         
                           
                             i 
                             1 
                           
                           , 
                           σ 
                         
                       
                        
                       
                         ( 
                         
                           
                             α 
                             
                               w 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                           + 
                           
                             β 
                             
                               w 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                           + 
                           
                             γ 
                             
                               w 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       α 
                       
                         w 
                         , 
                         
                           i 
                           2 
                         
                       
                     
                     = 
                     
                       
                         S 
                         
                           
                             i 
                             2 
                           
                           , 
                           σ 
                         
                       
                        
                       
                         ( 
                         
                           
                             α 
                             
                               w 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                           + 
                           
                             β 
                             
                               w 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                           + 
                           
                             γ 
                             
                               w 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
             
               
                 
                   
                     b 
                     
                       w 
                       , 
                       
                         i 
                         1 
                       
                     
                   
                   = 
                   
                     
                       S 
                       
                         
                           i 
                           1 
                         
                         , 
                         σ 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             α 
                             
                               w 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                            
                           
                             
                               p 
                               w 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             β 
                             
                               w 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                            
                           
                             
                               p 
                               w 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     2 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             γ 
                             
                               w 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                            
                           
                             
                               p 
                               w 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     4 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
             
               
                 
                   
                     b 
                     
                       w 
                       , 
                       
                         i 
                         2 
                       
                     
                   
                   = 
                   
                     
                       S 
                       
                         
                           i 
                           2 
                         
                         , 
                         σ 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             α 
                             
                               w 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                            
                           
                             
                               p 
                               w 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     2 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             β 
                             
                               w 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                            
                           
                             
                               p 
                               w 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     3 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             γ 
                             
                               w 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                            
                           
                             
                               p 
                               w 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     5 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       u 
                       
                         w 
                         , 
                         
                           i 
                           1 
                         
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 1 
                                 / 
                                 2 
                               
                               , 
                             
                           
                           
                             
                               
                                 
                                   b 
                                   
                                     w 
                                     , 
                                     
                                       i 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   b 
                                   
                                     w 
                                     , 
                                     
                                       i 
                                       2 
                                     
                                   
                                 
                               
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   b 
                                   
                                     w 
                                     , 
                                     
                                       i 
                                       2 
                                     
                                   
                                 
                                 / 
                                 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         w 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                     
                                     + 
                                     
                                       b 
                                       
                                         w 
                                         , 
                                         
                                           i 
                                           2 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               , 
                             
                           
                           
                             
                               
                                 
                                   b 
                                   
                                     w 
                                     , 
                                     
                                       i 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   b 
                                   
                                     w 
                                     , 
                                     
                                       i 
                                       2 
                                     
                                   
                                 
                               
                               ≠ 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       u 
                       
                         w 
                         , 
                         
                           i 
                           2 
                         
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 1 
                                 / 
                                 2 
                               
                               , 
                             
                           
                           
                             
                               
                                 
                                   b 
                                   
                                     w 
                                     , 
                                     
                                       i 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   b 
                                   
                                     w 
                                     , 
                                     
                                       i 
                                       2 
                                     
                                   
                                 
                               
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   b 
                                   
                                     o 
                                     , 
                                     
                                       i 
                                       1 
                                     
                                   
                                 
                                 / 
                                 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         w 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                     
                                     + 
                                     
                                       b 
                                       
                                         w 
                                         , 
                                         
                                           i 
                                           2 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               , 
                             
                           
                           
                             
                               
                                 
                                   b 
                                   
                                     w 
                                     , 
                                     
                                       i 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   b 
                                   
                                     w 
                                     , 
                                     
                                       i 
                                       2 
                                     
                                   
                                 
                               
                               ≠ 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     The vertex pressures in the formulas Eq.37 and Eq.34 above are treated by the inverse distance weighting method, i. e, 
     
       
         
           
             
               
                 
                   
                     
                       
                         p 
                         w 
                       
                        
                       
                         ( 
                         
                           
                             d 
                             j 
                           
                           , 
                           
                             t 
                             
                               n 
                               + 
                               1 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           ∈ 
                           
                             D 
                             j 
                           
                         
                       
                        
                       
                         
                           
                             || 
                             
                               
                                 d 
                                 j 
                               
                               - 
                               
                                 x 
                                 i 
                               
                             
                              
                             
                               || 
                               2 
                               l 
                             
                           
                           
                             
                               ∑ 
                               
                                 i 
                                 ∈ 
                                 
                                   D 
                                   j 
                                 
                               
                             
                              
                             
                               || 
                               
                                 
                                   d 
                                   j 
                                 
                                 - 
                                 
                                   x 
                                   i 
                                 
                               
                                
                               
                                 || 
                                 2 
                                 l 
                               
                             
                           
                         
                          
                         
                           
                             p 
                             w 
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 i 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     j 
                     = 
                     
                       j 
                       1 
                     
                   
                   , 
                   
                     j 
                     2 
                   
                   , 
                   
                     j 
                     3 
                   
                   , 
                   
                     j 
                     4 
                   
                   , 
                   
                     j 
                     5 
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     D j  denotes the number set of all discrete control body central points from the vertex d j , l is a real number less than 0. 
     The coefficients α w,i     1   , β w,i     1   , γ w,i     1    in the formulas [(31) and (33)] above are determined by the following formula 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             λ 
                             w 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 ϰ 
                                 
                                   i 
                                   1 
                                 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           n 
                           
                             
                               i 
                               1 
                             
                             , 
                             σ 
                           
                         
                       
                       
                         
                           B 
                           w 
                         
                          
                         
                           [ 
                           
                             
                               p 
                               w 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   
                                     i 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         
                           α 
                           
                             w 
                             , 
                             
                               i 
                               1 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 1 
                               
                             
                             - 
                             
                               ϰ 
                               
                                 i 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           β 
                           
                             w 
                             , 
                             
                               j 
                               1 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 2 
                               
                             
                             - 
                             
                               ϰ 
                               
                                 i 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           γ 
                           
                             w 
                             , 
                             
                               j 
                               1 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 4 
                               
                             
                             - 
                             
                               ϰ 
                               
                                 i 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       ϰ 
                       
                         i 
                         1 
                       
                     
                     ∈ 
                     
                       Ω 
                       I 
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     The subscripts of vertices coordinates j 1 , j 2 , j 4  are intersected by a ray in the direction of 
     
       
         
           
             
               
                 
                   λ 
                   w 
                   T 
                 
                  
                 
                   ( 
                   
                     
                       ϰ 
                       
                         i 
                         1 
                       
                     
                     , 
                     
                       t 
                       
                         n 
                         + 
                         1 
                       
                     
                   
                   ) 
                 
               
                
               
                 n 
                 
                   
                     i 
                     1 
                   
                   , 
                   σ 
                 
               
             
             
               
                 B 
                 w 
               
                
               
                 [ 
                 
                   
                     p 
                     w 
                   
                    
                   
                     ( 
                     
                       
                         ϰ 
                         
                           i 
                           1 
                         
                       
                       , 
                       
                         t 
                         
                           n 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
                 ] 
               
             
           
         
       
     
     through the starting point x i     1    with a sub-surface of the discrete control body Ω i     1   . The three vertices coordinates of the sub-surface are d j     1   , d j     2   , d j     4    as shown in  FIG. 5 . 
     The coefficients α w,i     2   , β w,i     2   , γ w,i     2   , in the formulas [(32) and (34)] above are determined by the following formula 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           λ 
                           w 
                           T 
                         
                          
                         
                           ( 
                           
                             
                               ϰ 
                               
                                 i 
                                 2 
                               
                             
                             , 
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         n 
                         
                           
                             i 
                             2 
                           
                           , 
                           σ 
                         
                       
                     
                     
                       
                         B 
                         w 
                       
                        
                       
                         [ 
                         
                           
                             p 
                             w 
                           
                            
                           
                             ( 
                             
                               
                                 ϰ 
                                 
                                   i 
                                   2 
                                 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         α 
                         
                           w 
                           , 
                           
                             i 
                             2 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               2 
                             
                           
                           - 
                           
                             ϰ 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         β 
                         
                           w 
                           , 
                           
                             j 
                             2 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               3 
                             
                           
                           - 
                           
                             ϰ 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         γ 
                         
                           w 
                           , 
                           
                             i 
                             2 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               5 
                             
                           
                           - 
                           
                             ϰ 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     The subscripts of vertices coordinates j 2 , j 3 , j 5  are intersected by a ray in the direction of 
     
       
         
           
             
               
                 
                   λ 
                   w 
                   T 
                 
                  
                 
                   ( 
                   
                     
                       ϰ 
                       
                         i 
                         2 
                       
                     
                     , 
                     
                       t 
                       
                         n 
                         + 
                         1 
                       
                     
                   
                   ) 
                 
               
                
               
                 n 
                 
                   
                     i 
                     2 
                   
                   , 
                   σ 
                 
               
             
             
               
                 B 
                 w 
               
                
               
                 [ 
                 
                   
                     p 
                     w 
                   
                    
                   
                     ( 
                     
                       
                         ϰ 
                         
                           i 
                           2 
                         
                       
                       , 
                       
                         t 
                         
                           n 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
                 ] 
               
             
           
         
       
     
     through the starting point x i     2    with a sub-surface of the discrete control body Ω i     2   . The three vertices coordinates of the sub-surface are d j     2   , d j     3   , d j     5    as shown in  FIG. 5 . 
     Gas phase governing equation considering source/sink term (Eq.3) is discretized as 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         σ 
                         ⋐ 
                         
                           ∂ 
                           
                             Ω 
                             
                               i 
                               1 
                             
                           
                         
                       
                     
                      
                     
                         
                     
                      
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   u 
                                   
                                     g 
                                     , 
                                     
                                       i 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         a 
                                         
                                           g 
                                           , 
                                           
                                             i 
                                             2 
                                           
                                         
                                       
                                        
                                       
                                         
                                           p 
                                           g 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               ϰ 
                                               
                                                 i 
                                                 2 
                                               
                                             
                                             , 
                                             
                                               t 
                                               
                                                 n 
                                                 + 
                                                 1 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         a 
                                         
                                           o 
                                           , 
                                           
                                             i 
                                             2 
                                           
                                         
                                         ′ 
                                       
                                        
                                       
                                         
                                           p 
                                           o 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               ϰ 
                                               
                                                 i 
                                                 2 
                                               
                                             
                                             , 
                                             
                                               t 
                                               
                                                 n 
                                                 + 
                                                 1 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 u 
                                 
                                   g 
                                   , 
                                   
                                     i 
                                     1 
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       a 
                                       
                                         g 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                     
                                      
                                     
                                       
                                         p 
                                         g 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             ϰ 
                                             
                                               i 
                                               1 
                                             
                                           
                                           , 
                                           
                                             t 
                                             
                                               n 
                                               + 
                                               1 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       a 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                       ′ 
                                     
                                      
                                     
                                       
                                         p 
                                         o 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             ϰ 
                                             
                                               i 
                                               1 
                                             
                                           
                                           , 
                                           
                                             t 
                                             
                                               n 
                                               + 
                                               1 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       } 
                     
                   
                   = 
                   
                       
                     
                       
                         [ 
                         
                             
                         
                          
                         
                           
                             
                               
                                 
                                   
                                     e 
                                     
                                       g 
                                       , 
                                       2 
                                     
                                   
                                    
                                   
                                     
                                       S 
                                       g 
                                     
                                      
                                     
                                       ( 
                                       
                                         ϰ 
                                         
                                           
                                             t 
                                             1 
                                           
                                           , 
                                           
                                             t 
                                             
                                               n 
                                               + 
                                               1 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     e 
                                     
                                       g 
                                       , 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       S 
                                       g 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ϰ 
                                           
                                             t 
                                             1 
                                           
                                         
                                         , 
                                         
                                           t 
                                           n 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     e 
                                     
                                       0 
                                       , 
                                       2 
                                     
                                     ′ 
                                   
                                    
                                   
                                     
                                       S 
                                       0 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ϰ 
                                           
                                             t 
                                             1 
                                           
                                         
                                         , 
                                         
                                           t 
                                           
                                             n 
                                             + 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     e 
                                     
                                       0 
                                       , 
                                       1 
                                     
                                     ′ 
                                   
                                    
                                   
                                     
                                       S 
                                       0 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ϰ 
                                           
                                             t 
                                             1 
                                           
                                         
                                         , 
                                         
                                           t 
                                           n 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       + 
                       
                         
                           V 
                           
                             i 
                             1 
                           
                         
                          
                         
                           
                             
                               q 
                               g 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   
                                     i 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             ρ 
                             gsc 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     Where x i     1   ∈Ω I , 
     
       
         
           
             
               
                 
                   
                     e 
                     
                       g 
                       , 
                       1 
                     
                   
                   = 
                   
                     
                       
                         V 
                         
                           i 
                           1 
                         
                       
                        
                       
                         φ 
                          
                         
                           [ 
                           
                             
                               p 
                               _ 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   
                                     i 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   n 
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                     
                     
                       
                         
                           B 
                           g 
                         
                          
                         
                           [ 
                           
                             
                               p 
                               g 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   
                                     i 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   n 
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                        
                       
                         ( 
                         
                           
                             t 
                             
                               n 
                               + 
                               1 
                             
                           
                           - 
                           
                             t 
                             n 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
             
               
                 
                   
                     e 
                     
                       g 
                       , 
                       2 
                     
                   
                   = 
                   
                     
                       
                         V 
                         
                           i 
                           1 
                         
                       
                        
                       
                         φ 
                          
                         
                           [ 
                           
                             
                               p 
                               _ 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   
                                     i 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                     
                     
                       
                         
                           B 
                           g 
                         
                          
                         
                           [ 
                           
                             
                               p 
                               g 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   
                                     i 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                        
                       
                         ( 
                         
                           
                             t 
                             
                               n 
                               + 
                               1 
                             
                           
                           - 
                           
                             t 
                             n 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
             
               
                 
                   
                     e 
                     
                       o 
                       , 
                       1 
                     
                     ′ 
                   
                   = 
                   
                     
                       
                         V 
                         
                           i 
                           1 
                         
                       
                        
                       
                         φ 
                          
                         
                           [ 
                           
                             
                               p 
                               _ 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   
                                     i 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   n 
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                     
                     
                       
                         
                           B 
                           o 
                         
                          
                         
                           [ 
                           
                             
                               
                                 p 
                                 o 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ϰ 
                                     
                                       i 
                                       1 
                                     
                                   
                                   , 
                                   
                                     t 
                                     n 
                                   
                                 
                                 ) 
                               
                             
                             , 
                             
                               p 
                               b 
                             
                           
                           ] 
                         
                       
                        
                       
                         ( 
                         
                           
                             t 
                             
                               n 
                               + 
                               1 
                             
                           
                           - 
                           
                             t 
                             n 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
             
               
                 
                   
                     e 
                     
                       o 
                       , 
                       2 
                     
                     ′ 
                   
                   = 
                   
                     
                       
                         V 
                         
                           i 
                           1 
                         
                       
                        
                       
                         φ 
                          
                         
                           [ 
                           
                             
                               p 
                               _ 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   
                                     i 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   n 
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                     
                     
                       
                         
                           B 
                           o 
                         
                          
                         
                           [ 
                           
                             
                               
                                 p 
                                 o 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ϰ 
                                     
                                       i 
                                       1 
                                     
                                   
                                   , 
                                   
                                     t 
                                     
                                       n 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             , 
                             
                               p 
                               b 
                             
                           
                           ] 
                         
                       
                        
                       
                         ( 
                         
                           
                             t 
                             
                               n 
                               + 
                               1 
                             
                           
                           - 
                           
                             t 
                             n 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
             
               
                 
                   
                     a 
                     
                       g 
                       , 
                       
                         i 
                         1 
                       
                     
                   
                   = 
                   
                     
                       S 
                       
                         
                           i 
                           1 
                         
                         , 
                         σ 
                       
                     
                      
                     
                       ( 
                       
                         
                           α 
                           
                             g 
                             , 
                             
                               i 
                               1 
                             
                           
                         
                         + 
                         
                           β 
                           
                             g 
                             , 
                             
                               i 
                               1 
                             
                           
                         
                         + 
                         
                           γ 
                           
                             g 
                             , 
                             
                               i 
                               1 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
             
               
                 
                   
                     a 
                     
                       g 
                       , 
                       
                         i 
                         2 
                       
                     
                   
                   = 
                   
                     
                       S 
                       
                         
                           i 
                           2 
                         
                         , 
                         σ 
                       
                     
                      
                     
                       ( 
                       
                         
                           α 
                           
                             g 
                             , 
                             
                               i 
                               2 
                             
                           
                         
                         + 
                         
                           β 
                           
                             g 
                             , 
                             
                               i 
                               2 
                             
                           
                         
                         + 
                         
                           γ 
                           
                             g 
                             , 
                             
                               i 
                               2 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
             
               
                 
                   
                     a 
                     
                       o 
                       , 
                       
                         i 
                         1 
                       
                     
                     ′ 
                   
                   = 
                   
                     
                       S 
                       
                         
                           i 
                           1 
                         
                         , 
                         σ 
                       
                     
                      
                     
                       ( 
                       
                         
                           α 
                           
                             o 
                             , 
                             
                               i 
                               1 
                             
                           
                           ′ 
                         
                         + 
                         
                           β 
                           
                             o 
                             , 
                             
                               i 
                               1 
                             
                           
                           ′ 
                         
                         + 
                         
                           γ 
                           
                             o 
                             , 
                             
                               i 
                               1 
                             
                           
                           ′ 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
             
               
                 
                   
                     a 
                     
                       o 
                       , 
                       
                         i 
                         2 
                       
                     
                     ′ 
                   
                   = 
                   
                     
                       S 
                       
                         
                           i 
                           2 
                         
                         , 
                         σ 
                       
                     
                      
                     
                       ( 
                       
                         
                           α 
                           
                             o 
                             , 
                             
                               i 
                               2 
                             
                           
                           ′ 
                         
                         + 
                         
                           β 
                           
                             o 
                             , 
                             
                               i 
                               2 
                             
                           
                           ′ 
                         
                         + 
                         
                           γ 
                           
                             o 
                             , 
                             
                               i 
                               2 
                             
                           
                           ′ 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
             
               
                 
                   
                     b 
                     
                       g 
                       , 
                       
                         i 
                         1 
                       
                     
                   
                   = 
                   
                     
                       S 
                       
                         
                           i 
                           1 
                         
                         , 
                         σ 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             α 
                             
                               g 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                            
                           
                             
                               p 
                               g 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             β 
                             
                               g 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                            
                           
                             
                               p 
                               g 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     3 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             γ 
                             
                               g 
                               , 
                               
                                 i 
                                 1 
                               
                             
                           
                            
                           
                             
                               p 
                               g 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
             
               
                 
                   
                     b 
                     
                       g 
                       , 
                       
                         i 
                         2 
                       
                     
                   
                   = 
                   
                     
                       S 
                       
                         
                           i 
                           2 
                         
                         , 
                         σ 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             α 
                             
                               g 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                            
                           
                             
                               p 
                               g 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     2 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             β 
                             
                               g 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                            
                           
                             
                               p 
                               g 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     3 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             γ 
                             
                               g 
                               , 
                               
                                 i 
                                 2 
                               
                             
                           
                            
                           
                             
                               p 
                               g 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     5 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
             
               
                 
                   
                     b 
                     
                       o 
                       , 
                       
                         i 
                         1 
                       
                     
                     ′ 
                   
                   = 
                   
                     
                       S 
                       
                         
                           i 
                           1 
                         
                         , 
                         σ 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             α 
                             
                               o 
                               , 
                               
                                 i 
                                 1 
                               
                             
                             ′ 
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             β 
                             
                               o 
                               , 
                               
                                 i 
                                 1 
                               
                             
                             ′ 
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     2 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             γ 
                             
                               o 
                               , 
                               
                                 i 
                                 1 
                               
                             
                             ′ 
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     4 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
             
               
                 
                   
                     b 
                     
                       o 
                       , 
                       
                         i 
                         2 
                       
                     
                     ′ 
                   
                   = 
                   
                     
                       S 
                       
                         
                           i 
                           2 
                         
                         , 
                         σ 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             α 
                             
                               o 
                               , 
                               
                                 i 
                                 2 
                               
                             
                             ′ 
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     2 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             β 
                             
                               o 
                               , 
                               
                                 i 
                                 2 
                               
                             
                             ′ 
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     3 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             γ 
                             
                               o 
                               , 
                               
                                 i 
                                 2 
                               
                             
                             ′ 
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   
                                     j 
                                     5 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
             
               
                 
                   
                     u 
                     
                       g 
                       , 
                       
                         i 
                         1 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               1 
                               / 
                               2 
                             
                             , 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         g 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                     
                                     + 
                                     
                                       b 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         g 
                                         , 
                                         
                                           i 
                                           2 
                                         
                                       
                                     
                                     + 
                                     
                                       b 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           2 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                               = 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     b 
                                     
                                       g 
                                       , 
                                       
                                         i 
                                         2 
                                       
                                     
                                   
                                   + 
                                   
                                     b 
                                     
                                       o 
                                       , 
                                       
                                         i 
                                         2 
                                       
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                               / 
                               
                                 ( 
                                 
                                   
                                     b 
                                     
                                       g 
                                       , 
                                       
                                         i 
                                         1 
                                       
                                     
                                   
                                   + 
                                   
                                     b 
                                     
                                       o 
                                       , 
                                       
                                         i 
                                         1 
                                       
                                     
                                     ′ 
                                   
                                   + 
                                   
                                     b 
                                     
                                       g 
                                       , 
                                       
                                         i 
                                         2 
                                       
                                     
                                   
                                   + 
                                   
                                     b 
                                     
                                       o 
                                       , 
                                       
                                         i 
                                         2 
                                       
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                             , 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         g 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                     
                                     + 
                                     
                                       b 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         g 
                                         , 
                                         
                                           i 
                                           2 
                                         
                                       
                                     
                                     + 
                                     
                                       b 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           2 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                               ≠ 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
             
               
                 
                   
                     u 
                     
                       g 
                       , 
                       
                         i 
                         2 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               1 
                               / 
                               2 
                             
                             , 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         g 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                     
                                     + 
                                     
                                       b 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         g 
                                         , 
                                         
                                           i 
                                           2 
                                         
                                       
                                     
                                     + 
                                     
                                       b 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           2 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                               = 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     b 
                                     
                                       g 
                                       , 
                                       
                                         i 
                                         1 
                                       
                                     
                                   
                                   + 
                                   
                                     b 
                                     
                                       o 
                                       , 
                                       
                                         i 
                                         1 
                                       
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                               / 
                               
                                 ( 
                                 
                                   
                                     b 
                                     
                                       g 
                                       , 
                                       
                                         i 
                                         1 
                                       
                                     
                                   
                                   + 
                                   
                                     b 
                                     
                                       o 
                                       , 
                                       
                                         i 
                                         1 
                                       
                                     
                                     ′ 
                                   
                                   + 
                                   
                                     b 
                                     
                                       g 
                                       , 
                                       
                                         i 
                                         2 
                                       
                                     
                                   
                                   + 
                                   
                                     b 
                                     
                                       o 
                                       , 
                                       
                                         i 
                                         2 
                                       
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                             , 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         g 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                     
                                     + 
                                     
                                       b 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           1 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         g 
                                         , 
                                         
                                           i 
                                           2 
                                         
                                       
                                     
                                     + 
                                     
                                       b 
                                       
                                         o 
                                         , 
                                         
                                           i 
                                           2 
                                         
                                       
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                               ≠ 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
           
         
       
     
     The vertex pressures in the formulas Eqs.49, 50, 51, 52 above are treated by the inverse distance weighting method, i.e, 
     
       
         
           
             
               
                 
                   
                     
                       
                         p 
                         g 
                       
                        
                       
                         ( 
                         
                           
                             d 
                             j 
                           
                           , 
                           
                             t 
                             
                               n 
                               + 
                               1 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           ∈ 
                           
                             D 
                             j 
                           
                         
                         
                             
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             
                                
                               
                                 
                                   d 
                                   j 
                                 
                                 - 
                                 
                                   ϰ 
                                   i 
                                 
                               
                                
                             
                             2 
                             l 
                           
                           
                             
                               ∑ 
                               
                                 i 
                                 ∈ 
                                 
                                   D 
                                   j 
                                 
                               
                               
                                   
                               
                             
                              
                             
                               
                                  
                                 
                                   
                                     d 
                                     j 
                                   
                                   - 
                                   
                                     ϰ 
                                     i 
                                   
                                 
                                  
                               
                               2 
                               l 
                             
                           
                         
                          
                         
                           
                             p 
                             g 
                           
                            
                           
                             ( 
                             
                               
                                 ϰ 
                                 i 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     j 
                     = 
                     
                       j 
                       1 
                     
                   
                   , 
                   
                     j 
                     2 
                   
                   , 
                   
                     j 
                     3 
                   
                   , 
                   
                     j 
                     4 
                   
                   , 
                   
                     j 
                     5 
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
           
         
       
     
     D j  denotes the number set of all discrete control body central points from the vertex d j , l is a real number less than 0. 
     The coefficients α o,i     1   , β o,i     1   , γ o,i     1   , α o,i     1   ′, β o,i     1   ′, γ o,i     1   ′ in the formulas Eqs.45, 47, 49, 51 above are determined by the following formula 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             λ 
                             g 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 ϰ 
                                 
                                   i 
                                   1 
                                 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           n 
                           
                             
                               i 
                               1 
                             
                             , 
                             σ 
                           
                         
                       
                       
                         
                           B 
                           g 
                         
                          
                         
                           [ 
                           
                             
                               p 
                               g 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   
                                     i 
                                     1 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         
                           α 
                           
                             g 
                             , 
                             
                               i 
                               1 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 1 
                               
                             
                             - 
                             
                               ϰ 
                               
                                 i 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           β 
                           
                             g 
                             . 
                             
                               i 
                               1 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 3 
                               
                             
                             - 
                             
                               ϰ 
                               
                                 i 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           γ 
                           
                             g 
                             , 
                             
                               i 
                               1 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 4 
                               
                             
                             - 
                             
                               ϰ 
                               
                                 i 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       ϰ 
                       
                         i 
                         1 
                       
                     
                     ∈ 
                     
                       Ω 
                       I 
                     
                   
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             R 
                             s 
                           
                            
                           
                             [ 
                             
                               
                                 p 
                                 o 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ϰ 
                                     
                                       i 
                                       1 
                                     
                                   
                                   , 
                                   
                                     t 
                                     
                                       n 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                          
                         
                           
                             λ 
                             o 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 ϰ 
                                 
                                   i 
                                   1 
                                 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           n 
                           
                             
                               i 
                               1 
                             
                             , 
                             σ 
                           
                         
                       
                       
                         
                           B 
                           o 
                         
                          
                         
                           [ 
                           
                             
                               
                                 p 
                                 o 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ϰ 
                                     
                                       i 
                                       1 
                                     
                                   
                                   , 
                                   
                                     t 
                                     
                                       n 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             , 
                             
                               p 
                               b 
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         
                           α 
                           
                             o 
                             , 
                             
                               i 
                               1 
                             
                           
                           ′ 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 1 
                               
                             
                             - 
                             
                               ϰ 
                               
                                 i 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           β 
                           
                             o 
                             , 
                             
                               i 
                               1 
                             
                           
                           ′ 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 2 
                               
                             
                             - 
                             
                               ϰ 
                               
                                 i 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           γ 
                           
                             o 
                             , 
                             
                               i 
                               1 
                             
                           
                           ′ 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 4 
                               
                             
                             - 
                             
                               ϰ 
                               
                                 i 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       ϰ 
                       
                         i 
                         1 
                       
                     
                     ∈ 
                     
                       Ω 
                       I 
                     
                   
                 
               
               
                 
                   ( 
                   57 
                   ) 
                 
               
             
           
         
       
     
     The subscripts of vertices coordinates j 1 , j 3 , j 4  are intersected by a ray in the direction of 
     
       
         
           
             
               
                 
                   λ 
                   g 
                   T 
                 
                  
                 
                   ( 
                   
                     
                       ϰ 
                       
                         i 
                         1 
                       
                     
                     , 
                     
                       t 
                       
                         n 
                         + 
                         1 
                       
                     
                   
                   ) 
                 
               
                
               
                 n 
                 
                   
                     i 
                     1 
                   
                   , 
                   σ 
                 
               
             
             
               
                 B 
                 g 
               
                
               
                 [ 
                 
                   
                     p 
                     g 
                   
                    
                   
                     ( 
                     
                       
                         ϰ 
                         
                           i 
                           1 
                         
                       
                       , 
                       
                         t 
                         
                           n 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
                 ] 
               
             
           
         
       
     
     through the starting point x i     1    with a sub-surface of the discrete control body Ω i     1   . The three vertices coordinates of the sub-surface are d j     1   , d j     3   , d j     4    as shown in  FIG. 6 . The subscripts of vertices coordinates j 1 , j 2 , j 4  are intersected by a ray in the direction of 
     
       
         
           
             
               
                 
                   R 
                   s 
                 
                  
                 
                   [ 
                   
                     
                       p 
                       o 
                     
                      
                     
                       ( 
                       
                         
                           ϰ 
                           
                             i 
                             1 
                           
                         
                         , 
                         
                           t 
                           
                             n 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                   ] 
                 
               
                
               
                 
                   λ 
                   o 
                   T 
                 
                  
                 
                   ( 
                   
                     
                       ϰ 
                       
                         i 
                         1 
                       
                     
                     , 
                     
                       t 
                       
                         n 
                         + 
                         1 
                       
                     
                   
                   ) 
                 
               
                
               
                 n 
                 
                   
                     i 
                     1 
                   
                   , 
                   σ 
                 
               
             
             
               
                 B 
                 o 
               
                
               
                 [ 
                 
                   
                     
                       p 
                       o 
                     
                      
                     
                       ( 
                       
                         
                           ϰ 
                           
                             i 
                             1 
                           
                         
                         , 
                         
                           t 
                           
                             n 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                     p 
                     b 
                   
                 
                 ] 
               
             
           
         
       
     
     through the starting point x i     1    with a sub-surface of the discrete control body Ω i     1   . The three vertices coordinates of the sub-surface are d j     1   , d j     2   , d j     4   , as shown in  FIG. 6 . 
     The coefficients α o,i     2   , β o,i     2   , γ o,i     2   , α o,i     2   ′, β o,i     2   ′, γ o,i     2   ′, in the formulas [(46),(48),(50),(52)] above are determined by the following formula 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           λ 
                           g 
                           T 
                         
                          
                         
                           ( 
                           
                             
                               ϰ 
                               
                                 i 
                                 2 
                               
                             
                             , 
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         n 
                         
                           
                             i 
                             2 
                           
                           , 
                           σ 
                         
                       
                     
                     
                       
                         B 
                         g 
                       
                        
                       
                         [ 
                         
                           
                             p 
                             g 
                           
                            
                           
                             ( 
                             
                               
                                 ϰ 
                                 
                                   i 
                                   2 
                                 
                               
                               , 
                               
                                 t 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         α 
                         
                           g 
                           , 
                           
                             i 
                             2 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               2 
                             
                           
                           - 
                           
                             ϰ 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         β 
                         
                           g 
                           . 
                           
                             i 
                             2 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               3 
                             
                           
                           - 
                           
                             ϰ 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         γ 
                         
                           g 
                           , 
                           
                             i 
                             2 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               5 
                             
                           
                           - 
                           
                             ϰ 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           R 
                           s 
                         
                          
                         
                           [ 
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   
                                     i 
                                     2 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                        
                       
                         
                           λ 
                           o 
                           T 
                         
                          
                         
                           ( 
                           
                             
                               ϰ 
                               
                                 i 
                                 2 
                               
                             
                             , 
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         n 
                         
                           
                             i 
                             2 
                           
                           , 
                           σ 
                         
                       
                     
                     
                       
                         B 
                         o 
                       
                        
                       
                         [ 
                         
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   
                                     i 
                                     2 
                                   
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           , 
                           
                             p 
                             b 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         α 
                         
                           o 
                           , 
                           
                             i 
                             2 
                           
                         
                         ′ 
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               2 
                             
                           
                           - 
                           
                             ϰ 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         β 
                         
                           o 
                           , 
                           
                             i 
                             1 
                           
                         
                         ′ 
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               3 
                             
                           
                           - 
                           
                             ϰ 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         γ 
                         
                           o 
                           , 
                           
                             i 
                             2 
                           
                         
                         ′ 
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               5 
                             
                           
                           - 
                           
                             ϰ 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   59 
                   ) 
                 
               
             
           
         
       
     
     The subscripts of vertices coordinates j 2 , j 3 , j 5 , are intersected by a ray in the direction of 
     
       
         
           
             
               
                 
                   λ 
                   g 
                   T 
                 
                  
                 
                   ( 
                   
                     
                       ϰ 
                       
                         i 
                         2 
                       
                     
                     , 
                     
                       t 
                       
                         n 
                         + 
                         1 
                       
                     
                   
                   ) 
                 
               
                
               
                 n 
                 
                   
                     i 
                     2 
                   
                   , 
                   σ 
                 
               
             
             
               
                 B 
                 g 
               
                
               
                 [ 
                 
                   
                     p 
                     g 
                   
                    
                   
                     ( 
                     
                       
                         ϰ 
                         
                           i 
                           2 
                         
                       
                       , 
                       
                         t 
                         
                           n 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
                 ] 
               
             
           
         
       
     
     through the starting point x i     2    with a sub-surface of the discrete control body Ω i     2   . The three vertices coordinates of the sub-surface are d j     2   , d j     3   , d j     5   , as shown in  FIG. 6 . The subscripts of vertices coordinates j 2 , j 3 , j 5  are intersected by a ray in the direction of 
     
       
         
           
             
               
                 
                   R 
                   s 
                 
                  
                 
                   [ 
                   
                     
                       p 
                       o 
                     
                      
                     
                       ( 
                       
                         
                           ϰ 
                           
                             i 
                             2 
                           
                         
                         , 
                         
                           t 
                           
                             n 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                   ] 
                 
               
                
               
                 
                   λ 
                   o 
                   T 
                 
                  
                 
                   ( 
                   
                     
                       ϰ 
                       
                         i 
                         2 
                       
                     
                     , 
                     
                       t 
                       
                         n 
                         + 
                         1 
                       
                     
                   
                   ) 
                 
               
                
               
                 n 
                 
                   
                     i 
                     2 
                   
                   , 
                   σ 
                 
               
             
             
               
                 B 
                 o 
               
                
               
                 [ 
                 
                   
                     
                       p 
                       o 
                     
                      
                     
                       ( 
                       
                         
                           ϰ 
                           
                             i 
                             2 
                           
                         
                         , 
                         
                           t 
                           
                             n 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                     p 
                     b 
                   
                 
                 ] 
               
             
           
         
       
     
     through the starting point x i     2    with a sub-surface of the discrete control body Ω i     2   . The three vertices coordinates of the sub-surface are d j     2   , d j     3   , d j     5    as shown in  FIG. 6 . 
     Auxiliary equations Eq.4, 5, 6 with saturation and capillary pressure are discretized as 
         S   o ( x   i     1     ,t   n )+ S   w ( x   t     1     ,t   n )+ S   g ( x   i     1     ,t   n )=1, x   i     1   ∈Ω I   UΩ   B   (60)
 
         p   cow [ S   w ( x   i     1     ,t   n )]= p   o ( x   i     1     ,t   n )− p   w ( x   i     1     ,t   n ), x   i     1   ∈Ω I   UΩ   B   (61)
 
         p   cgo [ S   g ( x   i     1     ,t   n )= p   g ( x   i     1     ,t   n )− p   o ( x   i     1     ,t   n ), x   i     1   ∈Ω I   UΩ   B   (62)
 
     Oil phase boundary condition equation is discretized as 
     
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             c 
                             
                               o 
                               , 
                               1 
                             
                           
                           - 
                           
                             α 
                             
                               o 
                               , 
                               
                                 i 
                                 1 
                               
                             
                             
                               ∂ 
                               Ω 
                             
                           
                           - 
                           
                             β 
                             
                               o 
                               , 
                               
                                 i 
                                 1 
                               
                             
                             
                               ∂ 
                               Ω 
                             
                           
                           - 
                           
                             γ 
                             
                               o 
                               , 
                               
                                 i 
                                 1 
                               
                             
                             
                               ∂ 
                               Ω 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           p 
                           o 
                         
                          
                         
                           ( 
                           
                             
                               ϰ 
                               
                                 i 
                                 1 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         α 
                         
                           o 
                           , 
                           
                             i 
                             1 
                           
                         
                         
                           ∂ 
                           Ω 
                         
                       
                        
                       
                         
                           p 
                           o 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 1 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         β 
                         
                           o 
                           , 
                           
                             i 
                             1 
                           
                         
                         
                           ∂ 
                           Ω 
                         
                       
                        
                       
                         
                           p 
                           o 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 2 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         γ 
                         
                           o 
                           , 
                           
                             i 
                             1 
                           
                         
                         
                           ∂ 
                           Ω 
                         
                       
                        
                       
                         
                           p 
                           o 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 4 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       g 
                       o 
                     
                      
                     
                       ( 
                       
                         
                           ϰ 
                           
                             i 
                             1 
                           
                         
                         , 
                         
                           t 
                           n 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           p 
                           o 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               j 
                             
                             , 
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             
                               D 
                               j 
                             
                           
                           
                               
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             
                               
                                  
                                 
                                   
                                     d 
                                     j 
                                   
                                   - 
                                   
                                     ϰ 
                                     i 
                                   
                                 
                                  
                               
                               2 
                               l 
                             
                             
                               
                                 ∑ 
                                 
                                   i 
                                   ∈ 
                                   
                                     D 
                                     j 
                                   
                                 
                                 
                                     
                                 
                               
                                
                               
                                 
                                    
                                   
                                     
                                       d 
                                       j 
                                     
                                     - 
                                     
                                       ϰ 
                                       i 
                                     
                                   
                                    
                                 
                                 2 
                                 l 
                               
                             
                           
                            
                           
                             
                               p 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   i 
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     , 
                     
                       j 
                       = 
                       
                         j 
                         1 
                       
                     
                     , 
                     
                       j 
                       2 
                     
                     , 
                     
                       j 
                       4 
                     
                   
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
           
         
       
     
     Where x i     1   ∈Ω B , D j  denotes the number set of all discrete control body central points from the vertex d j , l is a real number less than 0. 
     The coefficients α o,i     1     ∂Ω , β o,i     1     ∂Ω , γ o,i     1     ∂Ω  in the formulas above are determined by the formula below 
         c   o,2 λ o   T ( x   i     1     ,t   n ) n   i     1     ,σ   ∂Ω =α o,i     1     ∂Ω ( d   j     1     −x   i     1   )+β o,i     1     ∂Ω ( d   j     2     −x   i     1   )+γ o,i     1     ∂Ω ( d   j     4     −x   i     1   ), x   i     1   ∈Ω B   (65)
 
     The subscripts of vertices coordinates j 1 , j 2 , j 4  are intersected by a ray in the direction of c o,2 λ o   T (x i     1   ,t n )n i     1     ,σ   ∂Ω  and the starting point x i     1    with a sub-surface of the discrete control body Ω i     1   . The three vertices coordinates of the sub-surface are d j     1   , d j     2   , d j     4    as shown in  FIG. 7 . Water phase boundary condition equation Eq.8 is discretized as 
     
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             c 
                             
                               w 
                               , 
                               1 
                             
                           
                           - 
                           
                             α 
                             
                               w 
                               , 
                               
                                 i 
                                 1 
                               
                             
                             
                               ∂ 
                               Ω 
                             
                           
                           - 
                           
                             β 
                             
                               w 
                               , 
                               
                                 i 
                                 1 
                               
                             
                             
                               ∂ 
                               Ω 
                             
                           
                           - 
                           
                             γ 
                             
                               w 
                               , 
                               
                                 i 
                                 1 
                               
                             
                             
                               ∂ 
                               Ω 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           p 
                           w 
                         
                          
                         
                           ( 
                           
                             
                               ϰ 
                               
                                 i 
                                 1 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         α 
                         
                           w 
                           , 
                           
                             i 
                             1 
                           
                         
                         
                           ∂ 
                           Ω 
                         
                       
                        
                       
                         p 
                         w 
                       
                        
                       
                         ( 
                         
                           
                             d 
                             
                               j 
                               1 
                             
                           
                           , 
                           
                             t 
                             n 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         β 
                         
                           w 
                           , 
                           
                             i 
                             1 
                           
                         
                         
                           ∂ 
                           Ω 
                         
                       
                        
                       
                         
                           p 
                           w 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 2 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         γ 
                         
                           w 
                           , 
                           
                             i 
                             1 
                           
                         
                         
                           ∂ 
                           Ω 
                         
                       
                        
                       
                         
                           p 
                           w 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 3 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       g 
                       w 
                     
                      
                     
                       ( 
                       
                         
                           ϰ 
                           
                             i 
                             1 
                           
                         
                         , 
                         
                           t 
                           n 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           p 
                           w 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               j 
                             
                             , 
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             
                               D 
                               j 
                             
                           
                           
                               
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             
                               
                                  
                                 
                                   
                                     d 
                                     j 
                                   
                                   - 
                                   
                                     ϰ 
                                     i 
                                   
                                 
                                  
                               
                               2 
                               l 
                             
                             
                               
                                 ∑ 
                                 
                                   i 
                                   ∈ 
                                   
                                     D 
                                     j 
                                   
                                 
                                 
                                     
                                 
                               
                                
                               
                                 
                                    
                                   
                                     
                                       d 
                                       j 
                                     
                                     - 
                                     
                                       ϰ 
                                       i 
                                     
                                   
                                    
                                 
                                 2 
                                 l 
                               
                             
                           
                            
                           
                             
                               p 
                               w 
                             
                              
                             
                               ( 
                               
                                 
                                   ϰ 
                                   i 
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     , 
                     
                       j 
                       = 
                       
                         j 
                         1 
                       
                     
                     , 
                     
                       j 
                       2 
                     
                     , 
                     
                       j 
                       3 
                     
                   
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
           
         
       
     
     Where x i     1   ∈ΩB, D j  denotes the number set of all discrete control body central points from the vertex d j , l is areal numberless than 0. 
     The coefficients α w,i     1     ∂Ω , β w,i     1     ∂Ω , γ w,i     1     ∂Ω  in the formula Eq.66 above are determined by the formula below 
         c   w,2 λ w   T ( x   i     1     ,t   n ) n   i     1     ,σ   ∂Ω =α w,i     1     ∂Ω ( d   j     1     −x   i     1   )+β w,i     1     ∂Ω ( d   j     2     −x   i     1   )+γ w,i     1     ∂Ω ( d   j     3     −x   i     1   ), x   i     1   ∈Ω B   (68)
 
     The subscripts of vertices coordinates j 1 , j 2 , j 3  are intersected by a ray in the direction of c w,2 λ w   T (x i     1   ,t n )n i     1     ,σ   ∂Ω  through the starting point x i     1    with a sub-surface of the discrete control body Ω i     1   . The three vertices coordinates of the sub-surface are d j     1   , d j     2   , d j     3    as shown in  FIG. 8 . 
     Gas phase boundary condition equation is discretized as 
     
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             c 
                             
                               g 
                               , 
                               1 
                             
                           
                           - 
                           
                             α 
                             
                               g 
                               , 
                               
                                 i 
                                 1 
                               
                             
                             
                               ∂ 
                               Ω 
                             
                           
                           - 
                           
                             β 
                             
                               g 
                               , 
                               
                                 i 
                                 1 
                               
                             
                             
                               ∂ 
                               Ω 
                             
                           
                           - 
                           
                             γ 
                             
                               g 
                               , 
                               
                                 i 
                                 1 
                               
                             
                             
                               ∂ 
                               Ω 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           P 
                           g 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         α 
                         
                           g 
                           , 
                           
                             i 
                             1 
                           
                         
                         
                           ∂ 
                           Ω 
                         
                       
                        
                       
                         
                           P 
                           g 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 1 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         β 
                         
                           g 
                           , 
                           
                             i 
                             1 
                           
                         
                         
                           ∂ 
                           Ω 
                         
                       
                        
                       
                         
                           P 
                           g 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 2 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         γ 
                         
                           g 
                           , 
                           
                             i 
                             1 
                           
                         
                         
                           ∂ 
                           Ω 
                         
                       
                        
                       
                         
                           P 
                           g 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               
                                 j 
                                 4 
                               
                             
                             , 
                             
                               t 
                               n 
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       g 
                       g 
                     
                      
                     
                       ( 
                       
                         
                           x 
                           
                             i 
                             1 
                           
                         
                         , 
                         
                           t 
                           n 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   69 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           P 
                           g 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               j 
                             
                             , 
                             
                               t 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             
                               D 
                               j 
                             
                           
                         
                          
                         
                           
                             
                               || 
                               
                                 
                                   d 
                                   j 
                                 
                                 - 
                                 
                                   x 
                                   i 
                                 
                               
                                
                               
                                 || 
                                 2 
                                 l 
                               
                             
                             
                               
                                 ∑ 
                                 
                                   i 
                                   ∈ 
                                   
                                     D 
                                     j 
                                   
                                 
                               
                                
                               
                                 || 
                                 
                                   
                                     d 
                                     j 
                                   
                                   - 
                                   
                                     x 
                                     i 
                                   
                                 
                                  
                                 
                                   || 
                                   2 
                                   l 
                                 
                               
                             
                           
                            
                           
                             
                               P 
                               g 
                             
                              
                             
                               ( 
                               
                                 
                                   x 
                                   i 
                                 
                                 , 
                                 
                                   t 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     , 
                     
                       j 
                       = 
                       
                         j 
                         1 
                       
                     
                     , 
                     
                       j 
                       2 
                     
                     , 
                     
                       j 
                       4 
                     
                   
                 
               
               
                 
                   ( 
                   70 
                   ) 
                 
               
             
           
         
       
     
     Where x i     1   ∈Ω B , D j  denotes the number set of all discrete control body central points from the vertex d j , l is a real number less than 0. 
     The coefficients α g,i     1     ∂Ω , β g,i     1     ∂Ω , γ g,i     1     ∂Ω  in the formula Eq.69 above are determined by the formula below 
         c   g,2 λ g   T ( x   i     1     ,t   n ) n   i     1     ,σ   ∂Ω =α g,i     1     ∂Ω ( d   j     1     −x   i     1   )+β g,i     1     ∂Ω ( d   j     2     −x   i     1   )+γ g,i     1     ∂Ω ( d   j     4     −x   i     1   ), x   i     1   ∈Ω B   (71)
 
     The subscripts of vertices coordinates j 1 , j 2 , j 4  are intersected by a ray in the direction of c g,2 λ g   T (x i     1   ,t n )n i     1     ,σ   ∂Ω  through the starting point x i     1    with a sub-surface of the discrete control body Ω i     1   . The three vertices coordinates of the sub-surface are d j     1   , d j     2   , d j     4    as shown in  FIG. 9 . 
     Initial condition equations Eqs.10 and 11 are discretized as 
         S   o ( x   i     1   ,0)= l   o ( x   i     1   ), x   i     1   ∈Ω I   UΩ   B   (72)
 
         p   o ( x   i     1   ,0)=ƒ o ( x   i     1   ), x   i     1   ∈Ω I   UΩ   B   (73)
 
         S   w ( x   i     1   ,0)= l   w ( x   i     1   ), x   i     1   ∈Ω I   UΩ   B   (74)
 
         p   w ( x   i     1   ,0)=ƒ w ( x   i     1   ), x   i     1   ∈Ω I   UΩ   B   (75)
 
         S   g ( x   i     1   ,0)= l   g ( x   i     1   ), x   i     1   ∈Ω I   UΩ   B   (76)
 
         p   g ( x   i     1   ,0)=ƒ g ( x   i     1   ), x   i     1   ∈Ω I   UΩ   B   (77)
 
     Based on the above discrete control equations of oil, gas and water phase and the discrete boundary condition equations of oil, gas and water phase Eqs.16, 28, 40, 63, 66, 69, the following discrete equations are obtained 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           ∑ 
                                           
                                             σ 
                                             ⋐ 
                                             
                                               ∂ 
                                               
                                                 Ω 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                             
                                           
                                         
                                          
                                         
                                           
                                             u 
                                             
                                               w 
                                               , 
                                               
                                                 i 
                                                 2 
                                               
                                             
                                           
                                            
                                           
                                             a 
                                             
                                               w 
                                               , 
                                               
                                                 i 
                                                 2 
                                               
                                             
                                           
                                            
                                           
                                             
                                               P 
                                               w 
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   x 
                                                   
                                                     i 
                                                     2 
                                                   
                                                 
                                                 , 
                                                 
                                                   t 
                                                   
                                                     n 
                                                     + 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       - 
                                       
                                         
                                           u 
                                           
                                             w 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                         
                                          
                                         
                                           a 
                                           
                                             w 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                         
                                          
                                         
                                           
                                             P 
                                             w 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 x 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 
                                                   n 
                                                   + 
                                                   1 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     = 
                                     
                                       
                                         
                                           e 
                                           
                                             w 
                                             , 
                                             2 
                                           
                                         
                                          
                                         
                                           
                                             S 
                                             w 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 x 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 
                                                   n 
                                                   + 
                                                   1 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       - 
                                       
                                         
                                           e 
                                           
                                             w 
                                             , 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             S 
                                             w 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 x 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           V 
                                           
                                             i 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             
                                               q 
                                               w 
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   x 
                                                   
                                                     i 
                                                     1 
                                                   
                                                 
                                                 , 
                                                 
                                                   t 
                                                   
                                                     n 
                                                     + 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                           
                                             ρ 
                                             wsc 
                                           
                                         
                                       
                                     
                                   
                                   , 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     ∈ 
                                     
                                       Ω 
                                       I 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               c 
                                               
                                                 o 
                                                 , 
                                                 1 
                                               
                                             
                                             - 
                                             
                                               α 
                                               
                                                 o 
                                                 , 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               
                                                 ∂ 
                                                 Ω 
                                               
                                             
                                             - 
                                             
                                               β 
                                               
                                                 o 
                                                 , 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               
                                                 ∂ 
                                                 Ω 
                                               
                                             
                                             - 
                                             
                                               γ 
                                               
                                                 o 
                                                 , 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               
                                                 ∂ 
                                                 Ω 
                                               
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                           
                                             P 
                                             o 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 x 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           α 
                                           
                                             o 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                           
                                             ∂ 
                                             Ω 
                                           
                                         
                                          
                                         
                                           
                                             P 
                                             o 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 d 
                                                 
                                                   j 
                                                   1 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           β 
                                           
                                             o 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                           
                                             ∂ 
                                             Ω 
                                           
                                         
                                          
                                         
                                           
                                             P 
                                             o 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 d 
                                                 
                                                   j 
                                                   2 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           γ 
                                           
                                             o 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                           
                                             ∂ 
                                             Ω 
                                           
                                         
                                          
                                         
                                           
                                             P 
                                             o 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 d 
                                                 
                                                   j 
                                                   4 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     = 
                                     
                                       
                                         g 
                                         o 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             x 
                                             
                                               i 
                                               1 
                                             
                                           
                                           , 
                                           
                                             t 
                                             n 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   , 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     ∈ 
                                     
                                       Ω 
                                       B 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           ∑ 
                                           
                                             σ 
                                             ⋐ 
                                             
                                               ∂ 
                                               
                                                 Ω 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                             
                                           
                                         
                                          
                                         
                                           
                                             u 
                                             
                                               w 
                                               , 
                                               
                                                 i 
                                                 2 
                                               
                                             
                                           
                                            
                                           
                                             a 
                                             
                                               w 
                                               , 
                                               
                                                 i 
                                                 2 
                                               
                                             
                                           
                                            
                                           
                                             
                                               P 
                                               w 
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   x 
                                                   
                                                     i 
                                                     2 
                                                   
                                                 
                                                 , 
                                                 
                                                   t 
                                                   
                                                     n 
                                                     + 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       - 
                                       
                                         
                                           u 
                                           
                                             w 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                         
                                          
                                         
                                           a 
                                           
                                             w 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                         
                                          
                                         
                                           
                                             P 
                                             w 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 x 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 
                                                   n 
                                                   + 
                                                   1 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     = 
                                     
                                       
                                         
                                           e 
                                           
                                             w 
                                             , 
                                             2 
                                           
                                         
                                          
                                         
                                           
                                             S 
                                             w 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 x 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 
                                                   n 
                                                   + 
                                                   1 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       - 
                                       
                                         
                                           e 
                                           
                                             w 
                                             , 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             S 
                                             w 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 x 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           V 
                                           
                                             i 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             
                                               q 
                                               w 
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   x 
                                                   
                                                     i 
                                                     1 
                                                   
                                                 
                                                 , 
                                                 
                                                   t 
                                                   
                                                     n 
                                                     + 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                           
                                             ρ 
                                             wsc 
                                           
                                         
                                       
                                     
                                   
                                   , 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     ∈ 
                                     
                                       Ω 
                                       l 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               c 
                                               
                                                 w 
                                                 , 
                                                 1 
                                               
                                             
                                             - 
                                             
                                               α 
                                               
                                                 w 
                                                 , 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               
                                                 ∂ 
                                                 Ω 
                                               
                                             
                                             - 
                                             
                                               β 
                                               
                                                 w 
                                                 , 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               
                                                 ∂ 
                                                 Ω 
                                               
                                             
                                             - 
                                             
                                               γ 
                                               
                                                 w 
                                                 , 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               
                                                 ∂ 
                                                 Ω 
                                               
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                           
                                             P 
                                             w 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 x 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           α 
                                           
                                             w 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                           
                                             ∂ 
                                             Ω 
                                           
                                         
                                          
                                         
                                           
                                             P 
                                             w 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 d 
                                                 
                                                   j 
                                                   1 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           β 
                                           
                                             w 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                           
                                             ∂ 
                                             Ω 
                                           
                                         
                                          
                                         
                                           
                                             P 
                                             w 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 d 
                                                 
                                                   j 
                                                   2 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           γ 
                                           
                                             w 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                           
                                             ∂ 
                                             Ω 
                                           
                                         
                                          
                                         
                                           
                                             P 
                                             w 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 d 
                                                 
                                                   j 
                                                   4 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     = 
                                     
                                       
                                         g 
                                         w 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             x 
                                             
                                               i 
                                               1 
                                             
                                           
                                           , 
                                           
                                             t 
                                             n 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   , 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     ∈ 
                                     
                                       Ω 
                                       B 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ∑ 
                                         
                                           σ 
                                           ⋐ 
                                           
                                             ∂ 
                                             
                                               Ω 
                                               
                                                 i 
                                                 1 
                                               
                                             
                                           
                                         
                                       
                                        
                                       
                                         { 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     u 
                                                     
                                                       g 
                                                       , 
                                                       
                                                         i 
                                                         2 
                                                       
                                                     
                                                   
                                                    
                                                   
                                                     [ 
                                                     
                                                       
                                                         
                                                           a 
                                                           
                                                             g 
                                                             , 
                                                             
                                                             i 
                                                             2 
                                                             
                                                           
                                                         
                                                          
                                                         
                                                           
                                                             P 
                                                             g 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             x 
                                                             
                                                             i 
                                                             2 
                                                             
                                                             
                                                             , 
                                                             
                                                             t 
                                                             
                                                             n 
                                                             + 
                                                             1 
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                       
                                                       + 
                                                       
                                                         
                                                           a 
                                                           
                                                             o 
                                                             , 
                                                             
                                                             i 
                                                             2 
                                                             
                                                           
                                                         
                                                          
                                                         
                                                           
                                                             P 
                                                             o 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             x 
                                                             
                                                             i 
                                                             2 
                                                             
                                                             
                                                             , 
                                                             
                                                             t 
                                                             
                                                             n 
                                                             + 
                                                             1 
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 - 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   u 
                                                   
                                                     g 
                                                     , 
                                                     
                                                       i 
                                                       1 
                                                     
                                                   
                                                 
                                                  
                                                 
                                                   [ 
                                                   
                                                     
                                                       
                                                         a 
                                                         
                                                           g 
                                                           , 
                                                           
                                                             i 
                                                             1 
                                                           
                                                         
                                                       
                                                        
                                                       
                                                         
                                                           P 
                                                           g 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             x 
                                                             
                                                             i 
                                                             1 
                                                             
                                                             
                                                             , 
                                                             
                                                             t 
                                                             
                                                             n 
                                                             + 
                                                             1 
                                                             
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                     + 
                                                     
                                                       
                                                         a 
                                                         
                                                           o 
                                                           , 
                                                           
                                                             i 
                                                             1 
                                                           
                                                         
                                                       
                                                        
                                                       
                                                         
                                                           P 
                                                           o 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             x 
                                                             
                                                             i 
                                                             1 
                                                             
                                                             
                                                             , 
                                                             
                                                             t 
                                                             
                                                             n 
                                                             + 
                                                             1 
                                                             
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                   ] 
                                                 
                                               
                                             
                                           
                                         
                                         } 
                                       
                                     
                                     = 
                                     
                                       
                                         [ 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     e 
                                                     
                                                       g 
                                                       , 
                                                       2 
                                                     
                                                   
                                                    
                                                   
                                                     
                                                       S 
                                                       g 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         x 
                                                         
                                                           
                                                             i 
                                                             1 
                                                           
                                                           , 
                                                           
                                                             t 
                                                             
                                                             n 
                                                             + 
                                                             1 
                                                             
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 - 
                                                 
                                                   
                                                     e 
                                                     
                                                       g 
                                                       , 
                                                       1 
                                                     
                                                   
                                                    
                                                   
                                                     
                                                       S 
                                                       g 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           x 
                                                           
                                                             i 
                                                             1 
                                                           
                                                         
                                                         , 
                                                         
                                                           t 
                                                           n 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 + 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   
                                                     e 
                                                     
                                                       o 
                                                       , 
                                                       2 
                                                     
                                                   
                                                    
                                                   
                                                     
                                                       S 
                                                       o 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         x 
                                                         
                                                           
                                                             i 
                                                             1 
                                                           
                                                           , 
                                                           
                                                             t 
                                                             
                                                             n 
                                                             + 
                                                             1 
                                                             
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 - 
                                                 
                                                   
                                                     e 
                                                     
                                                       o 
                                                       , 
                                                       1 
                                                     
                                                   
                                                    
                                                   
                                                     
                                                       S 
                                                       o 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           x 
                                                           
                                                             i 
                                                             1 
                                                           
                                                         
                                                         , 
                                                         
                                                           t 
                                                           n 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                       + 
                                       
                                         
                                           V 
                                           
                                             i 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             
                                               q 
                                               g 
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   x 
                                                   
                                                     i 
                                                     1 
                                                   
                                                 
                                                 , 
                                                 
                                                   t 
                                                   
                                                     n 
                                                     + 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                           
                                             ρ 
                                             gsc 
                                           
                                         
                                       
                                     
                                   
                                   , 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     ∈ 
                                     
                                       Ω 
                                       f 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               c 
                                               
                                                 g 
                                                 , 
                                                 1 
                                               
                                             
                                             - 
                                             
                                               α 
                                               
                                                 g 
                                                 , 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               
                                                 ∂ 
                                                 Ω 
                                               
                                             
                                             - 
                                             
                                               β 
                                               
                                                 g 
                                                 , 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               
                                                 ∂ 
                                                 Ω 
                                               
                                             
                                             - 
                                             
                                               γ 
                                               
                                                 g 
                                                 , 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               
                                                 ∂ 
                                                 Ω 
                                               
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                           
                                             P 
                                             g 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 x 
                                                 
                                                   i 
                                                   1 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           α 
                                           
                                             g 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                           
                                             ∂ 
                                             Ω 
                                           
                                         
                                          
                                         
                                           
                                             P 
                                             g 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 d 
                                                 
                                                   j 
                                                   1 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           β 
                                           
                                             g 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                           
                                             ∂ 
                                             Ω 
                                           
                                         
                                          
                                         
                                           
                                             P 
                                             g 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 d 
                                                 
                                                   j 
                                                   2 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           γ 
                                           
                                             g 
                                             , 
                                             
                                               i 
                                               1 
                                             
                                           
                                           
                                             ∂ 
                                             Ω 
                                           
                                         
                                          
                                         
                                           
                                             P 
                                             g 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 d 
                                                 
                                                   j 
                                                   4 
                                                 
                                               
                                               , 
                                               
                                                 t 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     = 
                                     
                                       
                                         g 
                                         g 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             x 
                                             
                                               i 
                                               1 
                                             
                                           
                                           , 
                                           
                                             t 
                                             n 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   , 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     ∈ 
                                     
                                       Ω 
                                       
                                         B 
                                          
                                          
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   S 
                                   o 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     , 
                                     
                                       t 
                                       n 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   S 
                                   w 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     , 
                                     
                                       t 
                                       n 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   S 
                                   g 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     , 
                                     
                                       t 
                                       n 
                                     
                                   
                                   ) 
                                 
                               
                             
                             = 
                             1 
                           
                           , 
                           
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                             ∈ 
                             
                               
                                 Ω 
                                 I 
                               
                                
                               
                                 
                                   U 
                                    
                                   Ω 
                                 
                                 B 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 P 
                                 cow 
                               
                                
                               
                                 [ 
                                 
                                   
                                     S 
                                     w 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         
                                           i 
                                           1 
                                         
                                       
                                       , 
                                       
                                         t 
                                         n 
                                       
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                             = 
                             
                               
                                 
                                   P 
                                   o 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     , 
                                     
                                       t 
                                       n 
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   P 
                                   w 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     , 
                                     
                                       t 
                                       n 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           , 
                           
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                             ∈ 
                             
                               
                                 Ω 
                                 I 
                               
                                
                               
                                 
                                   U 
                                    
                                   Ω 
                                 
                                 B 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 P 
                                 cgo 
                               
                                
                               
                                 [ 
                                 
                                   
                                     S 
                                     g 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         
                                           i 
                                           1 
                                         
                                       
                                       , 
                                       
                                         t 
                                         n 
                                       
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                             = 
                             
                               
                                 
                                   P 
                                   g 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     , 
                                     
                                       t 
                                       n 
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   P 
                                   o 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       
                                         i 
                                         1 
                                       
                                     
                                     , 
                                     
                                       t 
                                       n 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           , 
                           
                             
                               x 
                               
                                 i 
                                 1 
                               
                             
                             ∈ 
                             
                               
                                 Ω 
                                 I 
                               
                                
                               
                                 
                                   U 
                                    
                                   Ω 
                                 
                                 B 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   78 
                   ) 
                 
               
             
           
         
       
     
     There are six unknowns p o , p w , p g , S o , S w , S g  to be found at any control point x i     1    (x i     1   ∈Ω I UΩ B ) in separating domain, and six equations are given by Eq.78 at any control point x i     1   . Therefore, the equations system is complete. The equations can be solved by the Newton-raphson algorithm, which can further be applied to describe the pressure and saturation distribution of oil, gas and water at any time point in the solved region. 
     Embodiment 2 
     This embodiment provides oil/gas/water three-phase flow simulation method through treating the wellbore as inner boundary for a partially open vertical well in a homogeneous reservoir. The physical model is shown in  FIG. 10 . 
     The general equation of oil/gas/water three-phase flow can be established by applying the mass conservation principle: 
     Oil phase governing equation: 
     
       
         
           
             
               
                 
                   
                     
                       ∇ 
                       
                         · 
                         
                           [ 
                           
                             
                               1 
                               
                                 B 
                                 o 
                               
                             
                              
                             
                               λ 
                               o 
                             
                              
                             
                               ∇ 
                               
                                 p 
                                 o 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         ∂ 
                         
                           ∂ 
                           t 
                         
                       
                        
                       
                         ( 
                         
                           
                             1 
                             
                               B 
                               o 
                             
                           
                            
                           
                             
                               φ 
                                
                               S 
                             
                             o 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   79 
                   ) 
                 
               
             
           
         
       
     
     Water phase governing equation: 
     
       
         
           
             
               
                 
                   
                     
                       ∇ 
                       
                         · 
                         
                           [ 
                           
                             
                               1 
                               
                                 B 
                                 w 
                               
                             
                              
                             
                               λ 
                               w 
                             
                              
                             
                               ∇ 
                               
                                 p 
                                 w 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         ∂ 
                         
                           ∂ 
                           t 
                         
                       
                        
                       
                         ( 
                         
                           
                             1 
                             
                               B 
                               w 
                             
                           
                            
                           
                             
                               φ 
                                
                               S 
                             
                             w 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   80 
                   ) 
                 
               
             
           
         
       
     
     Gas phase governing equation: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∇ 
                         
                           · 
                           
                             [ 
                             
                               
                                 1 
                                 
                                   B 
                                   g 
                                 
                               
                                
                               
                                 λ 
                                 g 
                               
                                
                               
                                 ∇ 
                                 
                                   p 
                                   g 
                                 
                               
                             
                             ] 
                           
                         
                       
                       + 
                       
                         ∇ 
                         
                           · 
                           
                             [ 
                             
                               
                                 
                                   R 
                                   s 
                                 
                                 
                                   B 
                                   o 
                                 
                               
                                
                               
                                 λ 
                                 o 
                               
                                
                               
                                 ∇ 
                                 
                                   p 
                                   o 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           ( 
                           
                             
                               1 
                               
                                 B 
                                 g 
                               
                             
                              
                             
                               
                                 φ 
                                  
                                 S 
                               
                               g 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 R 
                                 s 
                               
                               
                                 B 
                                 o 
                               
                             
                              
                             
                               
                                 φ 
                                  
                                 S 
                               
                               o 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   81 
                   ) 
                 
               
             
           
         
       
     
     Auxiliary equations with saturation and capillary pressure: 
         S   o   +S   w   +S   g =1,( x,t )∈Ω×(0, t   max ]  (82)
 
         p   cow ( S   w )= p   o   −p   w ( x,t )∈Ω×(0, t   max ]  (83)
 
         p   cgo ( S   g )= p   g   −p   o ( x,t )∈Ω×(0, t   max ]  (84)
 
     Boundary condition equations: 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               o 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             o 
                           
                         
                         + 
                         
                           
                             c 
                             
                               o 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             o 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 o 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         o 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   85 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               w 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             w 
                           
                         
                         + 
                         
                           
                             c 
                             
                               w 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             w 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 w 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         w 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   86 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               g 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             g 
                           
                         
                         + 
                         
                           
                             c 
                             
                               g 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             g 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 g 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         g 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   87 
                   ) 
                 
               
             
           
         
       
     
     Initial condition equations: 
         S   k ( x, 0)= l   k ( x ), x∈Ω   (88)
 
         p   k ( x, 0)=ƒ k ( x ), x∈Ω   (89)
 
       Where 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         S 
                         k 
                       
                       = 
                       
                         
                           S 
                           k 
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                     , 
                     
                       k 
                       = 
                       o 
                     
                     , 
                     w 
                     , 
                     g 
                   
                 
               
               
                 
                   ( 
                   90 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         p 
                         k 
                       
                       = 
                       
                         
                           p 
                           k 
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                     , 
                     
                       k 
                       = 
                       o 
                     
                     , 
                     w 
                     , 
                     g 
                   
                 
               
               
                 
                   ( 
                   91 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         q 
                         k 
                       
                       = 
                       
                         
                           q 
                           k 
                         
                          
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     , 
                     
                       k 
                       = 
                       o 
                     
                     , 
                     w 
                     , 
                     g 
                   
                 
               
               
                 
                   ( 
                   92 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       λ 
                       k 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 K 
                                 x 
                               
                                
                               
                                 
                                   K 
                                   
                                     Γ 
                                      
                                     k 
                                   
                                 
                                 
                                   μ 
                                   k 
                                 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                 K 
                                 y 
                               
                                
                               
                                 
                                   K 
                                   
                                     Γ 
                                      
                                     k 
                                   
                                 
                                 
                                   μ 
                                   k 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               
                                 K 
                                 z 
                               
                                
                               
                                 
                                   K 
                                   
                                     Γ 
                                      
                                     k 
                                   
                                 
                                 
                                   μ 
                                   k 
                                 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     
                                       ρ 
                                       k 
                                     
                                      
                                     g 
                                      
                                     
                                       
                                         ∂ 
                                         z 
                                       
                                       
                                         ∂ 
                                         p 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     k 
                     = 
                     o 
                   
                   , 
                   w 
                   , 
                   
                     g 
                     ; 
                     
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                       ∈ 
                       
                         Ω 
                         × 
                         
                           ( 
                           
                             0 
                             , 
                             
                               t 
                               max 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   93 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         c 
                         
                           k 
                           , 
                           1 
                         
                       
                       = 
                       0 
                     
                     , 
                     
                       k 
                       = 
                       o 
                     
                     , 
                     w 
                     , 
                     
                       g 
                       ; 
                       
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                         ∈ 
                         
                           
                             ∂ 
                             Ω 
                           
                           × 
                           
                             ( 
                             
                               0 
                               , 
                               
                                 t 
                                 max 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   94 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       c 
                       
                         k 
                         , 
                         2 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   
                                     - 
                                     2 
                                   
                                    
                                   
                                     
                                       π 
                                        
                                       r 
                                     
                                     w 
                                   
                                    
                                   h 
                                 
                                 , 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                                 ∈ 
                                 
                                   
                                     Γ 
                                     1 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       0 
                                       , 
                                       
                                         t 
                                         max 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 1 
                                 , 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                                 ∈ 
                                 
                                   
                                     Γ 
                                     2 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       0 
                                       , 
                                       
                                         t 
                                         max 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                         , 
                         
                           k 
                           = 
                           o 
                         
                         , 
                         w 
                         , 
                         g 
                       
                     
                   
                 
               
               
                 
                   ( 
                   95 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         g 
                         k 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   
                                     q 
                                     k 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 , 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                                 ∈ 
                                 
                                   
                                     Γ 
                                     1 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       0 
                                       , 
                                       
                                         t 
                                         max 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 0 
                                 , 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                                 ∈ 
                                 
                                   
                                     Γ 
                                     2 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       0 
                                       , 
                                       
                                         t 
                                         max 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                         , 
                         
                           k 
                           = 
                           o 
                         
                         , 
                         w 
                         , 
                         g 
                       
                     
                   
                 
               
               
                 
                   ( 
                   96 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       ∂ 
                       Ω 
                     
                     = 
                     
                       
                         Γ 
                         1 
                       
                        
                       
                         
                           U 
                            
                           Γ 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   97 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Γ 
                       1 
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                             | 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                           
                           = 
                           
                             r 
                             w 
                             2 
                           
                         
                         , 
                         
                           
                             h 
                             1 
                           
                           ≤ 
                           z 
                           ≤ 
                           
                             h 
                             2 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   98 
                   ) 
                 
               
             
             
               
                 
                   
                     Γ 
                     2 
                   
                   = 
                   
                     
                       { 
                       
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                             | 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                           
                           = 
                           
                             r 
                             w 
                             2 
                           
                         
                         , 
                         
                           0 
                           ≤ 
                           z 
                           &lt; 
                           
                             h 
                             1 
                           
                         
                         , 
                         
                           
                             h 
                             2 
                           
                           &lt; 
                           z 
                           ≤ 
                           H 
                         
                       
                       } 
                     
                      
                     U 
                      
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             &lt; 
                             
                               r 
                               w 
                               2 
                             
                           
                         
                         , 
                         
                           z 
                           = 
                           
                             
                               h 
                               1 
                             
                              
                             
                                 
                             
                              
                             or 
                              
                             
                                 
                             
                              
                             
                               h 
                               2 
                             
                           
                         
                       
                       } 
                     
                      
                     U 
                      
                     
                       { 
                       
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                             | 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                           
                           = 
                           
                             r 
                             e 
                             2 
                           
                         
                         , 
                         
                           0 
                           ≤ 
                           z 
                           ≤ 
                           H 
                         
                       
                       } 
                     
                      
                     U 
                      
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               r 
                               w 
                               2 
                             
                             &lt; 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             &lt; 
                             
                               r 
                               e 
                               2 
                             
                           
                         
                         , 
                         
                           z 
                           = 
                           
                             0 
                              
                             
                                 
                             
                              
                             or 
                              
                             
                                 
                             
                              
                             H 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   99 
                   ) 
                 
               
             
           
         
       
     
     Embodiment 3 
     The embodiment provides oil/gas/water three-phase flow simulation method through treating the wellbore as reservoir for a partially open vertical well in a homogeneous reservoir. The physical model is shown in  FIG. 11 . 
     The general equation of oil/gas/water three-phase flow can be established by applying the mass conservation principle: 
     Oil phase governing equation: 
     
       
         
           
             
               
                 
                   
                     
                       ∇ 
                       
                         · 
                         
                           [ 
                           
                             
                               1 
                               
                                 B 
                                 o 
                               
                             
                              
                             
                               λ 
                               o 
                             
                              
                             
                               ∇ 
                               
                                 p 
                                 o 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         ∂ 
                         
                           ∂ 
                           t 
                         
                       
                        
                       
                         ( 
                         
                           
                             1 
                             
                               B 
                               o 
                             
                           
                            
                           
                             
                               φ 
                                
                               S 
                             
                             o 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   100 
                   ) 
                 
               
             
           
         
       
     
     Water phase governing equation: 
     
       
         
           
             
               
                 
                   
                     
                       ∇ 
                       
                         · 
                         
                           [ 
                           
                             
                               1 
                               
                                 B 
                                 w 
                               
                             
                              
                             
                               λ 
                               w 
                             
                              
                             
                               ∇ 
                               
                                 p 
                                 w 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         ∂ 
                         
                           ∂ 
                           t 
                         
                       
                        
                       
                         ( 
                         
                           
                             1 
                             
                               B 
                               w 
                             
                           
                            
                           
                             
                               φ 
                                
                               S 
                             
                             w 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   101 
                   ) 
                 
               
             
           
         
       
     
     Gas phase governing equation: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∇ 
                         
                           · 
                           
                             [ 
                             
                               
                                 1 
                                 
                                   B 
                                   g 
                                 
                               
                                
                               
                                 λ 
                                 g 
                               
                                
                               
                                 ∇ 
                                 
                                   p 
                                   g 
                                 
                               
                             
                             ] 
                           
                         
                       
                       + 
                       
                         ∇ 
                         
                           · 
                           
                             [ 
                             
                               
                                 
                                   R 
                                   s 
                                 
                                 
                                   B 
                                   o 
                                 
                               
                                
                               
                                 λ 
                                 o 
                               
                                
                               
                                 ∇ 
                                 
                                   p 
                                   o 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           ( 
                           
                             
                               1 
                               
                                 B 
                                 g 
                               
                             
                              
                             
                               
                                 φ 
                                  
                                 S 
                               
                               g 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 R 
                                 s 
                               
                               
                                 B 
                                 o 
                               
                             
                              
                             
                               
                                 φ 
                                  
                                 S 
                               
                               o 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   102 
                   ) 
                 
               
             
           
         
       
     
     Auxiliary equations with saturation and capillary pressure: 
         S   o   +S   w   +S   g =1,( x,t )∈Ω×(0, t   max )  (103)
 
         p   cow ( S   w )= p   o   −p   w ( x,t )∈Ω×(0, t   max )  (104)
 
         p   cgo ( S   g )= p   g - p   o ,( x,t )∈Ω×(0, t   max ]  (105)
 
     Boundary condition equations: 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               o 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             o 
                           
                         
                         + 
                         
                           
                             c 
                             
                               o 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             o 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 o 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         o 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   106 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               w 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             w 
                           
                         
                         + 
                         
                           
                             c 
                             
                               w 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             w 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 w 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         w 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   107 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               g 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             g 
                           
                         
                         + 
                         
                           
                             c 
                             
                               g 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             g 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 g 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         g 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   108 
                   ) 
                 
               
             
           
         
       
     
     Initial condition equations: 
         S   k ( x, 0)= l   k ( x ), x∈Ω   (109)
 
         p   k ( x, 0)=ƒ k ( x ), x∈Ω   (110)
 
       Where 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         S 
                         k 
                       
                       = 
                       
                         
                           S 
                           k 
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                     , 
                     
                       k 
                       = 
                       o 
                     
                     , 
                     w 
                     , 
                     g 
                   
                 
               
               
                 
                   ( 
                   111 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         p 
                         k 
                       
                       = 
                       
                         
                           p 
                           k 
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                     , 
                     
                       k 
                       = 
                       o 
                     
                     , 
                     w 
                     , 
                     g 
                   
                 
               
               
                 
                   ( 
                   112 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         q 
                         k 
                       
                       = 
                       
                         
                           q 
                           k 
                         
                          
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     , 
                     
                       k 
                       = 
                       o 
                     
                     , 
                     w 
                     , 
                     g 
                   
                 
               
               
                 
                   ( 
                   113 
                   ) 
                 
               
             
             
               
                 
                   
                     λ 
                     k 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           K 
                                           
                                             x 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             K 
                                             
                                               Γ 
                                                
                                               k 
                                             
                                           
                                           
                                             μ 
                                             k 
                                           
                                         
                                       
                                     
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       
                                         
                                           K 
                                           
                                             y 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             K 
                                             
                                               Γ 
                                                
                                               k 
                                             
                                           
                                           
                                             μ 
                                             k 
                                           
                                         
                                       
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                     
                                       
                                         
                                           K 
                                           
                                             z 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             K 
                                             
                                               Γ 
                                                
                                               k 
                                             
                                           
                                           
                                             μ 
                                             k 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             1 
                                             - 
                                             
                                               
                                                 ρ 
                                                 k 
                                               
                                                
                                               g 
                                                
                                               
                                                 
                                                   ∂ 
                                                   z 
                                                 
                                                 
                                                   ∂ 
                                                   p 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                                 ∈ 
                                 
                                   
                                     Ω 
                                     1 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       0 
                                       , 
                                       
                                         t 
                                         max 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           K 
                                           
                                             x 
                                             2 
                                           
                                         
                                          
                                         
                                           
                                             K 
                                             
                                               Γ 
                                                
                                               k 
                                             
                                           
                                           
                                             μ 
                                             k 
                                           
                                         
                                       
                                     
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       
                                         
                                           K 
                                           
                                             y 
                                             2 
                                           
                                         
                                          
                                         
                                           
                                             K 
                                             
                                               Γ 
                                                
                                               k 
                                             
                                           
                                           
                                             μ 
                                             k 
                                           
                                         
                                       
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                     
                                       
                                         
                                           K 
                                           
                                             z 
                                             2 
                                           
                                         
                                          
                                         
                                           
                                             K 
                                             
                                               Γ 
                                                
                                               k 
                                             
                                           
                                           
                                             μ 
                                             k 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             1 
                                             - 
                                             
                                               
                                                 ρ 
                                                 k 
                                               
                                                
                                               g 
                                                
                                               
                                                 
                                                   ∂ 
                                                   z 
                                                 
                                                 
                                                   ∂ 
                                                   p 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                                 ∈ 
                                 
                                   
                                     Ω 
                                     2 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       0 
                                       , 
                                       
                                         t 
                                         max 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                       
                       , 
                       
                         k 
                         = 
                         o 
                       
                       , 
                       w 
                       , 
                       g 
                     
                   
                 
               
               
                 
                   ( 
                   114 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         c 
                         
                           k 
                           , 
                           1 
                         
                       
                       = 
                       0 
                     
                     , 
                     
                       k 
                       = 
                       o 
                     
                     , 
                     w 
                     , 
                     
                       g 
                       ; 
                       
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                         ∈ 
                         
                           
                             ∂ 
                             Ω 
                           
                           × 
                           
                             ( 
                             
                               0 
                               , 
                               
                                 t 
                                 max 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   115 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       c 
                       
                         k 
                         , 
                         2 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   
                                     - 
                                     
                                       
                                         π 
                                          
                                         r 
                                       
                                       w 
                                     
                                   
                                    
                                   h 
                                 
                                 , 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                                 ∈ 
                                 
                                   
                                     Γ 
                                     1 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       0 
                                       , 
                                       
                                         t 
                                         max 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 1 
                                 , 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                                 ∈ 
                                 
                                   
                                     Γ 
                                     2 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       0 
                                       , 
                                       
                                         t 
                                         max 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                         , 
                         
                           k 
                           = 
                           o 
                         
                         , 
                         w 
                         , 
                         g 
                       
                     
                   
                 
               
               
                 
                   ( 
                   116 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         g 
                         k 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   
                                     q 
                                     k 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 , 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                                 ∈ 
                                 
                                   
                                     Γ 
                                     1 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       0 
                                       , 
                                       
                                         t 
                                         max 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 0 
                                 , 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     x 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                                 ∈ 
                                 
                                   
                                     Γ 
                                     2 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       0 
                                       , 
                                       
                                         t 
                                         max 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                         , 
                         
                           k 
                           = 
                           o 
                         
                         , 
                         w 
                         , 
                         g 
                       
                     
                   
                 
               
               
                 
                   ( 
                   117 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     Ω 
                     = 
                     
                       
                         Ω 
                         1 
                       
                        
                       
                         
                           U 
                            
                           Ω 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   118 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Ω 
                       1 
                     
                     = 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             ≤ 
                             
                               r 
                               w 
                               2 
                             
                           
                         
                         , 
                         
                           0 
                           &lt; 
                           z 
                           &lt; 
                           
                             h 
                             2 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   119 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Ω 
                       2 
                     
                     = 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               r 
                               w 
                               2 
                             
                             &lt; 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             &lt; 
                             
                               r 
                               e 
                               2 
                             
                           
                         
                         , 
                         
                           
                             H 
                             1 
                           
                           &lt; 
                           z 
                           &lt; 
                           H 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   120 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       ∂ 
                       Ω 
                     
                     = 
                     
                       
                         Γ 
                         1 
                       
                        
                       
                         
                           U 
                            
                           Γ 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   121 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Γ 
                       1 
                     
                     = 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             ≤ 
                             
                               r 
                               w 
                               2 
                             
                           
                         
                         , 
                         
                           z 
                           = 
                           0 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   122 
                   ) 
                 
               
             
             
               
                 
                   
                     Γ 
                     2 
                   
                   = 
                   
                     
                       { 
                       
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                             | 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                           
                           = 
                           
                             r 
                             w 
                             2 
                           
                         
                         , 
                         
                           0 
                           &lt; 
                           z 
                           &lt; 
                           
                             h 
                             1 
                           
                         
                         , 
                         
                           
                             h 
                             2 
                           
                           &lt; 
                           z 
                           ≤ 
                           H 
                         
                       
                       } 
                     
                      
                     U 
                      
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             ≤ 
                             
                               r 
                               w 
                               2 
                             
                           
                         
                         , 
                         
                           z 
                           = 
                           
                             h 
                             1 
                           
                         
                       
                       } 
                     
                      
                     U 
                      
                     
                       { 
                       
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                             | 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                           
                           = 
                           
                             r 
                             e 
                             2 
                           
                         
                         , 
                         
                           
                             H 
                             1 
                           
                           ≤ 
                           z 
                           ≤ 
                           H 
                         
                       
                       } 
                     
                      
                     U 
                      
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               r 
                               w 
                               2 
                             
                             &lt; 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             &lt; 
                             
                               r 
                               e 
                               2 
                             
                           
                         
                         , 
                         
                           z 
                           = 
                           
                             
                               H 
                               1 
                             
                              
                             
                                 
                             
                              
                             or 
                              
                             
                                 
                             
                              
                             H 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   123 
                   ) 
                 
               
             
           
         
       
     
     Embodiment 4 
     This embodiment provides oil phase flow simulation method through treating the wellbore as inner boundary for a partially open vertical well in a homogeneous reservoir. The physical model is shown in  FIG. 12 . 
     The general equation of oil phase flow can be established by applying the mass conservation principle: 
     Oil phase governing equation: 
     
       
         
           
             
               
                 
                   
                     
                       ∇ 
                       
                         · 
                         
                           [ 
                           
                             
                               1 
                               
                                 B 
                                 o 
                               
                             
                              
                             
                               λ 
                               o 
                             
                              
                             
                               ∇ 
                               
                                 p 
                                 o 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         ∂ 
                         
                           ∂ 
                           t 
                         
                       
                        
                       
                         ( 
                         
                           
                             1 
                             
                               B 
                               o 
                             
                           
                            
                           
                             
                               φ 
                                
                               S 
                             
                             o 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   124 
                   ) 
                 
               
             
           
         
       
     
     Boundary condition equation: 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               o 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             o 
                           
                         
                         + 
                         
                           
                             c 
                             
                               o 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             o 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 o 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         o 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   125 
                   ) 
                 
               
             
           
         
       
     
     Initial condition equation: 
         p   o ( x, 0)=ƒ o ( x ), x∈Ω   (126)
 
       Where 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       p 
                       o 
                     
                     = 
                     
                       
                         p 
                         o 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   127 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       q 
                       o 
                     
                     = 
                     
                       
                         q 
                         o 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   128 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         λ 
                         o 
                       
                       = 
                       
                         
                           1 
                           μ 
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   K 
                                   Γ 
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   K 
                                   Γ 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   K 
                                   z 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                     , 
                     
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                       ∈ 
                       
                         Ω 
                         × 
                         
                           ( 
                           
                             0 
                             , 
                             
                               t 
                               max 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   129 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         c 
                         
                           o 
                           , 
                           1 
                         
                       
                       = 
                       0 
                     
                     , 
                     
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                       ∈ 
                       
                         
                           ∂ 
                           Ω 
                         
                         × 
                         
                           ( 
                           
                             0 
                             , 
                             
                               t 
                               max 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   130 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       c 
                       
                         o 
                         , 
                         2 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   - 
                                   2 
                                 
                                  
                                 
                                   
                                     π 
                                      
                                     r 
                                   
                                   w 
                                 
                                  
                                 h 
                               
                               , 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Γ 
                                   1 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         
                           
                             
                               1 
                               , 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Γ 
                                   2 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   131 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         g 
                         o 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 q 
                                 o 
                               
                               , 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Γ 
                                   1 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         
                           
                             
                               0 
                               , 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Γ 
                                   2 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   132 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       ∂ 
                       Ω 
                     
                     = 
                     
                       
                         Γ 
                         1 
                       
                        
                       
                         
                           U 
                            
                           Γ 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   133 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Γ 
                       1 
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                             | 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                           
                           = 
                           
                             r 
                             w 
                             2 
                           
                         
                         , 
                         
                           
                             h 
                             1 
                           
                           ≤ 
                           z 
                           ≤ 
                           
                             h 
                             2 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   134 
                   ) 
                 
               
             
             
               
                 
                   
                     Γ 
                     2 
                   
                   = 
                   
                     
                       { 
                       
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                             | 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                           
                           = 
                           
                             r 
                             w 
                             2 
                           
                         
                         , 
                         
                           0 
                           ≤ 
                           z 
                           &lt; 
                           
                             h 
                             1 
                           
                         
                         , 
                         
                           
                             h 
                             2 
                           
                           &lt; 
                           z 
                           ≤ 
                           H 
                         
                       
                       } 
                     
                      
                     U 
                      
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             &lt; 
                             
                               r 
                               w 
                               2 
                             
                           
                         
                         , 
                         
                           z 
                           = 
                           
                             
                               h 
                               1 
                             
                              
                             
                                 
                             
                              
                             or 
                              
                             
                                 
                             
                              
                             
                               h 
                               2 
                             
                           
                         
                       
                       } 
                     
                      
                     U 
                      
                     
                       { 
                       
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                             | 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                           
                           = 
                           
                             r 
                             e 
                             2 
                           
                         
                         , 
                         
                           0 
                           ≤ 
                           z 
                           ≤ 
                           H 
                         
                       
                       } 
                     
                      
                     U 
                      
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               r 
                               w 
                               2 
                             
                             &lt; 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             &lt; 
                             
                               r 
                               e 
                               2 
                             
                           
                         
                         , 
                         
                           z 
                           = 
                           
                             0 
                              
                             
                                 
                             
                              
                             or 
                              
                             
                                 
                             
                              
                             H 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   135 
                   ) 
                 
               
             
           
         
       
     
     Embodiment 5 
     This embodiment provides oil phase flow simulation method through treating the wellbore as reservoir for a partially open vertical well in a homogeneous reservoir. The physical model is shown in  FIG. 13 . 
     The general equation of oil phase flow can be established by applying the mass conservation principle:
         Oil phase governing equation:       

     
       
         
           
             
               
                 
                   
                     
                       ∇ 
                       
                         · 
                         
                           [ 
                           
                             
                               1 
                               
                                 B 
                                 o 
                               
                             
                              
                             
                               λ 
                               o 
                             
                              
                             
                               ∇ 
                               
                                 p 
                                 o 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         ∂ 
                         
                           ∂ 
                           t 
                         
                       
                        
                       
                         ( 
                         
                           
                             1 
                             
                               B 
                               o 
                             
                           
                            
                           φ 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   136 
                   ) 
                 
               
             
           
         
       
         
         
           
             Boundary condition Equation: 
           
         
       
    
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               o 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             o 
                           
                         
                         + 
                         
                           
                             c 
                             
                               o 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             o 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 o 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       g 
                       o 
                     
                   
                   , 
                   
                     ( 
                     
                       x 
                       , 
                       t 
                     
                     ) 
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   137 
                   ) 
                 
               
             
           
         
       
         
         
           
             Initial condition equation: 
           
         
       
    
         p   o ( x, 0)=ƒ o ( x ), x∈Ω   (138)
 
       where 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       p 
                       o 
                     
                     = 
                     
                       
                         p 
                         o 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   139 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       q 
                       o 
                     
                     = 
                     
                       
                         q 
                         o 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   140 
                   ) 
                 
               
             
             
               
                 
                   
                     λ 
                     o 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 1 
                                 μ 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                           
                                       
                                     
                                     
                                       
                                           
                                       
                                     
                                     
                                       
                                           
                                       
                                     
                                     
                                       
                                           
                                       
                                     
                                   
                                   
                                     
                                       
                                         K 
                                         
                                           r 
                                           1 
                                         
                                       
                                     
                                     
                                       0 
                                     
                                     
                                       
                                           
                                       
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       
                                         K 
                                         
                                           r 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                           
                                       
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                     
                                       
                                         K 
                                         
                                           z 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           
                                             
                                               ρ 
                                               o 
                                             
                                              
                                             g 
                                              
                                             
                                               
                                                 ∂ 
                                                 z 
                                               
                                               
                                                 ∂ 
                                                 p 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 t 
                               
                               ) 
                             
                             ∈ 
                             
                               
                                 Ω 
                                 1 
                               
                               × 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   
                                     t 
                                     max 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 1 
                                 μ 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         K 
                                         
                                           r 
                                           2 
                                         
                                       
                                     
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       
                                         K 
                                         
                                           r 
                                           2 
                                         
                                       
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                     
                                       
                                         K 
                                         
                                           z 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 t 
                               
                               ) 
                             
                             ∈ 
                             
                               
                                 Ω 
                                 2 
                               
                               × 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   
                                     t 
                                     max 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   141 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         c 
                         
                           o 
                           , 
                           1 
                         
                       
                       ≡ 
                       0 
                     
                     , 
                     
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                       ∈ 
                       
                         Ω 
                         × 
                         
                           ( 
                           
                             0 
                             , 
                             
                               t 
                               max 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   142 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       c 
                       
                         o 
                         , 
                         2 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   - 
                                   π 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   r 
                                   w 
                                   2 
                                 
                               
                               , 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Γ 
                                   1 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       
                                         ma 
                                          
                                         x 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         
                           
                             
                               1 
                               , 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Γ 
                                   2 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   143 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         g 
                         o 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               q 
                               , 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Γ 
                                   1 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         
                           
                             
                               0 
                               , 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Γ 
                                   2 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   144 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     Ω 
                     = 
                     
                       
                         Ω 
                         1 
                       
                        
                       U 
                        
                       
                           
                       
                        
                       
                         Ω 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   145 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Ω 
                       1 
                     
                     = 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             ≤ 
                             
                               r 
                               w 
                               2 
                             
                           
                         
                         , 
                         
                           0 
                           &lt; 
                           z 
                           &lt; 
                           H 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   146 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Ω 
                       2 
                     
                     = 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               r 
                               w 
                               2 
                             
                             &lt; 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             &lt; 
                             
                               r 
                               e 
                               2 
                             
                           
                         
                         , 
                         
                           
                             H 
                             1 
                           
                           &lt; 
                           z 
                           &lt; 
                           H 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   147 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       ∂ 
                       Ω 
                     
                     = 
                     
                       
                         Γ 
                         1 
                       
                        
                       U 
                        
                       
                           
                       
                        
                       
                         Γ 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   148 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Γ 
                       1 
                     
                     = 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             ≤ 
                             
                               r 
                               w 
                               2 
                             
                           
                         
                         , 
                         
                           z 
                           = 
                           0 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   149 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         Γ 
                         2 
                       
                       = 
                       
                         { 
                         
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                 
                                 ) 
                               
                               | 
                               
                                 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   y 
                                   2 
                                 
                               
                             
                             = 
                             
                               r 
                               w 
                               2 
                             
                           
                           , 
                           
                             0 
                             &lt; 
                             z 
                             &lt; 
                             
                               h 
                               1 
                             
                           
                           , 
                           
                             
                               h 
                               2 
                             
                             &lt; 
                             z 
                             ≤ 
                             H 
                           
                         
                         } 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       U 
                        
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                             | 
                             
                               
                                 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   y 
                                   2 
                                 
                               
                               ≤ 
                               
                                 r 
                                 w 
                                 2 
                               
                             
                           
                           , 
                           
                             z 
                             = 
                             
                               h 
                               2 
                             
                           
                         
                         } 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       U 
                        
                       
                         { 
                         
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                 
                                 ) 
                               
                               | 
                               
                                 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   y 
                                   2 
                                 
                               
                             
                             = 
                             
                               r 
                               e 
                               2 
                             
                           
                           , 
                           
                             
                               H 
                               1 
                             
                             ≤ 
                             z 
                             ≤ 
                             H 
                           
                         
                         } 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       U 
                        
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                             | 
                             
                               
                                 r 
                                 w 
                                 2 
                               
                               &lt; 
                               
                                 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   y 
                                   2 
                                 
                               
                               &lt; 
                               
                                 r 
                                 e 
                                 2 
                               
                             
                           
                           , 
                           
                             z 
                             = 
                             
                               
                                 H 
                                 1 
                               
                                
                               
                                   
                               
                                
                               or 
                                
                               
                                   
                               
                                
                               H 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   150 
                   ) 
                 
               
             
           
         
       
     
     Embodiment 6 
     This embodiment provides oil phase flow simulation method through treating the wellbore as inner boundary for a fully open vertical well in a homogeneous radial 2-zone composite reservoir. The physical model is shown in  FIG. 14 . 
     The general equation of oil phase flow can be established by applying the mass conservation principle:
         Oil phase governing equation:       

     
       
         
           
             
               
                 
                   
                     
                       ∇ 
                       
                         · 
                         
                           [ 
                           
                             
                               1 
                               
                                 B 
                                 o 
                               
                             
                              
                             
                               λ 
                               o 
                             
                              
                             
                               ∇ 
                               
                                 p 
                                 o 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         ∂ 
                         
                           ∂ 
                           t 
                         
                       
                        
                       
                         ( 
                         
                           
                             1 
                             
                               B 
                               o 
                             
                           
                            
                           
                             φS 
                             o 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   151 
                   ) 
                 
               
             
           
         
       
         
         
           
             Boundary condition equation: 
           
         
       
    
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               o 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             o 
                           
                         
                         + 
                         
                           
                             c 
                             
                               o 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             o 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 o 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       g 
                       o 
                     
                   
                   , 
                   
                     ( 
                     
                       x 
                       , 
                       t 
                     
                     ) 
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   152 
                   ) 
                 
               
             
           
         
       
         
         
           
             Initial condition equation: 
           
         
       
    
         p   o ( x, 0)=ƒ o ( x ), x∈Ω   (153)
 
       Where 
     
       
         
           
             
               
                 
                   
                     p 
                     o 
                   
                   = 
                   
                     
                       p 
                       o 
                     
                      
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   154 
                   ) 
                 
               
             
             
               
                 
                   
                     q 
                     o 
                   
                   = 
                   
                     
                       q 
                       o 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   155 
                   ) 
                 
               
             
             
               
                 
                   
                     λ 
                     o 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 1 
                                 
                                   μ 
                                   1 
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         K 
                                         
                                           r 
                                            
                                           
                                               
                                           
                                            
                                           1 
                                         
                                       
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       
                                         K 
                                         
                                           r 
                                            
                                           
                                               
                                           
                                            
                                           1 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 t 
                               
                               ) 
                             
                             ∈ 
                             
                               
                                 Ω 
                                 1 
                               
                               × 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   
                                     t 
                                     max 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 1 
                                 
                                   μ 
                                   2 
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         K 
                                         
                                           r 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                       
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       
                                         K 
                                         
                                           r 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 t 
                               
                               ) 
                             
                             ∈ 
                             
                               
                                 Ω 
                                 2 
                               
                               × 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   
                                     t 
                                     max 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   156 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       c 
                       
                         o 
                         , 
                         1 
                       
                     
                     ≡ 
                     0 
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   157 
                   ) 
                 
               
             
             
               
                 
                   
                     c 
                     
                       o 
                       , 
                       2 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 - 
                                 2 
                               
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 r 
                                 w 
                               
                                
                               h 
                             
                             , 
                           
                         
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 t 
                               
                               ) 
                             
                             ∈ 
                             
                               
                                 Γ 
                                 1 
                               
                               × 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   
                                     t 
                                     max 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             1 
                             , 
                           
                         
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 t 
                               
                               ) 
                             
                             ∈ 
                             
                               
                                 Γ 
                                 2 
                               
                               × 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   
                                     t 
                                     max 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   158 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       g 
                       o 
                     
                      
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             q 
                             , 
                           
                         
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 t 
                               
                               ) 
                             
                             ∈ 
                             
                               
                                 Γ 
                                 1 
                               
                               × 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   
                                     t 
                                     max 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       
                         
                           
                             0 
                             , 
                           
                         
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 t 
                               
                               ) 
                             
                             ∈ 
                             
                               
                                 Γ 
                                 2 
                               
                               × 
                               
                                 ( 
                                 
                                   0 
                                   , 
                                   
                                     t 
                                     max 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   159 
                   ) 
                 
               
             
             
               
                 
                   Ω 
                   = 
                   
                     
                       Ω 
                       1 
                     
                      
                     U 
                      
                     
                         
                     
                      
                     
                       Ω 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   160 
                   ) 
                 
               
             
             
               
                 
                   
                     Ω 
                     1 
                   
                   = 
                   
                     { 
                     
                       
                         ( 
                         
                           x 
                           , 
                           y 
                         
                         ) 
                       
                       | 
                       
                         
                           r 
                           w 
                           2 
                         
                         &lt; 
                         
                           
                             x 
                             2 
                           
                           + 
                           
                             y 
                             2 
                           
                         
                         ≤ 
                         
                           r 
                           1 
                           2 
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   161 
                   ) 
                 
               
             
             
               
                 
                   
                     Ω 
                     2 
                   
                   = 
                   
                     { 
                     
                       
                         ( 
                         
                           x 
                           , 
                           y 
                         
                         ) 
                       
                       | 
                       
                         
                           r 
                           1 
                           2 
                         
                         &lt; 
                         
                           
                             x 
                             2 
                           
                           + 
                           
                             y 
                             2 
                           
                         
                         &lt; 
                         
                           r 
                           e 
                           2 
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   162 
                   ) 
                 
               
             
             
               
                 
                   
                     ∂ 
                     Ω 
                   
                   = 
                   
                     
                       Γ 
                       1 
                     
                      
                     U 
                      
                     
                         
                     
                      
                     
                       Γ 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   163 
                   ) 
                 
               
             
             
               
                 
                   
                     Γ 
                     1 
                   
                   = 
                   
                     { 
                     
                       
                         
                           ( 
                           
                             x 
                             , 
                             y 
                           
                           ) 
                         
                         | 
                         
                           
                             x 
                             2 
                           
                           + 
                           
                             y 
                             2 
                           
                         
                       
                       = 
                       
                         r 
                         w 
                         2 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   164 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Γ 
                       2 
                     
                     = 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                             
                             ) 
                           
                           | 
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               y 
                               2 
                             
                           
                         
                         = 
                         
                           r 
                           e 
                           2 
                         
                       
                       } 
                     
                   
                    
                   
                       
                   
                 
               
               
                 
                   ( 
                   165 
                   ) 
                 
               
             
           
         
       
     
     Embodiment 7 
     This embodiment provides oil phase flow simulation method through treating the wellbore as reservoir for a fully open vertical well in a homogeneous radial 2-zone composite reservoir. The physical model is shown in  FIG. 15 . 
     The general equation of oil phase flow can be established by applying the mass conservation principle: 
     Oil phase governing equation: 
     
       
         
           
             
               
                 
                   
                     
                       ∇ 
                       
                         · 
                         
                           [ 
                           
                             
                               1 
                               
                                 B 
                                 o 
                               
                             
                              
                             
                               λ 
                               o 
                             
                              
                             
                               ∇ 
                               
                                 p 
                                 o 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         ∂ 
                         
                           ∂ 
                           t 
                         
                       
                        
                       
                         ( 
                         
                           
                             1 
                             
                               B 
                               o 
                             
                           
                            
                           φ 
                            
                           
                               
                           
                            
                           
                             S 
                             o 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   166 
                   ) 
                 
               
             
           
         
       
     
     Boundary condition equation: 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               o 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             o 
                           
                         
                         + 
                         
                           
                             c 
                             
                               o 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             o 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 o 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       g 
                       o 
                     
                   
                   , 
                   
                     ( 
                     
                       x 
                       , 
                       t 
                     
                     ) 
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   167 
                   ) 
                 
               
             
           
         
       
     
     Initial condition equation: 
         p   o ( x, 0)=ƒ o ( x ), x∈Ω   (168)
 
       Where 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       p 
                       o 
                     
                     = 
                     
                       
                         p 
                         o 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   169 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       q 
                       o 
                     
                     = 
                     
                       
                         q 
                         o 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   170 
                   ) 
                 
               
             
             
               
                 
                   
                     λ 
                     o 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               1 
                               μ 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                         
                                     
                                   
                                   
                                     
                                         
                                     
                                   
                                   
                                     
                                         
                                     
                                   
                                   
                                     
                                         
                                     
                                   
                                 
                                 
                                   
                                     
                                       K 
                                       
                                         r 
                                         1 
                                       
                                     
                                   
                                   
                                     0 
                                   
                                   
                                     
                                         
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                       K 
                                       
                                         r 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                         
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     
                                       K 
                                       
                                         z 
                                         1 
                                       
                                     
                                   
                                   
                                     
                                       ( 
                                       
                                         1 
                                         - 
                                         
                                           
                                             ρ 
                                             o 
                                           
                                            
                                           g 
                                            
                                           
                                             
                                               ∂ 
                                               z 
                                             
                                             
                                               ∂ 
                                               p 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             , 
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Ω 
                                   1 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               1 
                               μ 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       K 
                                       
                                         r 
                                         2 
                                       
                                     
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                       K 
                                       
                                         r 
                                         2 
                                       
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     
                                       K 
                                       
                                         z 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             , 
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Ω 
                                   2 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               1 
                               μ 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       K 
                                       
                                         r 
                                         3 
                                       
                                     
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                       K 
                                       
                                         r 
                                         3 
                                       
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     
                                       K 
                                       
                                         z 
                                         3 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             , 
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 Ω 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   171 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         c 
                         
                           o 
                           , 
                           1 
                         
                       
                       ≡ 
                       0 
                     
                     , 
                     
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                       ∈ 
                       
                         Ω 
                         × 
                         
                           ( 
                           
                             0 
                             , 
                             
                               t 
                               max 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   172 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       c 
                       
                         o 
                         , 
                         2 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   - 
                                   π 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   r 
                                   w 
                                   2 
                                 
                               
                               , 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Γ 
                                   1 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       
                                         ma 
                                          
                                         x 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         
                           
                             
                               1 
                               , 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Γ 
                                   2 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   173 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         g 
                         o 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               q 
                               , 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Γ 
                                   1 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         
                           
                             
                               0 
                               , 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               ∈ 
                               
                                 
                                   Γ 
                                   2 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     0 
                                     , 
                                     
                                       t 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   174 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     Ω 
                     = 
                     
                       
                         Ω 
                         1 
                       
                        
                       U 
                        
                       
                           
                       
                        
                       
                         Ω 
                         2 
                       
                        
                       U 
                        
                       
                           
                       
                        
                       
                         Ω 
                         3 
                       
                     
                   
                 
               
               
                 
                   ( 
                   175 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Ω 
                       1 
                     
                     = 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             ≤ 
                             
                               r 
                               w 
                               2 
                             
                           
                         
                         , 
                         
                           0 
                           &lt; 
                           z 
                           &lt; 
                           H 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   176 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Ω 
                       2 
                     
                     = 
                     
                       { 
                       
                         
                           ( 
                           
                             x 
                             , 
                             y 
                           
                           ) 
                         
                         | 
                         
                           
                             r 
                             w 
                             2 
                           
                           &lt; 
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               y 
                               2 
                             
                           
                           ≤ 
                           
                             r 
                             1 
                             2 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   177 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Ω 
                       3 
                     
                     = 
                     
                       { 
                       
                         
                           ( 
                           
                             x 
                             , 
                             y 
                           
                           ) 
                         
                         | 
                         
                           
                             r 
                             w 
                             2 
                           
                           &lt; 
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               y 
                               2 
                             
                           
                           ≤ 
                           
                             r 
                             e 
                             2 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   178 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       ∂ 
                       Ω 
                     
                     = 
                     
                       
                         Γ 
                         1 
                       
                        
                       U 
                        
                       
                           
                       
                        
                       
                         Γ 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   179 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       Γ 
                       1 
                     
                     = 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                           | 
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
                                 2 
                               
                             
                             ≤ 
                             
                               r 
                               w 
                               2 
                             
                           
                         
                         , 
                         
                           z 
                           = 
                           0 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   180 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         Γ 
                         2 
                       
                       = 
                       
                         { 
                         
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                 
                                 ) 
                               
                               | 
                               
                                 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   y 
                                   2 
                                 
                               
                             
                             = 
                             
                               r 
                               w 
                               2 
                             
                           
                           , 
                           
                             0 
                             &lt; 
                             z 
                             &lt; 
                             H 
                           
                         
                         } 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       U 
                        
                       
                         { 
                         
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   y 
                                   , 
                                   z 
                                 
                                 ) 
                               
                               | 
                               
                                 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   y 
                                   2 
                                 
                               
                             
                             = 
                             
                               r 
                               e 
                               2 
                             
                           
                           , 
                           
                             
                               H 
                               1 
                             
                             ≤ 
                             z 
                             ≤ 
                             H 
                           
                         
                         } 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       U 
                        
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                             | 
                             
                               
                                 r 
                                 w 
                                 2 
                               
                               &lt; 
                               
                                 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   y 
                                   2 
                                 
                               
                               &lt; 
                               
                                 r 
                                 e 
                                 2 
                               
                             
                           
                           , 
                           
                             z 
                             = 
                             
                               
                                 H 
                                 1 
                               
                                
                               
                                   
                               
                                
                               or 
                                
                               
                                   
                               
                                
                               H 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   181 
                   ) 
                 
               
             
           
         
       
     
     Embodiment 8 
     This embodiment provides flow simulation and transient pressure well testing analysis methods for tube-shaped reservoir. The physical model corresponding to the pressure drawdown well testing model is assumed as follows: 
     The reservoir is composed of single closed tube-shaped reservoir as shown in  FIG. 16 , 
     2) In the initial state, the pressure everywhere in the reservoir is the original reservoir pressure, 
     3) The fluid flow in the reservoir follows linear flow law, 
     4) The fluid and rock in the reservoir are slightly compressible, 
     5) The wellbore storage effect as well as the skin effect are not considered, 
     The dimensionless mathematical model corresponding to the pressure drawdown well test model is written as 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         2 
                       
                        
                       
                         p 
                         D 
                       
                     
                     
                       ∂ 
                       
                         x 
                         D 
                         2 
                       
                     
                   
                   = 
                   
                     
                       ∂ 
                       
                         p 
                         D 
                       
                     
                     
                       ∂ 
                       
                         t 
                         D 
                       
                     
                   
                 
               
               
                 
                   ( 
                   182 
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     D 
                   
                    
                   
                     ( 
                     
                       
                         x 
                         D 
                       
                       , 
                       
                         
                           t 
                           D 
                         
                         = 
                         0 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   183 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           A 
                           D 
                         
                         
                           2 
                            
                           π 
                         
                       
                        
                       
                         
                           ∂ 
                           
                             p 
                             D 
                           
                         
                         
                           ∂ 
                           
                             x 
                             D 
                           
                         
                       
                     
                      
                     
                       | 
                       
                         
                           x 
                           D 
                         
                         = 
                         0 
                       
                     
                   
                   = 
                   
                     - 
                     
                       q 
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   184 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           p 
                           D 
                         
                       
                       
                         ∂ 
                         
                           x 
                           D 
                         
                       
                     
                      
                     
                       | 
                       
                         
                           x 
                           D 
                         
                         = 
                         
                           L 
                           D 
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   185 
                   ) 
                 
               
             
           
         
       
     
     Dimensionless variables are defined as: 
     
       
         
           
             
               
                 
                   
                     t 
                     D 
                   
                   = 
                   
                     
                       λ 
                        
                       
                           
                       
                        
                       t 
                     
                     
                       φ 
                        
                       
                           
                       
                        
                       
                         C 
                         t 
                       
                        
                       
                         l 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   186 
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     D 
                   
                   = 
                   
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         l 
                          
                         
                             
                         
                          
                         λ 
                       
                       
                         
                           q 
                           SC 
                         
                          
                         B 
                       
                     
                      
                     
                       ( 
                       
                         
                           p 
                           i 
                         
                         - 
                         p 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   187 
                   ) 
                 
               
             
             
               
                 
                   
                     L 
                     D 
                   
                   = 
                   
                     L 
                     l 
                   
                 
               
               
                 
                   ( 
                   188 
                   ) 
                 
               
             
             
               
                 
                   
                     x 
                     D 
                   
                   = 
                   
                     x 
                     l 
                   
                 
               
               
                 
                   ( 
                   189 
                   ) 
                 
               
             
             
               
                 
                   
                     q 
                     D 
                   
                   = 
                   
                     q 
                     / 
                     
                       q 
                       SC 
                     
                   
                 
               
               
                 
                   ( 
                   190 
                   ) 
                 
               
             
           
         
       
     
     Laplace transformation of equations Eqs182-185 are performed based on t D . The dimensionless pressure solution in the Laplace space is obtained as: 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       _ 
                     
                     D 
                   
                   = 
                   
                     
                       
                         q 
                         _ 
                       
                       D 
                     
                      
                     
                       
                         2 
                          
                         
                           πcosh 
                            
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     L 
                                     D 
                                   
                                   - 
                                   
                                     x 
                                     D 
                                   
                                 
                                 ) 
                               
                                
                               
                                 u 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           A 
                           D 
                         
                          
                         
                           u 
                         
                          
                         
                           sinh 
                            
                           
                             ( 
                             
                               
                                 L 
                                 D 
                               
                                
                               
                                 u 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   191 
                   ) 
                 
               
             
           
         
       
     
     especially, setting x D =0 as a reference point for bottom hole pressure, i. e., 
           p     wD   = p     D ( x   D =0)  (192)
 
     Typical type curves can be drawn by performing Stehfest numerical inversion as shown in  FIGS. 17-19 . 
     The specific meanings of the identifiers in embodiment 8 are shown as follows: 
     A—open flowing area of tube-shaped reservoir, m 2 ; B—volume factor, m 3 /m 3 ; C t —compressibility of tube-shaped reservoir, 1/Pa; l—reference length, m; L—length of tube-shaped reservoir, m; p—pressure of tube-shaped reservoir, Pa; p i —initial pressure of tube-shaped reservoir·Pa; p w —bottom hole pressure, Pa; q—flow rate, m 3 /s; q sc —reference flow rate, m 3 /s; t—time, s; u—Laplace variable of dimensionless time t D ; x—x-axis distance, m; λ—generalized mobility of tube-shaped reservoir, m 2 /Pa·s; ϕ—porosity of tube-shaped reservoir, %; ∂—partial differential operator. 
     Embodiment 9 
     This embodiment provides flow simulation and transient pressure well testing analysis methods for a cylindrical reservoir. The physical model corresponding to the pressure drawdown well testing model is assumed as follows: 
     The reservoir is a cylindrical closed system and the point source is located on the axis of the cylindrical reservoir, which is shown in  FIG. 20 ; 
     2) In the initial state, the pressure everywhere in the reservoir is the original reservoir pressure, 
     3) The fluid flow in the reservoir follows linear flow law, 
     4) The fluid and rock in the reservoir are slightly compressible, 
     5) The wellbore storage effect as well as the skin effect are not considered. 
     The dimensionless mathematical model corresponding to the pressure drawdown well test model is written as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         1 
                         
                           r 
                           D 
                         
                       
                        
                       
                         
                           ∂ 
                           
                               
                           
                         
                         
                           ∂ 
                           
                             r 
                             D 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             r 
                             D 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 D 
                               
                             
                             
                               ∂ 
                               
                                 r 
                                 D 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         κ 
                         D 
                       
                        
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             p 
                             D 
                           
                         
                         
                           ∂ 
                           
                             z 
                             D 
                             2 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∂ 
                       
                         p 
                         D 
                       
                     
                     
                       ∂ 
                       
                         t 
                         D 
                       
                     
                   
                 
               
               
                 
                   ( 
                   193 
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     D 
                   
                    
                   
                     ( 
                     
                       
                         r 
                         D 
                       
                       , 
                       
                         z 
                         D 
                       
                       , 
                       
                         
                           t 
                           D 
                         
                         = 
                         0 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   194 
                   ) 
                 
               
             
             
               
                 
                   
                     lim 
                     
                       
                         
                           ɛ 
                           D 
                         
                         → 
                         0 
                       
                       
                         
                           r 
                           D 
                         
                         → 
                         0 
                       
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           
                             ɛ 
                             D 
                           
                            
                           
                             r 
                             D 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 D 
                               
                             
                             
                               ∂ 
                               
                                 r 
                                 D 
                               
                             
                           
                         
                         - 
                         
                           { 
                           
                             
                               
                                 
                                   
                                     - 
                                     
                                       q 
                                       D 
                                     
                                   
                                   , 
                                 
                               
                               
                                 
                                   
                                      
                                     
                                       
                                         z 
                                         D 
                                       
                                       - 
                                       
                                         z 
                                         wD 
                                       
                                     
                                      
                                   
                                   ≤ 
                                   
                                     
                                       ɛ 
                                       D 
                                     
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 
                                   0 
                                   , 
                                 
                               
                               
                                 otherwise 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   195 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           p 
                           D 
                         
                       
                       
                         ∂ 
                         
                           z 
                           D 
                         
                       
                     
                      
                     
                       | 
                       
                         
                           z 
                           D 
                         
                         = 
                         0 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   196 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           p 
                           D 
                         
                       
                       
                         ∂ 
                         
                           z 
                           D 
                         
                       
                     
                      
                     
                       | 
                       
                         
                           z 
                           D 
                         
                         = 
                         
                           h 
                           D 
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   197 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           p 
                           D 
                         
                       
                       
                         ∂ 
                         
                           z 
                           D 
                         
                       
                     
                      
                     
                       | 
                       
                         
                           r 
                           D 
                         
                         = 
                         
                           r 
                           eD 
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   198 
                   ) 
                 
               
             
           
         
       
     
     Dimensionless variables are defined as: 
     
       
         
           
             
               
                 
                   
                     p 
                     D 
                   
                   = 
                   
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         l 
                          
                         
                             
                         
                          
                         
                           λ 
                           r 
                         
                       
                       
                         
                           q 
                           SC 
                         
                          
                         B 
                       
                     
                      
                     
                       ( 
                       
                         
                           p 
                           i 
                         
                         - 
                         p 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   199 
                   ) 
                 
               
             
             
               
                 
                   
                     t 
                     D 
                   
                   = 
                   
                     
                       
                         λ 
                         r 
                       
                        
                       t 
                     
                     
                       φ 
                        
                       
                           
                       
                        
                       
                         C 
                         t 
                       
                        
                       
                         l 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   200 
                   ) 
                 
               
             
             
               
                 
                   
                     r 
                     eD 
                   
                   = 
                   
                     
                       r 
                       e 
                     
                     l 
                   
                 
               
               
                 
                   ( 
                   201 
                   ) 
                 
               
             
             
               
                 
                   
                     h 
                     D 
                   
                   = 
                   
                     h 
                     l 
                   
                 
               
               
                 
                   ( 
                   202 
                   ) 
                 
               
             
             
               
                 
                   
                     z 
                     D 
                   
                   = 
                   
                     z 
                     l 
                   
                 
               
               
                 
                   ( 
                   203 
                   ) 
                 
               
             
             
               
                 
                   
                     z 
                     wD 
                   
                   = 
                   
                     
                       z 
                       w 
                     
                     l 
                   
                 
               
               
                 
                   ( 
                   204 
                   ) 
                 
               
             
             
               
                 
                   
                     r 
                     D 
                   
                   = 
                   
                     r 
                     l 
                   
                 
               
               
                 
                   ( 
                   205 
                   ) 
                 
               
             
             
               
                 
                   
                     ɛ 
                     D 
                   
                   = 
                   
                     ɛ 
                     l 
                   
                 
               
               
                 
                   ( 
                   206 
                   ) 
                 
               
             
             
               
                 
                   
                     κ 
                     D 
                   
                   = 
                   
                     
                       λ 
                       z 
                     
                     
                       λ 
                       r 
                     
                   
                 
               
               
                 
                   ( 
                   207 
                   ) 
                 
               
             
             
               
                 
                   
                     q 
                     D 
                   
                   = 
                   
                     q 
                     / 
                     
                       q 
                       SC 
                     
                   
                 
               
               
                 
                   ( 
                   208 
                   ) 
                 
               
             
           
         
       
     
     Laplace transformation of equations Eqs.193 and 198 are performed based on t D . The dimensionless pressure solution in the Laplace space by applying variable separation method is obtained as: 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       _ 
                     
                     D 
                   
                   = 
                   
                     
                       
                         q 
                         _ 
                       
                       D 
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         ∞ 
                       
                        
                       
                         
                           
                             a 
                             n 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   K 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       r 
                                       D 
                                     
                                      
                                     
                                       b 
                                       n 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   
                                     
                                       K 
                                       1 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           r 
                                           eD 
                                         
                                          
                                         
                                           b 
                                           n 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     
                                       I 
                                       1 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           r 
                                           eD 
                                         
                                          
                                         
                                           b 
                                           n 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                  
                                 
                                   
                                     I 
                                     0 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         r 
                                         D 
                                       
                                        
                                       
                                         b 
                                         n 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                          
                         cos 
                          
                         
                           
                             n 
                              
                             
                                 
                             
                              
                             
                               πz 
                               D 
                             
                           
                           
                             h 
                             D 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   209 
                   ) 
                 
               
             
           
         
       
     
     Where 
     
       
         
           
             
               
                 
                   
                     a 
                     n 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             1 
                             , 
                           
                         
                         
                           
                             n 
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               2 
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     z 
                                     wD 
                                   
                                 
                                 
                                   h 
                                   D 
                                 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             2 
                             , 
                             … 
                              
                             
                                 
                             
                             , 
                             ∞ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   210 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       b 
                       n 
                     
                     = 
                     
                       
                         u 
                         + 
                         
                           
                             λ 
                             z 
                           
                           / 
                           
                             
                               
                                 λ 
                                 r 
                               
                                
                               
                                 ( 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   
                                     π 
                                     / 
                                     
                                       h 
                                       D 
                                     
                                   
                                 
                                 ) 
                               
                             
                             2 
                           
                         
                       
                     
                   
                   , 
                   
                     n 
                     = 
                     0 
                   
                   , 
                   1 
                   , 
                   2 
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   ∞ 
                 
               
               
                 
                   ( 
                   211 
                   ) 
                 
               
             
           
         
       
     
     I 0 (g) and I 1 (g) are the category I modified zero-order and first-order Bessel functions respectively, K 0 (g) and K 1 (g) are the category II modified zero-order and first-order Bessel functions respectively. 
     Especially, setting r D =1, z D =z wD  as reference points of bottom hole pressure, i. e., 
           p     wD   = p     D ( r   D =1, z   D   =z   wD )  (212)
 
     Typical type curves can be drawn by performing Stehfest numerical inversion as shown in  FIGS. 21-26 . 
     The specific meanings of the identifiers in embodiment 9 are shown as follows: 
     B—volume factor, m 3 /m; C t —compressibility of cylindrical reservoir, 1/Pa; h—height of cylindrical reservoir, m; l—reference length, m; p—pressure of cylindrical reservoir, Pa; p i —initial pressure of cylindrical reservoir, Pa; p w —bottom hole pressure, Pa; q—flow rate, m 3 /s; q sc —reference flow rate, m 3 /s; r e —radius of cylindrical reservoir, m; t—time, s; u—Laplace variable corresponding to dimensionless time t D ; z—z-axis distance, m; z w —central point position of point source along z-axis, m; ε—height of the point source (tends to zero), m; λ r —radial generalized mobility of cylindrical reservoir, m 2 /Pa·s; λ z —vertical generalized mobility of cylindrical reservoir, m 2 /Pa·s; ϕ—porosity of cylindrical reservoir, %; ∂—partial differential operator. 
     If necessary, the line source solution and cylindrical plane source solutions can be obtained through integrating the point source solution Eq.209. For example,
         Horizontal well (line source):       

     
       
         
           
             
               
                 
                   
                     
                       p 
                       _ 
                     
                     LD 
                   
                   = 
                   
                     
                       
                         
                           q 
                           _ 
                         
                         
                           D 
                            
                           
                               
                           
                         
                       
                       
                         
                           2 
                            
                           
                             L 
                             D 
                           
                         
                          
                         
                             
                         
                       
                     
                      
                     
                       
                         ∫ 
                         
                           - 
                           
                             L 
                             D 
                           
                         
                         
                           L 
                           D 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           ∞ 
                         
                          
                         
                           
                             
                               a 
                               n 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         
                                           K 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         x 
                                                         D 
                                                       
                                                       - 
                                                       
                                                         x 
                                                         wD 
                                                       
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                                 - 
                                                 
                                                   y 
                                                   D 
                                                   2 
                                                 
                                               
                                             
                                              
                                             
                                               b 
                                               n 
                                             
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           
                                             K 
                                             1 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 r 
                                                 eD 
                                               
                                                
                                               
                                                 b 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                         
                                           
                                             I 
                                             1 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 r 
                                                 eD 
                                               
                                                
                                               
                                                 b 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         
                                           I 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         x 
                                                         D 
                                                       
                                                       - 
                                                       
                                                         x 
                                                         wD 
                                                       
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                                 - 
                                                 
                                                   y 
                                                   D 
                                                   2 
                                                 
                                               
                                             
                                              
                                             
                                               b 
                                               n 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               n 
                                
                               
                                   
                               
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 z 
                                 D 
                               
                             
                             
                               h 
                               D 
                             
                           
                            
                           d 
                            
                           
                               
                           
                            
                           
                             x 
                             wD 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   213 
                   ) 
                 
               
             
           
         
       
     
     Where L D  is the dimensionless half length of horizontal well. 
     Partially open hydraulic fractured vertical well (cylindrical plane source): 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       _ 
                     
                     fD 
                   
                   = 
                   
                     
                       
                         
                           q 
                           _ 
                         
                         D 
                       
                       
                         2 
                          
                         
                           
                             x 
                             fD 
                           
                            
                           
                             ( 
                             
                               
                                 z 
                                 bD 
                               
                               - 
                               
                                 z 
                                 aD 
                               
                             
                             ) 
                           
                         
                       
                     
                      
                     
                       
                         ∫ 
                         
                           z 
                           aD 
                         
                         
                           z 
                           bD 
                         
                       
                        
                       
                         
                           ∫ 
                           
                             - 
                             
                               x 
                               fD 
                             
                           
                           
                             x 
                             fD 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             ∞ 
                           
                            
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         
                                           K 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         x 
                                                         D 
                                                       
                                                       - 
                                                       
                                                         x 
                                                         wD 
                                                       
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                                 - 
                                                 
                                                   y 
                                                   D 
                                                   2 
                                                 
                                               
                                             
                                              
                                             
                                               b 
                                               n 
                                             
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           
                                             K 
                                             1 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 r 
                                                 eD 
                                               
                                                
                                               
                                                 b 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                         
                                           
                                             I 
                                             1 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 r 
                                                 eD 
                                               
                                                
                                               
                                                 b 
                                                 n 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         
                                           I 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         x 
                                                         D 
                                                       
                                                       - 
                                                       
                                                         x 
                                                         wD 
                                                       
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                                 - 
                                                 
                                                   y 
                                                   D 
                                                   2 
                                                 
                                               
                                             
                                              
                                             
                                               b 
                                               n 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   z 
                                   D 
                                 
                               
                               
                                 h 
                                 D 
                               
                             
                              
                             
                               dx 
                               wD 
                             
                              
                             
                               dz 
                               wD 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   214 
                   ) 
                 
               
             
           
         
       
     
     Where x fD  is the dimensionless half length of hydraulic fractures, z aD , z bD  are the top and bottom positions of hydraulic fractures. 
     Embodiment 10 
     This embodiment provides flow simulation and transient pressure well testing analysis methods for a spherical reservoir. The physical model corresponding to the pressure drawdown well testing model is assumed as follows: 
     1) The reservoir is a spherical closed system and the point source is located inside the spherical reservoir as shown in  FIG. 27 ,
 
2) In the initial state, the pressure everywhere in the reservoir is the original reservoir pressure,
 
3) The fluid flow in the reservoir follows linear flow law,
 
4) The fluid and rock in the reservoir are slightly compressible,
 
5) The wellbore storage effect as well as the skin effect are not considered,
 
     The dimensionless mathematical model corresponding to the pressure drawdown well test model is written as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         1 
                         
                           r 
                           D 
                           2 
                         
                       
                        
                       
                         ∂ 
                         
                           ∂ 
                           
                             r 
                             D 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             r 
                             D 
                             2 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 D 
                               
                             
                             
                               ∂ 
                               
                                 r 
                                 D 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         1 
                         
                           
                             r 
                             D 
                             2 
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           θ 
                         
                       
                        
                       
                         ∂ 
                         
                           ∂ 
                           θ 
                         
                       
                        
                       
                         ( 
                         
                           sin 
                            
                           
                               
                           
                            
                           θ 
                            
                           
                               
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 D 
                               
                             
                             
                               ∂ 
                               θ 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       ∂ 
                       
                         p 
                         D 
                       
                     
                     
                       ∂ 
                       
                         t 
                         D 
                       
                     
                   
                 
               
               
                 
                   ( 
                   215 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       p 
                       D 
                     
                      
                     
                       ( 
                       
                         
                           r 
                           D 
                         
                         , 
                         θ 
                         , 
                         
                           
                             t 
                             D 
                           
                           = 
                           0 
                         
                       
                       ) 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   216 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       lim 
                       
                         
                           r 
                           D 
                         
                         -&gt; 
                         0 
                       
                     
                      
                     
                       2 
                        
                       
                         r 
                         D 
                         2 
                       
                        
                       
                         
                           ∂ 
                           
                             p 
                             D 
                           
                         
                         
                           ∂ 
                           
                             r 
                             D 
                           
                         
                       
                     
                   
                   = 
                   
                     - 
                     
                       q 
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   217 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         p 
                         D 
                       
                     
                     
                       ∂ 
                       
                         r 
                         D 
                       
                     
                   
                    
                   
                     
                        
                       
                         
                           r 
                           D 
                         
                         = 
                         
                           
                             
                               r 
                               eD 
                               2 
                             
                             + 
                             
                               r 
                               aD 
                               2 
                             
                             - 
                             
                               2 
                                
                               
                                 r 
                                 eD 
                               
                                
                               
                                 r 
                                 aD 
                               
                                
                               co 
                                
                               
                                   
                               
                                
                               s 
                                
                               
                                   
                               
                                
                               Φ 
                             
                           
                         
                       
                     
                      
                     
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   218 
                   ) 
                 
               
             
           
         
       
         
         
           
             Dimensionless variables are defined as: 
           
         
       
    
     
       
         
           
             
               
                 
                   
                     t 
                     D 
                   
                   = 
                   
                     
                       λ 
                        
                       
                           
                       
                        
                       t 
                     
                     
                       φ 
                        
                       
                           
                       
                        
                       
                         C 
                         t 
                       
                        
                       
                         l 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   219 
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     D 
                   
                   = 
                   
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         λ 
                          
                         
                             
                         
                          
                         l 
                       
                       
                         
                           q 
                           sc 
                         
                          
                         B 
                       
                     
                      
                     
                       ( 
                       
                         
                           p 
                           i 
                         
                         - 
                         p 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   220 
                   ) 
                 
               
             
             
               
                 
                   
                     r 
                     aD 
                   
                   = 
                   
                     
                       r 
                       a 
                     
                     l 
                   
                 
               
               
                 
                   ( 
                   221 
                   ) 
                 
               
             
             
               
                 
                   
                     r 
                     bD 
                   
                   = 
                   
                     
                       r 
                       b 
                     
                     l 
                   
                 
               
               
                 
                   ( 
                   222 
                   ) 
                 
               
             
             
               
                 
                   
                     r 
                     eD 
                   
                   = 
                   
                     
                       r 
                       e 
                     
                     l 
                   
                 
               
               
                 
                   ( 
                   223 
                   ) 
                 
               
             
             
               
                 
                   
                     r 
                     D 
                   
                   = 
                   
                     
                       
                         r 
                         bD 
                         2 
                       
                       + 
                       
                         r 
                         aD 
                         2 
                       
                       - 
                       
                         2 
                          
                         
                           r 
                           bD 
                         
                          
                         
                           r 
                           aD 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         Φ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   224 
                   ) 
                 
               
             
             
               
                 
                   θ 
                   = 
                   
                     arcsin 
                      
                     
                         
                     
                      
                     
                       
                         r 
                         D 
                       
                       
                         
                           r 
                           bD 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         Φ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   225 
                   ) 
                 
               
             
             
               
                 
                   
                     q 
                     D 
                   
                   = 
                   
                     q 
                     / 
                     
                       q 
                       sc 
                     
                   
                 
               
               
                 
                   ( 
                   226 
                   ) 
                 
               
             
           
         
       
     
     The dimensionless pressure solution of Eqs.215-218 in the Laplace space by applying Laplace integral transformation and boundary construction method is obtained as: 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       _ 
                     
                     D 
                   
                   = 
                   
                     
                       
                         q 
                         _ 
                       
                       D 
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         ∞ 
                       
                        
                       
                         
                           
                             n 
                             + 
                             
                               1 
                               / 
                               2 
                             
                           
                           
                             
                               
                                 r 
                                 aD 
                               
                                
                               
                                 r 
                                 bD 
                               
                             
                           
                         
                          
                         
                           
                             I 
                             
                               n 
                               + 
                               
                                 1 
                                 2 
                               
                             
                           
                            
                           
                             ( 
                             
                               
                                 r 
                                 aD 
                               
                                
                               
                                 u 
                               
                             
                             ) 
                           
                         
                          
                         
                             
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       K 
                                       
                                         n 
                                         + 
                                         
                                           1 
                                           2 
                                         
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           r 
                                           bD 
                                         
                                          
                                         
                                           u 
                                         
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         
                                           
                                             nK 
                                             
                                               n 
                                               + 
                                               
                                                 1 
                                                 2 
                                               
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 r 
                                                 eD 
                                               
                                                
                                               
                                                 u 
                                               
                                             
                                             ) 
                                           
                                         
                                         - 
                                         
                                           
                                             r 
                                             eD 
                                           
                                            
                                           
                                             u 
                                           
                                            
                                           
                                             
                                               K 
                                               
                                                 n 
                                                 + 
                                                 
                                                   3 
                                                   2 
                                                 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   r 
                                                   eD 
                                                 
                                                  
                                                 
                                                   u 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             nI 
                                             
                                               n 
                                               + 
                                               
                                                 1 
                                                 2 
                                               
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 r 
                                                 eD 
                                               
                                                
                                               
                                                 u 
                                               
                                             
                                             ) 
                                           
                                         
                                         + 
                                         
                                           
                                             r 
                                             eD 
                                           
                                            
                                           
                                             u 
                                           
                                            
                                           
                                             
                                               I 
                                               
                                                 n 
                                                 + 
                                                 
                                                   3 
                                                   2 
                                                 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   r 
                                                   eD 
                                                 
                                                  
                                                 
                                                   u 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                      
                                     
                                       
                                         I 
                                         
                                           n 
                                           + 
                                           
                                             1 
                                             2 
                                           
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             r 
                                             bD 
                                           
                                            
                                           
                                             u 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ] 
                               
                                
                               
                                 
                                   P 
                                   n 
                                 
                                  
                                 
                                   ( 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     Φ 
                                   
                                   ) 
                                 
                               
                             
                             , 
                             
                               
                                 r 
                                 bD 
                               
                               &gt; 
                               
                                 r 
                                 aD 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   227 
                   ) 
                 
               
             
           
         
       
     
     Where 
     
       
         
           
             
               I 
               
                 n 
                 + 
                 
                   1 
                   2 
                 
               
             
              
             
               ( 
               g 
               ) 
             
           
         
       
     
     is the category I modified n+½ order Bessel function, 
     
       
         
           
             
               
                 K 
                 
                   n 
                   + 
                   
                     1 
                     2 
                   
                 
               
                
               
                 ( 
                 g 
                 ) 
               
             
              
             
                 
             
              
             and 
              
             
                 
             
              
             
               
                 K 
                 
                   n 
                   + 
                   
                     3 
                     2 
                   
                 
               
                
               
                 ( 
                 g 
                 ) 
               
             
           
         
       
     
     are the category II modified n+½ order and n+3/2 order Bessel functions respectively. P n  is the category I nth order Legendre function. 
     In particular, setting r aD =0, r bD =1 as the reference points of bottom hole pressure, i. e., 
           p     wD   = p     D ( r   aD =0, r   bD =1)  (228)
 
     Typical type curves can be drawn by performing Stehfest numerical inversion as shown in  FIGS. 28-30 . 
     The specific meanings of the identifiers in embodiment 10 are shown as follows: 
     B—volume factor, m 3 /m 3 ; C t —compressibility of spherical reservoir, 1/Pa; l—reference length, m; p—pressure of spherical reservoir, Pa; p i —initial pressure of spherical reservoir, Pa; p w —bottom hole pressure, Pa; q—flow rate, m 3 /s; q sc —reference flow rate, m 3 /s; r—distance from the point source to the observation point, m; r a —eccentric distance, m; r b —the distance from the center of the spherical reservoir to the observation point, m; r e —radius of spherical reservoir, m; t—time, s; u—Laplace variable of dimensionless time t D ; λ—generalized mobility of spherical reservoir, m 2 /Pa·s; ϕ—porosity of spherical reservoir, %; Φ—angle between r a  and r b ; ∂—partial differential operator. 
     If necessary, the line source solution and spherical plane source solutions can be obtained through integrating the point source solution Eq.227, but in the process of integration, we should pay attention to r bD &gt;r aD . 
     Embodiment 11 
     This embodiment provides a composite flow simulation and transient pressure well testing analysis methods for a compound body of hollow cylinder and cylindrical reservoir. The physical model corresponding to the pressure drawdown well testing model is assumed as follows: 
     1) The reservoir is a composite reservoir of hollow cylinder and cylinder. The cylindrical reservoir is nested in the middle of the hollow cylindrical reservoir, and the point source is located inside the cylindrical reservoir, which is shown in  FIG. 31 ;
 
2) In the initial state, the pressure everywhere in the reservoir is the original reservoir pressure,
 
3) The fluid flow in the reservoir follows linear flow law,
 
4) The fluid and rock in the reservoir are slightly compressible,
 
5) The wellbore storage effect as well as the skin effect are not considered,
 
     The dimensionless mathematical model corresponding to the pressure drawdown well test model is written as: 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             p 
                             
                               1 
                                
                               D 
                             
                           
                         
                         
                           ∂ 
                           
                             r 
                             D 
                             2 
                           
                         
                       
                       + 
                       
                         
                           1 
                           
                             r 
                             D 
                           
                         
                          
                         
                           
                             ∂ 
                             
                               p 
                               
                                 1 
                                  
                                 D 
                               
                             
                           
                           
                             ∂ 
                             
                               r 
                               D 
                             
                           
                         
                       
                       + 
                       
                         
                           1 
                           
                             h 
                             
                               1 
                                
                               D 
                             
                             2 
                           
                         
                          
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               p 
                               
                                 1 
                                  
                                 D 
                               
                             
                           
                           
                             ∂ 
                             
                               z 
                               D 
                               2 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         ∂ 
                         
                           p 
                           
                             1 
                              
                             D 
                           
                         
                       
                       
                         ∂ 
                         
                           t 
                           D 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   229 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               p 
                               
                                 2 
                                  
                                 D 
                               
                             
                           
                           
                             ∂ 
                             
                               r 
                               D 
                               2 
                             
                           
                         
                         + 
                         
                           
                             1 
                             
                               r 
                               D 
                             
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 
                                   2 
                                    
                                   D 
                                 
                               
                             
                             
                               ∂ 
                               
                                 r 
                                 D 
                               
                             
                           
                         
                         + 
                         
                           
                             1 
                             
                               h 
                               
                                 2 
                                  
                                 D 
                               
                               2 
                             
                           
                            
                           
                             
                               
                                 ∂ 
                                 2 
                               
                                
                               
                                 p 
                                 
                                   2 
                                    
                                   D 
                                 
                               
                             
                             
                               ∂ 
                               
                                 z 
                                 D 
                                 2 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           1 
                           σ 
                         
                          
                         
                           
                             ∂ 
                             
                               p 
                               
                                 2 
                                  
                                 D 
                               
                             
                           
                           
                             ∂ 
                             
                               t 
                               D 
                             
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           p 
                           
                             1 
                              
                             D 
                           
                         
                          
                         
                           ( 
                           
                             
                               r 
                               D 
                             
                             , 
                             
                               z 
                               D 
                             
                             , 
                             0 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           
                             p 
                             
                               2 
                                
                               D 
                             
                           
                            
                           
                             ( 
                             
                               
                                 r 
                                 D 
                               
                               , 
                               
                                 z 
                                 D 
                               
                               , 
                               0 
                             
                             ) 
                           
                         
                         = 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   230 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       lim 
                       
                         
                           
                             ɛ 
                             D 
                           
                           -&gt; 
                           0 
                         
                         
                           
                             r 
                             D 
                           
                           -&gt; 
                           0 
                         
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             ∫ 
                             
                               
                                 z 
                                 wD 
                               
                               - 
                               
                                 
                                   ɛ 
                                   D 
                                 
                                 / 
                                 2 
                               
                             
                             
                               
                                 z 
                                 wD 
                               
                               + 
                               
                                 
                                   ɛ 
                                   D 
                                 
                                 / 
                                 2 
                               
                             
                           
                            
                           
                             
                               r 
                               D 
                             
                              
                             
                               
                                 ∂ 
                                 
                                   p 
                                   
                                     1 
                                      
                                     D 
                                   
                                 
                               
                               
                                 ∂ 
                                 
                                   r 
                                   D 
                                 
                               
                             
                              
                             
                               dz 
                               D 
                             
                           
                         
                         - 
                         
                           { 
                           
                             
                               
                                 
                                   
                                     - 
                                     
                                       q 
                                       D 
                                     
                                   
                                   , 
                                 
                               
                               
                                 
                                   
                                     
                                        
                                       
                                         
                                           z 
                                           D 
                                         
                                         - 
                                         
                                           z 
                                           wD 
                                         
                                       
                                        
                                     
                                     ≤ 
                                     
                                       
                                         ɛ 
                                         D 
                                       
                                       / 
                                       2 
                                     
                                   
                                   ; 
                                 
                               
                             
                             
                               
                                 
                                   0 
                                   , 
                                 
                               
                               
                                 
                                   else 
                                   . 
                                 
                               
                             
                           
                           } 
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   231 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           ∂ 
                           
                             p 
                             
                               1 
                                
                               D 
                             
                           
                         
                         
                           ∂ 
                           
                             z 
                             D 
                           
                         
                       
                        
                       
                         
                           
                              
                             
                               
                                 z 
                                 D 
                               
                               = 
                               
                                 z 
                                 
                                   j2 
                                    
                                   
                                       
                                   
                                    
                                   D 
                                 
                               
                             
                           
                            
                           
                             = 
                             
                               
                                 ∂ 
                                 
                                   p 
                                   
                                     2 
                                      
                                     D 
                                   
                                 
                               
                               
                                 ∂ 
                                 
                                   z 
                                   D 
                                 
                               
                             
                           
                            
                         
                         
                           
                             z 
                             D 
                           
                           = 
                           
                             h 
                             D 
                           
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   232 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           ∂ 
                           
                             p 
                             
                               1 
                                
                               D 
                             
                           
                         
                         
                           ∂ 
                           
                             z 
                             D 
                           
                         
                       
                        
                       
                         
                           
                              
                             
                               
                                 z 
                                 D 
                               
                               = 
                               
                                 z 
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                    
                                   D 
                                 
                               
                             
                           
                            
                           
                             = 
                             
                               
                                 ∂ 
                                 
                                   p 
                                   
                                     2 
                                      
                                     D 
                                   
                                 
                               
                               
                                 ∂ 
                                 
                                   z 
                                   D 
                                 
                               
                             
                           
                            
                         
                         
                           
                             z 
                             D 
                           
                           = 
                           0 
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   233 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   ∂ 
                                   
                                     p 
                                     
                                       1 
                                        
                                       D 
                                     
                                   
                                 
                                 
                                   ∂ 
                                   
                                     r 
                                     
                                       D 
                                        
                                       
                                           
                                       
                                     
                                   
                                 
                               
                                
                               
                                 
                                   
                                      
                                     
                                       
                                         r 
                                         D 
                                       
                                       = 
                                       
                                         r 
                                         
                                           1 
                                            
                                           D 
                                         
                                       
                                     
                                   
                                    
                                   
                                     = 
                                     
                                       
                                         κ 
                                         D 
                                       
                                        
                                       
                                         
                                           ∂ 
                                           
                                             p 
                                             
                                               2 
                                                
                                               D 
                                             
                                           
                                         
                                         
                                           ∂ 
                                           
                                             r 
                                             D 
                                           
                                         
                                       
                                     
                                   
                                    
                                 
                                 
                                   
                                     r 
                                     D 
                                   
                                   = 
                                   
                                     r 
                                     
                                       1 
                                        
                                       D 
                                     
                                   
                                 
                               
                             
                             , 
                             
                               ( 
                               
                                 
                                   z 
                                   
                                     j 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                      
                                     D 
                                   
                                 
                                 ≤ 
                                 
                                   z 
                                   D 
                                 
                                 ≤ 
                                 
                                   z 
                                   
                                     j 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     D 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 ∂ 
                                 
                                   p 
                                   
                                     2 
                                      
                                     D 
                                   
                                 
                               
                               
                                 ∂ 
                                 
                                   r 
                                   D 
                                 
                               
                             
                              
                             
                               
                                  
                                 
                                   
                                     r 
                                     D 
                                   
                                   = 
                                   
                                     r 
                                     
                                       1 
                                        
                                       D 
                                     
                                   
                                 
                               
                                
                               
                                 
                                   = 
                                   0 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     0 
                                     ≤ 
                                     
                                       z 
                                       D 
                                     
                                     &lt; 
                                     
                                       z 
                                       
                                         j 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                          
                                         
                                             
                                         
                                          
                                         D 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 ∂ 
                                 
                                   p 
                                   
                                     2 
                                      
                                     D 
                                   
                                 
                               
                               
                                 ∂ 
                                 D 
                               
                             
                              
                             
                               
                                  
                                 
                                   
                                     r 
                                     D 
                                   
                                   = 
                                   
                                     r 
                                     
                                       1 
                                        
                                       D 
                                     
                                   
                                 
                               
                                
                               
                                 
                                   = 
                                   0 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       z 
                                       
                                         j 
                                          
                                         
                                             
                                         
                                          
                                         2 
                                          
                                         D 
                                       
                                     
                                     &lt; 
                                     
                                       z 
                                       D 
                                     
                                     ≤ 
                                     
                                       h 
                                       D 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   234 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       p 
                       
                         1 
                          
                         D 
                       
                     
                      
                     
                       
                         
                            
                           
                             
                               r 
                               D 
                             
                             = 
                             
                               r 
                               
                                 1 
                                  
                                 D 
                               
                             
                           
                         
                          
                         
                           = 
                           
                             p 
                             
                               2 
                                
                               D 
                             
                           
                         
                          
                       
                       
                         
                           r 
                           D 
                         
                         = 
                         
                           r 
                           
                             1 
                              
                             D 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   235 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         ∂ 
                         
                           p 
                           
                             2 
                              
                             D 
                           
                         
                       
                       
                         ∂ 
                         
                           r 
                           D 
                         
                       
                     
                      
                     
                       
                          
                         
                           
                             r 
                             D 
                           
                           = 
                           
                             r 
                             
                               2 
                                
                               D 
                             
                           
                         
                       
                        
                       
                         = 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   236 
                   ) 
                 
               
             
           
         
       
     
     Dimensionless variables are defined as 
     
       
         
           
             
               
                 
                   
                     p 
                     
                       1 
                        
                       D 
                     
                   
                   = 
                   
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           λ 
                           
                             r 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                           h 
                           1 
                         
                       
                       
                         
                           q 
                           
                             s 
                              
                             
                                 
                             
                              
                             c 
                           
                         
                          
                         B 
                       
                     
                      
                     
                       ( 
                       
                         
                           p 
                           i 
                         
                         - 
                         
                           p 
                           1 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   237 
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     
                       2 
                        
                       D 
                     
                   
                   = 
                   
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           λ 
                           
                             r 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                           h 
                           1 
                         
                       
                       
                         
                           q 
                           sc 
                         
                          
                         B 
                       
                     
                      
                     
                       ( 
                       
                         
                           p 
                           i 
                         
                         - 
                         
                           p 
                           2 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   238 
                   ) 
                 
               
             
             
               
                 
                   
                     r 
                     D 
                   
                   = 
                   
                     r 
                     l 
                   
                 
               
               
                 
                   ( 
                   239 
                   ) 
                 
               
             
             
               
                 
                   
                     t 
                     D 
                   
                   = 
                   
                     
                       
                         λ 
                         
                           r 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       t 
                     
                     
                       
                         φ 
                         1 
                       
                        
                       
                         C 
                         
                           t 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       
                         l 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   240 
                   ) 
                 
               
             
             
               
                 
                   
                     z 
                     D 
                   
                   = 
                   
                     z 
                     
                       h 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   241 
                   ) 
                 
               
             
             
               
                 
                   
                     z 
                     wD 
                   
                   = 
                   
                     
                       z 
                       w 
                     
                     
                       h 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   242 
                   ) 
                 
               
             
             
               
                 
                   
                     h 
                     
                       1 
                        
                       D 
                     
                   
                   = 
                   
                     
                       
                         h 
                         1 
                       
                       l 
                     
                      
                     
                       
                         
                           λ 
                           
                             r 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                         
                           λ 
                           
                             z 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   243 
                   ) 
                 
               
             
             
               
                 
                   
                     h 
                     
                       2 
                        
                       D 
                     
                   
                   = 
                   
                     
                       
                         h 
                         1 
                       
                       l 
                     
                      
                     
                       
                         
                           λ 
                           
                             r 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                         
                           λ 
                           
                             z 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   244 
                   ) 
                 
               
             
             
               
                 
                   σ 
                   = 
                   
                     
                       
                         λ 
                         
                           r 
                            
                           
                               
                           
                            
                           2 
                         
                       
                       
                         
                           φ 
                           2 
                         
                          
                         
                           C 
                           
                             t 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                     
                     / 
                     
                       
                         λ 
                         
                           r 
                            
                           
                               
                           
                            
                           1 
                         
                       
                       
                         
                           φ 
                           1 
                         
                          
                         
                           C 
                           
                             t 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   245 
                   ) 
                 
               
             
             
               
                 
                   
                     κ 
                     D 
                   
                   = 
                   
                     
                       λ 
                       
                         r 
                          
                         
                             
                         
                          
                         2 
                       
                     
                     
                       λ 
                       
                         r 
                          
                         
                             
                         
                          
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   246 
                   ) 
                 
               
             
             
               
                 
                   
                     z 
                     
                       j 
                        
                       
                           
                       
                        
                       1 
                        
                       D 
                     
                   
                   = 
                   
                     
                       z 
                       j 
                     
                     
                       h 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   247 
                   ) 
                 
               
             
             
               
                 
                   
                     z 
                     
                       j 
                        
                       
                           
                       
                        
                       2 
                        
                       D 
                     
                   
                   = 
                   
                     
                       
                         h 
                         1 
                       
                       + 
                       
                         z 
                         j 
                       
                     
                     
                       h 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   248 
                   ) 
                 
               
             
             
               
                 
                   
                     h 
                     D 
                   
                   = 
                   
                     
                       h 
                       2 
                     
                     
                       h 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   249 
                   ) 
                 
               
             
             
               
                 
                   
                     q 
                     D 
                   
                   = 
                   
                     q 
                     / 
                     
                       q 
                       sc 
                     
                   
                 
               
               
                 
                   ( 
                   250 
                   ) 
                 
               
             
           
         
       
     
     The dimensionless pressure solution of Eqs.229-236 in the Laplace space by applying Laplace integral transformation and Fourier finite cosine transformation is obtained as: 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       _ 
                     
                     
                       1 
                        
                       D 
                     
                   
                   = 
                   
                     
                       
                         q 
                         _ 
                       
                       D 
                     
                      
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   K 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       r 
                                       D 
                                     
                                      
                                     
                                       δ 
                                       10 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   g 
                                   0 
                                 
                                  
                                 
                                   
                                     I 
                                     0 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         r 
                                         D 
                                       
                                        
                                       
                                         δ 
                                         10 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         = 
                                         1 
                                       
                                       ∞ 
                                     
                                      
                                     
                                       2 
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               ξ 
                                               
                                                 1 
                                                  
                                                 m 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   z 
                                                   
                                                     j 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     1 
                                                      
                                                     D 
                                                   
                                                 
                                                 - 
                                                 
                                                   z 
                                                   wd 
                                                 
                                               
                                               ) 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               ξ 
                                               
                                                 1 
                                                  
                                                 m 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   z 
                                                   
                                                     j 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     1 
                                                      
                                                     D 
                                                   
                                                 
                                                 - 
                                                 
                                                   z 
                                                   D 
                                                 
                                               
                                               ) 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         
                                           K 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               r 
                                               D 
                                             
                                              
                                             
                                               δ 
                                               
                                                 1 
                                                  
                                                 m 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       + 
                                       
                                         
                                           g 
                                           m 
                                         
                                          
                                         
                                           
                                             I 
                                             0 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 r 
                                                 D 
                                               
                                                
                                               
                                                 δ 
                                                 
                                                   1 
                                                    
                                                   m 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   251 
                   ) 
                 
               
             
           
         
       
     
     Where 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       g 
                       m 
                     
                     = 
                     
                       
                         
                           
                             F 
                             
                               n 
                                
                               
                                   
                               
                                
                               m 
                             
                           
                            
                           
                             
                               K 
                               1 
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   
                                     1 
                                      
                                     D 
                                   
                                 
                                  
                                 
                                   δ 
                                   
                                     1 
                                      
                                     m 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   ξ 
                                   
                                     1 
                                      
                                     m 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       z 
                                       D 
                                     
                                     - 
                                     
                                       z 
                                       
                                         j 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                          
                                         D 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               K 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   
                                     1 
                                      
                                     D 
                                   
                                 
                                  
                                 
                                   δ 
                                   
                                     1 
                                      
                                     m 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               n 
                                
                               
                                   
                               
                                
                               m 
                             
                           
                            
                           
                             
                               I 
                               1 
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   
                                     1 
                                      
                                     D 
                                   
                                 
                                  
                                 
                                   δ 
                                   
                                     1 
                                      
                                     m 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   ξ 
                                   
                                     1 
                                      
                                     m 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       z 
                                       D 
                                     
                                     - 
                                     
                                       z 
                                       
                                         j 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                          
                                         D 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               I 
                               0 
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   
                                     1 
                                      
                                     D 
                                   
                                 
                                  
                                 
                                   δ 
                                   
                                     1 
                                      
                                     m 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   252 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     
                       n 
                        
                       
                           
                       
                        
                       m 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         0 
                       
                       ∞ 
                     
                      
                     
                       
                         
                           4 
                            
                           
                             
                               ξ 
                               
                                 2 
                                  
                                 n 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                           ξ 
                                           
                                             1 
                                              
                                             m 
                                           
                                         
                                          
                                         
                                           sin 
                                            
                                           
                                             ( 
                                             
                                               ξ 
                                               
                                                 1 
                                                  
                                                 m 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               
                                                 ξ 
                                                 
                                                   2 
                                                    
                                                   n 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     z 
                                                     
                                                       j 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       1 
                                                        
                                                       D 
                                                     
                                                   
                                                   + 
                                                   1 
                                                 
                                                 ) 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ξ 
                                         
                                           2 
                                            
                                           n 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             sin 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   ξ 
                                                   
                                                     2 
                                                      
                                                     n 
                                                   
                                                 
                                                  
                                                 
                                                   z 
                                                   
                                                     j 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     1 
                                                      
                                                     D 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                           - 
                                           
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 
                                                   ξ 
                                                   
                                                     1 
                                                      
                                                     m 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               sin 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ξ 
                                                     
                                                       2 
                                                        
                                                       n 
                                                     
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         z 
                                                         
                                                           j 
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           1 
                                                            
                                                           D 
                                                         
                                                       
                                                       + 
                                                       1 
                                                     
                                                     ) 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             ( 
                             
                               
                                 ξ 
                                 
                                   1 
                                    
                                   m 
                                 
                               
                               - 
                               
                                 ξ 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 ξ 
                                 
                                   1 
                                    
                                   m 
                                 
                               
                               + 
                               
                                 ξ 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                 
                                   h 
                                   D 
                                 
                                  
                                 
                                   ξ 
                                   
                                     2 
                                      
                                     n 
                                   
                                 
                               
                               + 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     2 
                                      
                                     
                                       h 
                                       D 
                                     
                                      
                                     
                                       ξ 
                                       
                                         2 
                                          
                                         n 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                       × 
                       
                         
                           
                             
                               δ 
                               
                                 1 
                                  
                                 m 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                           K 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               r 
                                               
                                                 1 
                                                  
                                                 D 
                                               
                                             
                                              
                                             
                                               δ 
                                               
                                                 2 
                                                  
                                                 n 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           
                                             K 
                                             1 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 r 
                                                 eD 
                                               
                                                
                                               
                                                 δ 
                                                 
                                                   2 
                                                    
                                                   n 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                         / 
                                         
                                           
                                             I 
                                             1 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 r 
                                                 eD 
                                               
                                                
                                               
                                                 δ 
                                                 
                                                   2 
                                                    
                                                   n 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         
                                           I 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               r 
                                               
                                                 1 
                                                  
                                                 D 
                                               
                                             
                                              
                                             
                                               δ 
                                               
                                                 2 
                                                  
                                                 n 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   ξ 
                                   
                                     2 
                                      
                                     n 
                                   
                                 
                                  
                                 
                                   z 
                                   D 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             κ 
                             D 
                           
                            
                           
                             
                               δ 
                               
                                 2 
                                  
                                 n 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     K 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         r 
                                         
                                           1 
                                            
                                           D 
                                         
                                       
                                        
                                       
                                         δ 
                                         
                                           2 
                                            
                                           n 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       
                                         K 
                                         1 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             r 
                                             eD 
                                           
                                            
                                           
                                             δ 
                                             
                                               2 
                                                
                                               n 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     / 
                                     
                                       
                                         I 
                                         1 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             r 
                                             eD 
                                           
                                            
                                           
                                             δ 
                                             
                                               2 
                                                
                                               n 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     
                                       I 
                                       1 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           r 
                                           
                                             1 
                                              
                                             D 
                                           
                                         
                                          
                                         
                                           δ 
                                           
                                             2 
                                              
                                             n 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   253 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       δ 
                       
                         1 
                          
                         m 
                       
                     
                     = 
                     
                       
                         u 
                         + 
                         
                           
                             ξ 
                             
                               1 
                                
                               m 
                             
                             2 
                           
                           
                             h 
                             
                               1 
                                
                               D 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   254 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       δ 
                       
                         2 
                          
                         n 
                       
                     
                     = 
                     
                       
                         
                           u 
                           σ 
                         
                         + 
                         
                           
                             ξ 
                             
                               2 
                                
                               n 
                             
                             2 
                           
                           
                             h 
                             
                               2 
                                
                               D 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   255 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       ξ 
                       1 
                     
                     = 
                     
                       m 
                        
                       
                           
                       
                        
                       π 
                     
                   
                 
               
               
                 
                   ( 
                   256 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       ξ 
                       
                         2 
                          
                         
                             
                         
                       
                     
                     = 
                     
                       
                         n 
                          
                         
                             
                         
                          
                         π 
                       
                       
                         h 
                         D 
                       
                     
                   
                 
               
               
                 
                   ( 
                   257 
                   ) 
                 
               
             
           
         
       
     
     I 0 (g) and I 1 (g) are the category I modified zero-order and first-order Bessel functions respectively, K 0 (g) and K 1 (g) are the category II modified zero-order and first-order Bessel functions respectively. 
     Especially, setting r D =1, z D =z wD  as reference points of bottom hole pressure, i. e., 
           p     wD   = p     D ( r   D =1, z   D   =z   wD )  (258)
 
     Typical type curves can be drawn by performing Stehfest numerical inversion as shown in  FIGS. 32-37 . 
     The specific meanings of the identifiers in embodiment 11 are shown as follows: 
     B—volume factor, m 3 /m 3 ; C t1 ,C t2 —compressibility of cylindrical reservoir and hollow cylindrical reservoir respectively, 1/Pa; h 1 ,h 2 —height of cylindrical reservoir and hollow cylindrical reservoir respectively, m; l—reference length, m; p 1 , p 2 —pressure of cylindrical reservoir and hollow cylindrical reservoir, Pa; p i —initial pressure of reservoir, Pa: p w —bottom hole pressure, Pa: q—flow rate, m 3 /s; q sc —reference flow rate, m 3 /s; r—radial distance from the point source to the axis of cylindrical reservoir, m: r 1 ,r 2 —radius of cylindrical reservoir and hollow cylindrical reservoir, m; t—time, s; u—Laplace variable of dimensionless time t D ; z—z-axis distance, m; z w —central point position of point source on z-axis, m; z j —vertical position of the low end of the cylindrical reservoir, m; ε—height of the point source (tends to zero), m; λ r1 ,λ r2 —generalized mobility of cylindrical reservoir and hollow cylindrical reservoir respectively, m 2 /Pa·s; ϕ 1 ,ϕ 2 —porosity of cylindrical reservoir and hollow cylindrical reservoir, %; ∂—partial differential operator. 
     If necessary, the line source solution and spherical plane source solutions can be obtained through integrating the point source solution Eq.251. 
     Embodiment 12 
     This embodiment provides a composite flow simulation and transient pressure well testing analysis methods for a compound body of tube-shaped and spherical reservoir. The physical model corresponding to the pressure drawdown well testing model is assumed as follows: 
     1) The reservoir is a composite reservoir of a compound body of tube-shaped and spherical reservoir as shown in  FIG. 38 ;
 
2) In the initial state, the pressure everywhere in the reservoir is the original reservoir pressure,
 
3) The fluid flow in the reservoir follows linear flow law,
 
4) The fluid and rock in the reservoir are slightly compressible,
 
5) The wellbore storage effect as well as the skin effect are not considered,
 
     The dimensionless mathematical model corresponding to the pressure drawdown well test model is written as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         1 
                         
                           r 
                           D 
                           2 
                         
                       
                        
                       
                         ∂ 
                         
                           ∂ 
                           
                             r 
                             D 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             r 
                             D 
                             2 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 vD 
                               
                             
                             
                               ∂ 
                               
                                 r 
                                 D 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         1 
                         
                           
                             r 
                             D 
                             2 
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           θ 
                         
                       
                        
                       
                         ∂ 
                         
                           ∂ 
                           θ 
                         
                       
                        
                       
                         ( 
                         
                           sin 
                            
                           
                               
                           
                            
                           θ 
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 vD 
                               
                             
                             
                               ∂ 
                               θ 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       1 
                       σ 
                     
                      
                     
                       
                         ∂ 
                         
                           p 
                           vD 
                         
                       
                       
                         ∂ 
                         
                           t 
                           D 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   259 
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     vD 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           r 
                           D 
                         
                         , 
                         
                           θ 
                           = 
                           0 
                         
                         , 
                         
                           
                             t 
                             D 
                           
                           = 
                           0 
                         
                       
                       ) 
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   260 
                   ) 
                 
               
             
             
               
                 
                   
                     - 
                     
                       q 
                       vD 
                     
                   
                   = 
                   
                     
                       lim 
                       
                         
                           r 
                           D 
                         
                         -&gt; 
                         0 
                       
                     
                      
                     
                       2 
                        
                       
                         r 
                         D 
                         2 
                       
                        
                       
                         
                           λ 
                           v 
                         
                         
                           λ 
                           1 
                         
                       
                        
                       
                         
                           ∂ 
                           
                             p 
                             vD 
                           
                         
                         
                           ∂ 
                           
                             r 
                             D 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   261 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         p 
                         vd 
                       
                     
                     
                       ∂ 
                       
                         r 
                         D 
                       
                     
                   
                    
                   
                     
                        
                       
                         
                           r 
                           D 
                         
                         = 
                         
                           
                             
                               r 
                               eD 
                               2 
                             
                             + 
                             
                               r 
                               aD 
                               2 
                             
                             - 
                             
                               2 
                                
                               
                                 r 
                                 eD 
                               
                                
                               
                                 r 
                                 aD 
                               
                                
                               co 
                                
                               
                                   
                               
                                
                               s 
                                
                               
                                   
                               
                                
                               Φ 
                             
                           
                         
                       
                     
                      
                     
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   262 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∂ 
                         2 
                       
                        
                       
                         p 
                         
                           1 
                            
                           D 
                         
                       
                     
                     
                       ∂ 
                       
                         x 
                         D 
                         2 
                       
                     
                   
                   = 
                   
                     
                       ∂ 
                       
                         p 
                         
                           1 
                            
                           D 
                         
                       
                     
                     
                       ∂ 
                       
                         t 
                         D 
                       
                     
                   
                 
               
               
                 
                   ( 
                   263 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       p 
                       
                         1 
                          
                         D 
                       
                     
                      
                     
                       ( 
                       
                         
                           x 
                           D 
                         
                         , 
                         
                           
                             t 
                             D 
                           
                           = 
                           0 
                         
                       
                       ) 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   264 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       D 
                     
                     
                       2 
                        
                       π 
                     
                   
                    
                   
                     
                       ∂ 
                       
                         p 
                         
                           1 
                            
                           D 
                         
                       
                     
                     
                       ∂ 
                       
                         x 
                         D 
                       
                     
                   
                    
                   
                     
                        
                       
                         
                           x 
                           D 
                         
                         = 
                         0 
                       
                     
                      
                     
                       = 
                       
                         - 
                         
                           q 
                           D 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   265 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       D 
                     
                     
                       2 
                        
                       π 
                     
                   
                    
                   
                     
                       ∂ 
                       
                         p 
                         
                           1 
                            
                           D 
                         
                       
                     
                     
                       ∂ 
                       
                         x 
                         D 
                       
                     
                   
                    
                   
                     
                        
                       
                         
                           x 
                           D 
                         
                         = 
                         
                           L 
                           D 
                         
                       
                     
                      
                     
                       = 
                       
                         - 
                         
                           q 
                           vD 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   266 
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     
                       1 
                        
                       D 
                     
                   
                    
                   
                     
                        
                       
                         
                           x 
                           D 
                         
                         = 
                         
                           L 
                           D 
                         
                       
                     
                      
                     
                       = 
                       
                         
                           p 
                           vD 
                         
                          
                         
                            
                           
                             
                               
                                 r 
                                 bD 
                               
                               = 
                               
                                 r 
                                 eD 
                               
                             
                             , 
                             
                               
                                 r 
                                 aD 
                               
                               = 
                               
                                 
                                   r 
                                   eD 
                                 
                                 - 
                                 1 
                               
                             
                             , 
                             
                               Φ 
                               = 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   267 
                   ) 
                 
               
             
           
         
       
     
     Dimensionless variables are defined as 
     
       
         
           
             
               
                 
                   
                     t 
                     D 
                   
                   = 
                   
                     
                       
                         λ 
                         1 
                       
                        
                       t 
                     
                     
                       
                         φ 
                         1 
                       
                        
                       
                         C 
                         
                           t 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       
                         l 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   268 
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     
                       1 
                        
                       D 
                     
                   
                   = 
                   
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           λ 
                           1 
                         
                          
                         l 
                       
                       
                         
                           q 
                           sc 
                         
                          
                         B 
                       
                     
                      
                     
                       ( 
                       
                         
                           p 
                           i 
                         
                         - 
                         
                           p 
                           1 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   269 
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     vD 
                   
                   = 
                   
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           λ 
                           1 
                         
                          
                         l 
                       
                       
                         
                           q 
                           sc 
                         
                          
                         B 
                       
                     
                      
                     
                       ( 
                       
                         
                           p 
                           i 
                         
                         - 
                         
                           p 
                           v 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   270 
                   ) 
                 
               
             
             
               
                 
                   σ 
                   = 
                   
                     
                       
                         λ 
                         v 
                       
                       
                         
                           φ 
                           v 
                         
                          
                         
                           C 
                           tv 
                         
                       
                     
                     / 
                     
                       
                         λ 
                         1 
                       
                       
                         
                           φ 
                           1 
                         
                          
                         
                           C 
                           
                             t 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   271 
                   ) 
                 
               
             
             
               
                 
                   
                     A 
                     D 
                   
                   = 
                   
                     A 
                     l 
                   
                 
               
               
                 
                   ( 
                   272 
                   ) 
                 
               
             
             
               
                 
                   
                     x 
                     D 
                   
                   = 
                   
                     x 
                     l 
                   
                 
               
               
                 
                   ( 
                   273 
                   ) 
                 
               
             
             
               
                 
                   
                     L 
                     D 
                   
                   = 
                   
                     L 
                     l 
                   
                 
               
               
                 
                   ( 
                   274 
                   ) 
                 
               
             
             
               
                 
                   
                     r 
                     aD 
                   
                   = 
                   
                     
                       r 
                       a 
                     
                     l 
                   
                 
               
               
                 
                   ( 
                   275 
                   ) 
                 
               
             
             
               
                 
                   
                     r 
                     bD 
                   
                   = 
                   
                     
                       r 
                       b 
                     
                     l 
                   
                 
               
               
                 
                   ( 
                   276 
                   ) 
                 
               
             
             
               
                 
                   
                     r 
                     eD 
                   
                   = 
                   
                     
                       r 
                       e 
                     
                     l 
                   
                 
               
               
                 
                   ( 
                   277 
                   ) 
                 
               
             
             
               
                 
                   
                     r 
                     D 
                   
                   = 
                   
                     
                       
                         r 
                         bD 
                         2 
                       
                       + 
                       
                         r 
                         aD 
                         2 
                       
                       - 
                       
                         2 
                          
                         
                           r 
                           bD 
                         
                          
                         
                           r 
                           aD 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         Φ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   278 
                   ) 
                 
               
             
             
               
                 
                   θ 
                   = 
                   
                     arcsin 
                      
                     
                         
                     
                      
                     
                       
                         r 
                         D 
                       
                       
                         
                           r 
                           bD 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         Φ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   279 
                   ) 
                 
               
             
             
               
                 
                   
                     q 
                     vD 
                   
                   = 
                   
                     
                       q 
                       v 
                     
                     / 
                     
                       q 
                       
                         s 
                          
                         
                             
                         
                          
                         c 
                       
                     
                   
                 
               
               
                 
                   ( 
                   280 
                   ) 
                 
               
             
             
               
                 
                   
                     q 
                     D 
                   
                   = 
                   
                     q 
                     / 
                     
                       q 
                       sc 
                     
                   
                 
               
               
                 
                   ( 
                   281 
                   ) 
                 
               
             
           
         
       
     
     The dimensionless pressure solution of equations Eqs.259 and 267 in the Laplace space by applying Laplace integral transformation and boundary construction method is obtained as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         p 
                         _ 
                       
                       D 
                     
                     = 
                     
                       
                         
                           q 
                           _ 
                         
                         D 
                       
                        
                       
                         
                           2 
                            
                           
                             π 
                              
                             
                               ( 
                               
                                 
                                   
                                     A 
                                     D 
                                   
                                    
                                   
                                     f 
                                     _ 
                                   
                                    
                                   
                                     u 
                                   
                                    
                                   
                                     cosh 
                                      
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               L 
                                               D 
                                             
                                             - 
                                             
                                               x 
                                               D 
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                           u 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sinh 
                                      
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               L 
                                               D 
                                             
                                             - 
                                             
                                               x 
                                               D 
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                           u 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             A 
                             D 
                           
                            
                           
                             u 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   A 
                                   D 
                                 
                                  
                                 
                                   f 
                                   _ 
                                 
                                  
                                 
                                   u 
                                 
                                  
                                 
                                   sinh 
                                    
                                   
                                     ( 
                                     
                                       
                                         L 
                                         D 
                                       
                                        
                                       
                                         u 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   cosh 
                                    
                                   
                                     ( 
                                     
                                       
                                         L 
                                         D 
                                       
                                        
                                       
                                         u 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   282 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       f 
                       _ 
                     
                     = 
                     
                       
                         - 
                         
                           
                             λ 
                             
                               ? 
                             
                           
                           
                             λ 
                             
                               ? 
                             
                           
                         
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           ∞ 
                         
                          
                         
                           
                             
                               n 
                               + 
                               
                                 1 
                                 / 
                                 2 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     
                                       r 
                                       eD 
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 
                                   r 
                                   eD 
                                 
                               
                             
                           
                            
                           
                             
                               
                                 I 
                                 
                                   n 
                                   + 
                                   
                                     1 
                                     2 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         r 
                                         eD 
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   
                                     
                                       u 
                                       / 
                                       σ 
                                     
                                   
                                 
                                 ) 
                               
                             
                             [ 
                             
                               
                                 
                                   K 
                                   
                                     n 
                                     + 
                                     
                                       1 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       r 
                                       eD 
                                     
                                      
                                     
                                       
                                         u 
                                         / 
                                         σ 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   
                                     
                                       
                                         nK 
                                         
                                           n 
                                           + 
                                           
                                             1 
                                             2 
                                           
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             r 
                                             eD 
                                           
                                            
                                           
                                             
                                               u 
                                               / 
                                               σ 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         r 
                                         eD 
                                       
                                        
                                       
                                         
                                           
                                             u 
                                             / 
                                             σ 
                                           
                                            
                                           
                                               
                                           
                                         
                                       
                                        
                                       
                                         
                                           K 
                                           
                                             n 
                                             + 
                                             
                                               3 
                                               2 
                                             
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               r 
                                               eD 
                                             
                                              
                                             
                                               
                                                 u 
                                                 / 
                                                 σ 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         nI 
                                         
                                           n 
                                           + 
                                           
                                             1 
                                             2 
                                           
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             r 
                                             eD 
                                           
                                            
                                           
                                             
                                               u 
                                               / 
                                               σ 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         r 
                                         eD 
                                       
                                        
                                       
                                         
                                           u 
                                           / 
                                           σ 
                                         
                                       
                                        
                                       
                                         
                                           I 
                                           
                                             n 
                                             + 
                                             
                                               3 
                                               2 
                                             
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               r 
                                               eD 
                                             
                                              
                                             
                                               
                                                 u 
                                                 / 
                                                 σ 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                  
                                 
                                   
                                     I 
                                     
                                       n 
                                       + 
                                       
                                         1 
                                         2 
                                       
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         r 
                                         eD 
                                       
                                        
                                       
                                         
                                           u 
                                           / 
                                           σ 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ? 
                     
                      
                     
                       indicates text missing or illegible when filed 
                     
                   
                 
               
               
                 
                   ( 
                   283 
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               I 
               
                 n 
                 + 
                 
                   1 
                   2 
                 
               
             
              
             
               ( 
               g 
               ) 
             
           
         
       
     
     is the category I modified n+½ order Bessel function, 
     
       
         
           
             
               
                 K 
                 
                   n 
                   + 
                   
                     1 
                     2 
                   
                 
               
                
               
                 ( 
                 g 
                 ) 
               
             
              
             
                 
             
              
             and 
              
             
                 
             
              
             
               
                 K 
                 
                   n 
                   + 
                   
                     3 
                     2 
                   
                 
               
                
               
                 ( 
                 g 
                 ) 
               
             
           
         
       
     
     are the category II modified n+½ order and n+3/2 order Bessel functions respectively. 
     Especially, setting x D =0 as the reference points of bottom hole pressure, i. e., 
           p     wD   = p     D ( x   D =0)  (284)
 
     Typical type curves can be drawn by performing Stehfest numerical inversion as shown in  FIG. 39-41 . 
     The specific meanings of the identifiers in embodiment 12 are shown as follows: 
     A—open flowing area of tube-shaped reservoir, m 2 : B—volume factor, m 3 /m 3 ; C t1 ,C tv —compressibility of tube-shaped reservoir and spherical reservoir respectively, 1/Pa; l—reference length, m; L—length of tube-shaped reservoir, m; p 1 ,p v —pressure of tube-shaped reservoir and spherical reservoir, Pa; p i —initial pressure of tube-shaped and spherical reservoir, Pa; p w —bottom hole pressure, Pa; q—flow rate, m 3 /s; q sc —reference flow rate, m 3 /s; q v —flow rate from spherical reservoir to tube-shaped reservoir, m 3 /s; r—distance from the point source to the observation point, m; r a —eccentric distance, m; r b —the distance from the center of the spherical reservoir to the observation point, m; r e —radius of spherical reservoir, m; t—time, s; u—Laplace variable of dimensionless time t D ; x—x-axis distance, m; λ 1 ,λ v —generalized mobility of tube-shaped reservoir and spherical reservoir respectively, m 2 /Pa·s; ϕ 1 ,ϕ 2 —porosity of tube-shaped reservoir and spherical reservoir, %; Φ—angle between r a  and r b ; ∂—partial differential operator. 
     Embodiment 13 
     This embodiment provides flow simulation and transient pressure well testing analysis methods for one-dimensional tube-shaped reservoir. The physical model corresponding to the pressure drawdown well testing model is assumed as follows: 
     1) The reservoir is composed of a single tube-shaped reservoir as shown in  FIG. 46 ;
 
2) In the initial state, the pressure everywhere in the reservoir is the original reservoir pressure,
 
3) The fluid flow in the reservoir follows linear flow law or non-linear flow law (in this case, the non-linear flow refers to high-speed non-Darcy flow, but the turbulent flow and non-Newton flow can be solved in the similar way),
 
4) The fluid and rock in the reservoir are slightly compressible,
 
5) The wellbore storage effect as well as the skin effect are not considered,
 
     The dimensionless mathematical model corresponding to the pressure drawdown well test model is written as: 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         x 
                       
                     
                      
                     
                       ( 
                       
                         ρλ 
                          
                         
                           
                             ∂ 
                             p 
                           
                           
                             ∂ 
                             x 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∂ 
                       
                         ( 
                         ρφ 
                         ) 
                       
                     
                     
                       ∂ 
                       t 
                     
                   
                 
               
               
                 
                   ( 
                   285 
                   ) 
                 
               
             
             
               
                 
                   λ 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               1 
                               
                                 
                                   
                                     μ 
                                     K 
                                   
                                   + 
                                   βρ 
                                 
                                 | 
                                 v 
                                 | 
                               
                             
                             , 
                           
                         
                         
                           
                             0 
                             ≤ 
                             x 
                             &lt; 
                             
                               X 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             
                               α 
                                
                               
                                 K 
                                 μ 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               X 
                               1 
                             
                             ≤ 
                             x 
                             ≤ 
                             
                               X 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   286 
                   ) 
                 
               
             
             
               
                 
                   ρ 
                   = 
                   
                     
                       ρ 
                       1 
                     
                      
                     
                       e 
                       
                         
                           C 
                           L 
                         
                          
                         
                           ( 
                           
                             p 
                             - 
                             
                               p 
                               i 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   287 
                   ) 
                 
               
             
             
               
                 
                   φ 
                   = 
                   
                     
                       φ 
                       i 
                     
                      
                     
                       e 
                       
                         
                           C 
                           f 
                         
                          
                         
                           ( 
                           
                             p 
                             - 
                             
                               p 
                               i 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   288 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         A 
                          
                         λ 
                       
                        
                       
                         
                           ∂ 
                           p 
                         
                         
                           ∂ 
                           x 
                         
                       
                     
                      
                     
                       | 
                       
                         x 
                         = 
                         0 
                       
                     
                   
                   = 
                   
                     
                       q 
                       0 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   289 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         A 
                          
                         λ 
                       
                        
                       
                         
                           ∂ 
                           p 
                         
                         
                           ∂ 
                           x 
                         
                       
                     
                      
                     
                       | 
                       
                         x 
                         = 
                         
                           x 
                           2 
                         
                       
                     
                   
                   = 
                   
                     
                       q 
                       2 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   290 
                   ) 
                 
               
             
             
               
                 
                   
                     P 
                      
                     
                       | 
                       
                         t 
                         = 
                         0 
                       
                     
                   
                   = 
                   
                     p 
                     i 
                   
                 
               
               
                 
                   ( 
                   291 
                   ) 
                 
               
             
           
         
       
     
     Eq.292 can be obtained by Substituting Eqs.287-288 into Eq.285 and simplifying 
     
       
         
           
             
               
                 
                   
                     
                       λ 
                        
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           p 
                         
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                     
                     + 
                     
                       
                         
                           ∂ 
                           λ 
                         
                         
                           ∂ 
                           x 
                         
                       
                        
                       
                         
                           ∂ 
                           p 
                         
                         
                           ∂ 
                           x 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         φ 
                         i 
                       
                        
                       
                         ( 
                         
                           
                             C 
                             f 
                           
                           + 
                           
                             C 
                             L 
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         ∂ 
                         p 
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                 
               
               
                 
                   ( 
                   293 
                   ) 
                 
               
             
           
         
       
     
     Note: When the piecewise generalized mobility function is discontinuous at certain point x 0 , 
     
       
         
           
             
               ∂ 
               λ 
             
             
               ∂ 
               x 
             
           
         
       
     
     takes the average value of the left and right limits at x 0 . The above equations are discretized as follows 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                             
                               A 
                                
                               λ 
                             
                             1 
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                            
                           
                             
                               
                                 p 
                                 2 
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 p 
                                 1 
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 x 
                                 2 
                               
                               - 
                               
                                 x 
                                 1 
                               
                             
                           
                         
                         = 
                         
                           q 
                           0 
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       ; 
                       
                         k 
                         = 
                         1 
                       
                     
                     , 
                     2 
                     , 
                     … 
                     , 
                     
                       K 
                       - 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   294 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                             
                               A 
                                
                               λ 
                             
                             M 
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                            
                           
                             
                               
                                 p 
                                 M 
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 p 
                                 
                                   M 
                                   - 
                                   1 
                                 
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 x 
                                 M 
                               
                               - 
                               
                                 x 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                         = 
                         
                           q 
                           2 
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       ; 
                       
                         k 
                         = 
                         1 
                       
                     
                     , 
                     2 
                     , 
                     … 
                     , 
                     
                       K 
                       - 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   295 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           P 
                           m 
                           
                             ( 
                             1 
                             ) 
                           
                         
                         = 
                         
                           p 
                           i 
                         
                       
                       ; 
                       
                         m 
                         = 
                         1 
                       
                     
                     , 
                     2 
                     , 
                     … 
                     , 
                     M 
                   
                 
               
               
                 
                   ( 
                   296 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             λ 
                             
                               m 
                               + 
                               1 
                             
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                            
                           
                             [ 
                             
                               
                                 
                                   2 
                                    
                                   
                                     p 
                                     m 
                                     
                                       ( 
                                       
                                         k 
                                         + 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         m 
                                       
                                       - 
                                       
                                         x 
                                         
                                           m 
                                           + 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         m 
                                       
                                       - 
                                       
                                         x 
                                         
                                           m 
                                           + 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   2 
                                    
                                   
                                     p 
                                     
                                       m 
                                       + 
                                       1 
                                     
                                     
                                       ( 
                                       
                                         k 
                                         + 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         
                                           m 
                                           + 
                                           1 
                                         
                                       
                                       - 
                                       
                                         x 
                                         m 
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         
                                           m 
                                           + 
                                           1 
                                         
                                       
                                       - 
                                       
                                         x 
                                         
                                           m 
                                           + 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   2 
                                    
                                   
                                     p 
                                     
                                       m 
                                       + 
                                       2 
                                     
                                     
                                       ( 
                                       
                                         k 
                                         + 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         
                                           m 
                                           + 
                                           2 
                                         
                                       
                                       - 
                                       
                                         x 
                                         m 
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         
                                           m 
                                           + 
                                           2 
                                         
                                       
                                       - 
                                       
                                         x 
                                         
                                           m 
                                           + 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         + 
                         
                           
                             ξ 
                              
                             
                               [ 
                               
                                 λ 
                                 
                                   m 
                                   + 
                                   1 
                                 
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                            
                           
                             ξ 
                              
                             
                               [ 
                               
                                 p 
                                 
                                   m 
                                   + 
                                   1 
                                 
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                       
                       = 
                       
                         
                           
                             φ 
                             i 
                           
                            
                           
                             ( 
                             
                               
                                 C 
                                 f 
                               
                               + 
                               
                                 C 
                                 L 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             
                               p 
                               
                                 m 
                                 + 
                                 1 
                               
                               
                                 ( 
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             
                               p 
                               
                                 m 
                                 + 
                                 1 
                               
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           
                             
                               t 
                               
                                 k 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               t 
                               k 
                             
                           
                         
                       
                     
                     ; 
                     
                       m 
                       = 
                       1 
                     
                   
                   , 
                   2 
                   , 
                   … 
                   , 
                   
                     
                       M 
                       - 
                       2 
                     
                     ; 
                     
                       k 
                       = 
                       1 
                     
                   
                   , 
                   2 
                   , 
                   … 
                   , 
                   
                     K 
                     - 
                     1 
                   
                 
               
               
                 
                   ( 
                   297 
                   ) 
                 
               
             
           
         
       
     
     Where 
     
       
         
           
             
               
                 
                   
                     ξ 
                      
                     
                       [ 
                       
                         f 
                         
                           m 
                           + 
                           1 
                         
                         
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               x 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               x 
                               
                                 m 
                                 + 
                                 2 
                               
                             
                           
                           ) 
                         
                          
                         
                           f 
                           m 
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           ( 
                           
                             
                               x 
                               m 
                             
                             - 
                             
                               x 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               m 
                             
                             - 
                             
                               x 
                               
                                 m 
                                 + 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             
                               x 
                               m 
                             
                             - 
                             
                               2 
                                
                               
                                 x 
                                 
                                   m 
                                   + 
                                   1 
                                 
                               
                             
                             + 
                             
                               x 
                               
                                 m 
                                 + 
                                 2 
                               
                             
                           
                           ) 
                         
                          
                         
                           f 
                           
                             m 
                             + 
                             1 
                           
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           ( 
                           
                             
                               x 
                               m 
                             
                             - 
                             
                               x 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               x 
                               
                                 m 
                                 + 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             
                               x 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               x 
                               m 
                             
                           
                           ) 
                         
                          
                         
                           f 
                           
                             m 
                             + 
                             2 
                           
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           ( 
                           
                             
                               x 
                               
                                 m 
                                 + 
                                 2 
                               
                             
                             - 
                             
                               x 
                               m 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               
                                 m 
                                 + 
                                 2 
                               
                             
                             - 
                             
                               x 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   298 
                   ) 
                 
               
             
           
         
       
     
     Eq. 286 is discretized as follows 
     
       
         
           
             
               
                 
                   
                     λ 
                     
                       m 
                       + 
                       1 
                     
                     
                       ( 
                       
                         k 
                         + 
                         1 
                       
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         
                                           μ 
                                           K 
                                         
                                         + 
                                         
                                           βρ 
                                           i 
                                         
                                       
                                       | 
                                       
                                         v 
                                         
                                           m 
                                           + 
                                           1 
                                         
                                         
                                           ( 
                                           
                                             k 
                                             + 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       | 
                                     
                                     ] 
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                                 , 
                               
                             
                             
                               
                                 0 
                                 ≤ 
                                 
                                   x 
                                   
                                     m 
                                     + 
                                     1 
                                   
                                 
                                 &lt; 
                                 
                                   X 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   α 
                                    
                                   
                                     K 
                                     μ 
                                   
                                 
                                 , 
                               
                             
                             
                               
                                 
                                   X 
                                   1 
                                 
                                 ≤ 
                                 
                                   x 
                                   
                                     m 
                                     + 
                                     1 
                                   
                                 
                                 ≤ 
                                 
                                   X 
                                   2 
                                 
                               
                             
                           
                         
                         ; 
                         
                           m 
                           = 
                           0 
                         
                       
                       , 
                       1 
                       , 
                       2 
                       , 
                       … 
                       , 
                       
                         
                           M 
                           - 
                           1 
                         
                         ; 
                         
                           k 
                           = 
                           1 
                         
                       
                       , 
                       2 
                       , 
                       … 
                       , 
                       
                         κ 
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   299 
                   ) 
                 
               
             
           
         
       
     
     Since the flow velocity v m+1   (k+1)  at k+1 step in Eq.298 is unknown, the exact value of λ m+1   (k+1)  cannot be obtained, which needs further approximate evaluation. For simplicity, the flow rate v m+1   (k)  at k step is used to approximate the flow rate v m+1   (k+1)  at k+1 step, i.e., 
         v   m+1   (k+1)   ≈v   m+1   (k) =−λ m=1   (k) ξ[ p   m+1   (k+1) ]; m =0,2 , . . . ,M −1 ;k =1,2, . . . ,κ−1  (300)
 
     Furthermore 
     
       
         
           
             
               
                 
                   
                     λ 
                     
                       m 
                       + 
                       1 
                     
                     
                       ( 
                       
                         k 
                         + 
                         1 
                       
                       ) 
                     
                   
                   ≈ 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         
                                           μ 
                                           K 
                                         
                                         + 
                                         
                                           βρ 
                                           i 
                                         
                                       
                                       | 
                                       
                                         
                                           λ 
                                           
                                             m 
                                             + 
                                             1 
                                           
                                           
                                             ( 
                                             k 
                                             ) 
                                           
                                         
                                          
                                         
                                           ξ 
                                            
                                           
                                             [ 
                                             
                                               p 
                                               
                                                 m 
                                                 + 
                                                 1 
                                               
                                               
                                                 ( 
                                                 
                                                   k 
                                                   + 
                                                   1 
                                                 
                                                 ) 
                                               
                                             
                                             ] 
                                           
                                         
                                       
                                       | 
                                     
                                     ] 
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                                 , 
                               
                             
                             
                               
                                 0 
                                 ≤ 
                                 
                                   x 
                                   
                                     m 
                                     + 
                                     1 
                                   
                                 
                                 &lt; 
                                 
                                   X 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   α 
                                    
                                   
                                     K 
                                     μ 
                                   
                                 
                                 , 
                               
                             
                             
                               
                                 
                                   X 
                                   1 
                                 
                                 ≤ 
                                 
                                   x 
                                   
                                     m 
                                     + 
                                     1 
                                   
                                 
                                 ≤ 
                                 
                                   X 
                                   2 
                                 
                               
                             
                           
                         
                         ; 
                         
                           m 
                           = 
                           0 
                         
                       
                       , 
                       1 
                       , 
                       2 
                       , 
                       … 
                       , 
                       
                         
                           M 
                           - 
                           1 
                         
                         ; 
                         
                           k 
                           = 
                           1 
                         
                       
                       , 
                       2 
                       , 
                       … 
                       , 
                       
                         κ 
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   301 
                   ) 
                 
               
             
           
         
       
     
     Eq.300 is substituted into Eqs.293-296 to obtain p m   (k+1) =1, 2, . . . , M is the M-order linear equations of the unknown variables. The pressure distribution at k+1 is obtained by solving the above equations. The relationship between time and pressure can be plotted as shown in  FIG. 47-49 . 
     The specific meanings of the identifiers in embodiment 13 ar shown as follows: 
     A—open flowing area of tube-shaped reservoir, m 2 ; C f —compressibility of rock, 1/Pa; C L —compressibility of fluid, 1/Pa; L—length of tube-shaped reservoir, m; P—pressure of tube-shaped reservoir, Pa; p i —initial pressure of tube-shaped reservoir, Pa; q 0 —flow rate at x=0, m 3 /s; q 2 —flow rate at x=X 2 , m 3 /s; t—time, s; x—x-axis distance, m; λ—generalized mobility of tube-shaped reservoir, m 2 /Pa·s; K—permeability, m 2 ; μ—viscosity of the fluid, Pa·s; α—equation coefficient; β—non-Darcy flow coefficient, 1/m; v—fluid flow velocity, m/s; ϕ, ϕ i —porosity and initial porosity in tube-shaped reservoir respectively, %; ρ, ρ i —fluid density and initial density in tube-shaped reservoir respectively, kg/m 3 ; m, k—distance and time index number respectively; M, κ—maximum distance and time index number respectively; ∂—partial differential operator. 
     The shut-in pressure build-up type curves in embodiment 8-12 are calculated by the formula (301) and plotted by performing Stehfest numerical inversion 
           p     sD ( u )=   p     wD ( u )+   p     wD ( u   t     pD   )−   p     wD ( u   t     pD     +t     D   )  (302)
 
     where t pD  is the dimensionless production time before shut-in, explanation of  p   wD  refers embodiments 8 to 12. 
     The wellbore storage effect and skin effect are considered in the bottom hole pressure expression in embodiment 8-12 by Duhamel principle as 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       _ 
                     
                     
                       wD 
                       , 
                       
                         C 
                         D 
                       
                       , 
                       S 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             P 
                             _ 
                           
                           wD 
                         
                          
                         u 
                       
                       + 
                       S 
                     
                     
                       u 
                       + 
                       
                         
                           u 
                           2 
                         
                          
                         
                           
                             C 
                             D 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     P 
                                     _ 
                                   
                                   wD 
                                 
                                  
                                 u 
                               
                               + 
                               S 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   303 
                   ) 
                 
               
             
           
         
       
     
     where explanation of  p   wD  refers embodiments 8 to 12. 
     By applying this method, typical pressure draw-down type curves with consideration of skin effect and wellbore storage effects from embodiment 8 to 12 can be plotted as  FIG. 19 ,  FIG. 23 ,  FIG. 26 .  FIG. 30 ,  FIG. 34 ,  FIG. 37 , and  FIG. 41 . Typical pressure build-up type curves can also be plotted in the same way by using Eq.301. 
     Embodiment 14 
     This embodiment provides a multi-component condensate gas flow simulation analysis method for complex reservoirs. The model assumptions are as follows: 
     (1) Oil/gas/water three phases fluids are in the reservoir. The fluid flow is an isothermal process,
 
(2) The phase equilibrium of hydrocarbon components in oil and gas phase is completed in a flash,
 
(3) Water is an independent component and there is no mass exchange between hydrocarbons and water.
 
(4) Rocks are slightly compressible.
 
(5) Heterogeneity and anisotropy of the reservoirs are considered.
 
     The multi-component condensate gas flow simulation analysis for complex reservoirs are created based on the assumptions above: 
     Water phase governing equation: 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         ∇ 
                         
                           · 
                           
                             ( 
                             
                               
                                 ρ 
                                 w 
                               
                                
                               
                                 λ 
                                 w 
                               
                                
                               
                                 ∇ 
                                 
                                   p 
                                   w 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           [ 
                           
                             φ 
                              
                             
                               ( 
                               
                                 
                                   ρ 
                                   w 
                                 
                                  
                                 
                                   S 
                                   w 
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                       + 
                       
                         q 
                         w 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   304 
                   ) 
                 
               
             
           
         
       
     
     Hydrocarbon components governing equation in oil and gas phase 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         ∇ 
                         
                           · 
                           
                             
                               ∑ 
                               
                                 
                                   k 
                                   = 
                                   o 
                                 
                                 , 
                                 g 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   ρ 
                                   k 
                                 
                                  
                                 
                                   C 
                                   ik 
                                 
                                  
                                 
                                   λ 
                                   k 
                                 
                                  
                                 
                                   ∇ 
                                   
                                     p 
                                     k 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           [ 
                           
                             φ 
                              
                             
                               
                                 ∑ 
                                 
                                   
                                     k 
                                     = 
                                     o 
                                   
                                   , 
                                   g 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ρ 
                                     k 
                                   
                                    
                                   
                                     S 
                                     k 
                                   
                                    
                                   
                                     C 
                                     k 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             
                               k 
                               = 
                               o 
                             
                             , 
                             g 
                           
                         
                          
                         
                           q 
                           ik 
                         
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   … 
                   , 
                   
                     n 
                     c 
                   
                 
               
               
                 
                   ( 
                   305 
                   ) 
                 
               
             
           
         
       
     
     Auxiliary equations with saturation and capillary pressure: 
     Saturation equation 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           
                             k 
                             = 
                             o 
                           
                           , 
                           g 
                           , 
                           w 
                         
                       
                        
                       
                         S 
                         k 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   306 
                   ) 
                 
               
             
           
         
       
     
     Capillary pressure equation at α-β phase interface 
         p   cαβ ( S   w )= p   α   p   β ,( x,t )∈Ω×(0, t   max ],α= o,g,w;β=o,g,w   (307)
 
     Phase equilibrium equation: 
     The equilibrium constant of hydrocarbon component i in oil and gas phase: 
         K   iog   =C   io   /C   ig   ,i= 1, . . . , n   c   (308)
 
     The total mole percent of hydrocarbon component i in oil and gas phase: 
     
       
         
           
             
               
                 
                   
                     
                       Z 
                       i 
                     
                     = 
                     
                       
                         ∑ 
                         
                           
                             k 
                             = 
                             o 
                           
                           , 
                           g 
                         
                       
                        
                       
                         
                           C 
                           ik 
                         
                          
                         
                           n 
                           k 
                           % 
                         
                       
                     
                   
                   , 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   … 
                   , 
                   
                     n 
                     c 
                   
                 
               
               
                 
                   ( 
                   309 
                   ) 
                 
               
             
           
         
       
     
     Mole percentage normalization condition of hydrocarbon components in each phase: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           n 
                           c 
                         
                       
                        
                       
                           
                       
                        
                       
                         C 
                         ik 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     k 
                     = 
                     o 
                   
                   , 
                   g 
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
     Normalizing conditions for ratio of moles of each phase to the whole system: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           n 
                           c 
                         
                       
                        
                       
                           
                       
                        
                       
                         n 
                         k 
                         % 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     k 
                     = 
                     o 
                   
                   , 
                   g 
                 
               
               
                 
                   ( 
                   311 
                   ) 
                 
               
             
           
         
       
     
     Boundary condition equation: 
     Boundary condition equation of each phase 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               k 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             k 
                           
                         
                         + 
                         
                           
                             c 
                             
                               k 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             k 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 k 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         k 
                       
                        
                       
                         ( 
                         
                           ϰ 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         ϰ 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     k 
                     = 
                     0 
                   
                   , 
                   g 
                   , 
                   w 
                 
               
               
                 
                   ( 
                   312 
                   ) 
                 
               
             
           
         
       
     
     Initial saturation equation: 
     Initial saturation equation of each phase 
         S   k ( x, 0)= l   k ( x ), x∈Ω,k=o,g,w   (313)
 
     Initial pressure equation: 
     Initial pressure equation of each phase 
         p   k ( x, 0)=ƒ k ( x ), x∈Ω,k=o,g,w   (314)
 
     Variable symbols description in multi-component multi-phase flow simulation analysis equations: ϕ is porosity, which is the function of average pressure  p , %; n c  is the total component number, dimensionless; λ k  is the generalized mobility of k-phase, m 2 /(Pa·s); ρ k  is density of k-phase, kg/m 3 ; S k  is the k-phase saturation, t is time, s; t max  is the total time, s; p k  is the k-phase pressure, Pa; l k  is the k-phase initial saturation distribution function; q ik  is the source/sink term of hydrocarbon component i in k-phase, kg/(m 3 ·s); ƒ k  is the k-phase initial pressure distribution function, Pa; K iog  is the hydrocarbon equilibrium constant of component i in α and β phase, dimensionless; C ik  is the mole percent of hydrocarbon component i in k-phase, dimensionless; Z i  is the total mole percent of component i, dimensionless;   is the ratio of k-phase to the whole system, dimensionless; g k  is the boundary functions of k-phase on reservoir boundary, dimensionless; p cαβ  is the capillary pressure at α-β phase interface, Pa; Ω is reservoir space; ∂Ω is reservoir boundary including internal boundary and outer boundary; c k,1  is the pressure term coefficients of k-phase on reservoir boundary, 1/Pa; c k,2  is the k-phase coefficients of directional derivative term along the normal direction outside the boundary on reservoir boundary condition, s/m; n ∂Ω  is outer normal direction on reservoir boundary, m; ∇ is the Hamilton operator; ∂ is the partial derivative sign. 
     Embodiment 15 
     This embodiment provides a multi-component carbon dioxide driven flow simulation analysis method for complex reservoirs. The model assumptions are as follows: 
     (1) Oil/gas/water three phases fluids are in the reservoir. The fluid flow is a isothermal process,
 
(2) The phase equilibrium of hydrocarbon components in oil and gas phase is completed in a flash,
 
(3) Rocks are slightly compressible.
 
(4) Heterogeneity and anisotropy of the reservoirs are considered.
 
(5) Carbon dioxide diffusion is considered.
 
     The multi-component carbon dioxide drive flow simulation analysis for complex reservoirs are created based on the assumptions above: 
     
       
         
           
             
               
                 
                   
                     
                       
                         [ 
                         
                           ∇ 
                           
                             · 
                             
                               
                                 ∑ 
                                 
                                   
                                     k 
                                     = 
                                     o 
                                   
                                   , 
                                   g 
                                   , 
                                   w 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   
                                     ρ 
                                     k 
                                   
                                    
                                   
                                     C 
                                     ik 
                                   
                                    
                                   
                                     λ 
                                     k 
                                   
                                    
                                   
                                     ∇ 
                                     
                                       p 
                                       k 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       + 
                       
                         [ 
                         
                           ∇ 
                           
                             · 
                             
                               
                                 ∑ 
                                 
                                   
                                     k 
                                     = 
                                     o 
                                   
                                   , 
                                   g 
                                   , 
                                   w 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   φ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ρ 
                                     k 
                                   
                                    
                                   
                                     S 
                                     k 
                                   
                                    
                                   
                                     D 
                                     ik 
                                   
                                    
                                   
                                     ∇ 
                                     
                                       C 
                                       ik 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           [ 
                           
                             φ 
                              
                             
                               
                                 ∑ 
                                 
                                   
                                     k 
                                     = 
                                     o 
                                   
                                   , 
                                   g 
                                   , 
                                   w 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   
                                     ρ 
                                     k 
                                   
                                    
                                   
                                     S 
                                     k 
                                   
                                    
                                   
                                     C 
                                     ik 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             
                               k 
                               = 
                               o 
                             
                             , 
                             g 
                             , 
                             w 
                           
                         
                          
                         
                             
                         
                          
                         
                           q 
                           ik 
                         
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         ϰ 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     n 
                     c 
                   
                 
               
               
                 
                   ( 
                   315 
                   ) 
                 
               
             
           
         
       
     
     Auxiliary equations with saturation and capillary pressure: 
     Saturation equation 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           
                             k 
                             = 
                             o 
                           
                           , 
                           g 
                           , 
                           w 
                         
                       
                        
                       
                         S 
                         k 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     
                       ( 
                       
                         ϰ 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   316 
                   ) 
                 
               
             
           
         
       
     
     Capillary pressure equation at α-β phase interface 
         p   cαβ ( S   w )= p   a - p   β ,( x,t )∈Ω×(0, t   max ],α= o,g,w;β=o,g,w   (317)
 
     Phase equilibrium equation: 
     The equilibrium constant of component i in α and β phase: 
         K   iαβ   =C   iα   /C   iβ   ,α=o,g,w;β=o,g,w;i= 1, . . . , n   c   (318)
 
     The total mole percent of component i: 
     
       
         
           
             
               
                 
                   
                     
                       Z 
                       i 
                     
                     = 
                     
                       
                         ∑ 
                         
                           
                             k 
                             = 
                             o 
                           
                           , 
                           g 
                           , 
                           w 
                         
                       
                        
                       
                         
                           C 
                           lk 
                         
                          
                         
                           n 
                           k 
                           % 
                         
                       
                     
                   
                   , 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     n 
                     c 
                   
                 
               
               
                 
                   ( 
                   319 
                   ) 
                 
               
             
           
         
       
     
     Mole percentage normalization condition of components in each phase: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           n 
                           c 
                         
                       
                        
                       
                           
                       
                        
                       
                         C 
                         ik 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     k 
                     = 
                     o 
                   
                   , 
                   g 
                   , 
                   w 
                 
               
               
                 
                   ( 
                   320 
                   ) 
                 
               
             
           
         
       
     
     Normalizing conditions for ratio of moles of each phase to the whole system: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           n 
                           c 
                         
                       
                        
                       
                         n 
                         k 
                         % 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     k 
                     = 
                     o 
                   
                   , 
                   g 
                   , 
                   w 
                 
               
               
                 
                   ( 
                   321 
                   ) 
                 
               
             
           
         
       
     
     Boundary condition equation: 
     Boundary condition equation of each phase 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               k 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             k 
                           
                         
                         + 
                         
                           
                             c 
                             
                               k 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             k 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 k 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         k 
                       
                        
                       
                         ( 
                         
                           ϰ 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         ϰ 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     k 
                     = 
                     o 
                   
                   , 
                   g 
                   , 
                   w 
                 
               
               
                 
                   ( 
                   322 
                   ) 
                 
               
             
           
         
       
     
     Initial saturation equation: 
     Initial saturation equation of each phase 
         S   k ( x, 0)= l   k ( x ), x∈Ω,k=o,g,w   (323)
 
     Initial pressure equation: 
     Initial pressure equation of each phase 
         p   k ( x, 0)=ƒ k ( x ), x∈Ω,k=o,g,w   (324)
 
     Variable symbols description in multi-component multi-phase flow simulation analysis equations: ϕ is porosity, which is the function of average pressure  p , %; n c  is the total component number, dimensionless; λ k  is the generalized mobility of k-phase, m 2 /(Pa·s); ρ k  is density of k-phase, kg/m 3 ; V i  is adsorption amount of component i, dimensionless; S k  is the k-phase saturation; t is time, s; t max  is the total time, s; p k  is the k-phase pressure, Pa; l k  is the k-phase initial saturation distribution function; q ik  is the source/sink term of component i in k-phase, kg/(m 3 ·s); ƒ k  is the k-phase initial pressure distribution function, Pa; K ioβ  is the equilibrium constant of component i in α and β phase, dimensionless; C ik  is the mole percent of component i in k-phase, dimensionless; D ik  is the diffusion coefficient of component i in k-phase, m 2 /s; Z i  is the total mole percent of component i, dimensionless;   is the ratio of k-phase to the whole system, dimensionless; g k  is the boundary functions of k-phase on reservoir boundary, dimensionless; p cαβ  is the capillary pressure at α-β phase interface, Pa; Ω is reservoir space; ∂Ω is reservoir boundary including internal boundary and outer boundary; c k,1  is the pressure term coefficients of k-phase on reservoir boundary, 1/Pa; c k,2  is the k-phase coefficients of directional derivative term along the normal direction outside the boundary on reservoir boundary condition, s/m; n ∂Ω  is outer normal direction on reservoir boundary, m; ∇ is the Hamilton operator; ∂ is the partial derivative sign. 
     Embodiment 16 
     This embodiment provides a multi-component polymer driven flow simulation analysis method for complex reservoirs. The model assumptions are as follows: 
     (1) Oil/gas/water three phases fluids are in the reservoir. The fluid flow is a isothermal process,
 
(2) The fluid flow phase equilibrium is completed in a flash,
 
(3) Rocks are slightly compressible.
 
(4) Heterogeneity and anisotropy of the reservoirs are considered.
 
(5) Diffusion and adsorption of polymer are considered.
 
     The multi-component polymer driven flow simulation analysis for complex reservoirs are created based on the assumptions above: 
     
       
         
           
             
               
                 
                   
                     
                       
                         [ 
                         
                           ∇ 
                           
                             · 
                             
                               
                                 ∑ 
                                 
                                   
                                     k 
                                     = 
                                     o 
                                   
                                   , 
                                   g 
                                   , 
                                   w 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   
                                     ρ 
                                     k 
                                   
                                    
                                   
                                     C 
                                     ik 
                                   
                                    
                                   
                                     λ 
                                     k 
                                   
                                    
                                   
                                     ∇ 
                                     
                                       p 
                                       k 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       + 
                       
                         [ 
                         
                           ∇ 
                           
                             · 
                             
                               
                                 ∑ 
                                 
                                   
                                     k 
                                     = 
                                     o 
                                   
                                   , 
                                   g 
                                   , 
                                   w 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   φ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ρ 
                                     k 
                                   
                                    
                                   
                                     S 
                                     k 
                                   
                                    
                                   
                                     D 
                                     ik 
                                   
                                    
                                   
                                     ∇ 
                                     
                                       C 
                                       ik 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           [ 
                           
                             φ 
                              
                             
                               
                                 ∑ 
                                 
                                   
                                     k 
                                     = 
                                     o 
                                   
                                   , 
                                   g 
                                   , 
                                   w 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   
                                     ρ 
                                     k 
                                   
                                    
                                   
                                     S 
                                     k 
                                   
                                    
                                   
                                     C 
                                     ik 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           [ 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 φ 
                               
                               ) 
                             
                              
                             
                               ρ 
                               r 
                             
                              
                             
                               V 
                               i 
                             
                           
                           ] 
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             
                               k 
                               = 
                               o 
                             
                             , 
                             g 
                             , 
                             w 
                           
                         
                          
                         
                             
                         
                          
                         
                           q 
                           ik 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       
                         ϰ 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     n 
                     c 
                   
                 
               
               
                 
                   ( 
                   325 
                   ) 
                 
               
             
           
         
       
     
     Auxiliary equations with saturation and capillary pressure: 
     Saturation equation 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           
                             k 
                             = 
                             o 
                           
                           , 
                           g 
                           , 
                           w 
                         
                       
                        
                       
                         S 
                         k 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     
                       ( 
                       
                         ϰ 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   326 
                   ) 
                 
               
             
           
         
       
     
     Capillary pressure equation at α-β phase interface 
         p   cαβ ( S   w )=ρ α -ρ β ,( x,t )∈Ω×(0, t   max ],α= o,g,w;β=o,g,w   (327)
 
     Phase equilibrium equation: 
     The equilibrium constant of component i in α and β phase: 
         K   iαβ   =C   iα   /C   iβ   ,α=o,g,w,β=o,g,w,i=, . . . ,n   c   (328)
 
     The total mole percent of component i: 
     
       
         
           
             
               
                 
                   
                     
                       Z 
                       i 
                     
                     = 
                     
                       
                         ∑ 
                         
                           
                             k 
                             = 
                             o 
                           
                           , 
                           g 
                           , 
                           w 
                         
                         
                             
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           c 
                           lk 
                         
                          
                         
                           n 
                           k 
                           % 
                         
                       
                     
                   
                   , 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     n 
                     c 
                   
                 
               
               
                 
                   ( 
                   329 
                   ) 
                 
               
             
           
         
       
     
     Mole percentage normalization condition of components in each phase: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           n 
                           c 
                         
                       
                        
                       
                           
                       
                        
                       
                         C 
                         ik 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     k 
                     = 
                     o 
                   
                   , 
                   g 
                   , 
                   w 
                 
               
               
                 
                   ( 
                   330 
                   ) 
                 
               
             
           
         
       
     
     Normalizing conditions for ratio of moles of each phase to the whole system: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           n 
                           c 
                         
                       
                        
                       
                           
                       
                        
                       
                         n 
                         k 
                         % 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     k 
                     = 
                     o 
                   
                   , 
                   g 
                   , 
                   w 
                 
               
               
                 
                   ( 
                   331 
                   ) 
                 
               
             
           
         
       
     
     Boundary condition equation 
     Boundary condition equation of each phase 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               k 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             k 
                           
                         
                         + 
                         
                           
                             c 
                             
                               k 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             k 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 k 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         k 
                       
                        
                       
                         ( 
                         
                           ϰ 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         ϰ 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     k 
                     = 
                     o 
                   
                   , 
                   g 
                   , 
                   w 
                 
               
               
                 
                   ( 
                   332 
                   ) 
                 
               
             
           
         
       
         
         
           
             Initial saturation equations: 
             Initial saturation equation of each phase 
           
         
       
    
         S   k ( x, 0)= l   k ( x ), x∈Ω,k=o,g,w   (333)
         Initial pressure equation:   Initial pressure equation of each phase       

         p   k ( x, 0)=ƒ k ( x ), x∈Ω,k=o,g,w   (334)
 
     Variable symbols description in multi-component multi-phase flow simulation analysis equations: ϕ is porosity, which is the function of average pressure  p , %; n c  is the total component number, dimensionless; λ k  is the generalized mobility of k-phase, m 2 /(Pa·s); ρ k , ρ r  are densities of k-phase and rock, kg/m 3 ; V i  is adsorption amount of component i, dimensionless; S k  is the k-phase saturation; t is time, s; t max  is the total time, s; p k  is the k-phase pressure, Pa; l k  is the k-phase initial saturation distribution function; q ik  is the source/sink term of component i in k-phase, kg/(m 3 ·s); ƒ k  is the k-phase initial pressure distribution function, Pa; K iαβ  is the equilibrium constant of component i in α and β phase, dimensionless; C ik  is the mole percent of component i in k-phase, dimensionless; D ik  is the diffusion coefficient of component i in k-phase, m 2 /s; Z i  is the total mole percent of component i, dimensionless;   is the ratio of k-phase to the whole system, dimensionless; g k  is the boundary functions of k-phase on reservoir boundary, dimensionless; p cαβ  is the capillary pressure at α-β phase interface, Pa; Ω is reservoir space; ∂Ω is reservoir boundary including internal boundary and outer boundary; c k,1  is the pressure term coefficients of k-phase on reservoir boundary, 1/Pa; c k,2  is the k-phase coefficients of directional derivative term along the normal direction outside the boundary on reservoir boundary condition, s/m; n ∂Ω  is outer normal direction on reservoir boundary, m; ∇ is the Hamilton operator; ∂ is the partial derivative sign. 
     Embodiment 17 
     This embodiment provides a multi-component three-phase flow simulation analysis method considering non-isothermal process in complex reservoirs. 
     (1) Oil/gas/water three phases fluids are in the reservoir. The fluid flow is a non-isothermal process,
 
(2) The fluid flow phase equilibrium is is completed in a flash,
 
(3) Rocks are slightly compressible,
 
(4) Heterogeneity and anisotropy of the reservoirs are considered.
 
     The multi-component three-phase flow simulation analysis for complex reservoirs considering temperature change are created based on the assumptions above: 
     Component governing equations: 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         ∇ 
                         
                           · 
                           
                             
                               ∑ 
                               
                                 
                                   k 
                                   = 
                                   o 
                                 
                                 , 
                                 g 
                                 , 
                                 w 
                               
                             
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 
                                   ρ 
                                   k 
                                 
                                  
                                 
                                   C 
                                   ik 
                                 
                                  
                                 
                                   λ 
                                   k 
                                 
                                  
                                 
                                   ∇ 
                                   
                                     p 
                                     k 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         ∂ 
                         
                           ∂ 
                           t 
                         
                       
                        
                       
                         [ 
                         
                           φ 
                            
                           
                             
                               ∑ 
                               
                                 
                                   k 
                                   = 
                                   o 
                                 
                                 , 
                                 g 
                                 , 
                                 w 
                               
                             
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 
                                   ρ 
                                   k 
                                 
                                  
                                 
                                   S 
                                   k 
                                 
                                  
                                 
                                   C 
                                   ik 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       ( 
                       
                         ϰ 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     n 
                     c 
                   
                 
               
               
                 
                   ( 
                   335 
                   ) 
                 
               
             
           
         
       
     
     Energy conservation equation: 
     
       
         
           
             
               
                 
                   
                     
                       
                         [ 
                         
                           ∇ 
                           
                             · 
                             
                               
                                 ∑ 
                                 
                                   
                                     k 
                                     = 
                                     0 
                                   
                                   , 
                                   w 
                                   , 
                                   g 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     π 
                                     k 
                                   
                                    
                                   
                                     ρ 
                                     k 
                                   
                                    
                                   
                                     λ 
                                     k 
                                   
                                    
                                   
                                     ∇ 
                                     
                                       p 
                                       k 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       + 
                       
                         [ 
                         
                           
                             ∇ 
                             
                               · 
                               κ 
                             
                           
                            
                           
                             ∇ 
                             T 
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                           [ 
                           
                             φ 
                              
                             
                               
                                 ∑ 
                                 
                                   
                                     k 
                                     = 
                                     o 
                                   
                                   , 
                                   w 
                                   , 
                                   g 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ρ 
                                     k 
                                   
                                    
                                   
                                     S 
                                     k 
                                   
                                    
                                   
                                     U 
                                     k 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         
                           ∂ 
                           
                             ∂ 
                             t 
                           
                         
                         [ 
                         
                           
                             ( 
                             
                               1 
                               - 
                               φ 
                             
                             ) 
                           
                            
                           
                             ρ 
                             r 
                           
                            
                           
                             U 
                             r 
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   336 
                   ) 
                 
               
             
           
         
       
     
     Auxiliary equations with saturation and capillary pressure: 
     Saturation equation 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           
                             k 
                             = 
                             o 
                           
                           , 
                           w 
                           , 
                           g 
                         
                       
                        
                       
                         S 
                         k 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       Ω 
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           T 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   337 
                   ) 
                 
               
             
           
         
       
     
     Capillary pressure equation at α-β phase interface 
         p   cαβ ( S   w )= p   α   −p   β ,( x,t )∈Ω×(0, T ],α= o,w,g   p   ;β=o,w,g   (338)
 
     Phase equilibrium equation: 
     The equilibrium constant of component i in α and β phase: 
         K   iαβ   =C   iα   /C   iβ   ,α=o,w,g;β=o,w,g;i=, . . . ,n   c   (339)
 
     Mole percentage normalization condition of components in each phase: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           n 
                           c 
                         
                       
                        
                       
                         C 
                         ik 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     k 
                     = 
                     o 
                   
                   , 
                   w 
                   , 
                   g 
                 
               
               
                 
                   ( 
                   340 
                   ) 
                 
               
             
           
         
       
     
     The total mole percent of component i: 
     
       
         
           
             
               
                 
                   
                     
                       Z 
                       i 
                     
                     = 
                     
                       
                         ∑ 
                         
                           
                             k 
                             = 
                             o 
                           
                           , 
                           w 
                           , 
                           g 
                         
                       
                        
                       
                         
                           C 
                           ik 
                         
                          
                       
                     
                   
                   , 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     n 
                     c 
                   
                 
               
               
                 
                   ( 
                   341 
                   ) 
                 
               
             
           
         
       
     
     Normalizing conditions for ratio of moles of each phase to the whole system: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           n 
                           c 
                         
                       
                        
                     
                     = 
                     1 
                   
                   , 
                   
                     k 
                     = 
                     o 
                   
                   , 
                   w 
                   , 
                   g 
                 
               
               
                 
                   ( 
                   342 
                   ) 
                 
               
             
           
         
       
     
     Boundary condition equations: 
     Boundary condition equation of pressure for each phase 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             c 
                             
                               k 
                               , 
                               1 
                             
                           
                            
                           
                             p 
                             k 
                           
                         
                         + 
                         
                           
                             c 
                             
                               k 
                               , 
                               2 
                             
                           
                            
                           
                             λ 
                             k 
                           
                            
                           
                             
                               ∂ 
                               
                                 p 
                                 k 
                               
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         g 
                         k 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     k 
                     = 
                     o 
                   
                   , 
                   w 
                   , 
                   g 
                 
               
               
                 
                   ( 
                   343 
                   ) 
                 
               
             
           
         
       
     
     Boundary condition equation of temperature 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             d 
                             1 
                           
                            
                           T 
                         
                         + 
                         
                           
                             d 
                             2 
                           
                            
                           κ 
                            
                           
                             
                               ∂ 
                               T 
                             
                             
                               ∂ 
                               
                                 n 
                                 
                                   ∂ 
                                   Ω 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       w 
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                     ∈ 
                     
                       
                         ∂ 
                         Ω 
                       
                       × 
                       
                         ( 
                         
                           0 
                           , 
                           
                             t 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   344 
                   ) 
                 
               
             
           
         
       
     
     Initial saturation equation: 
     Initial saturation equation of each phase 
         S   k ( x, 0)= l   k ( x ), x∈Ω,k=o,w,g   (345)
 
     Initial pressure equation: 
     Initial pressure equation of each phase 
         p   k ( x, 0)=ƒ k ( x ), x∈Ω,k=o,w,g   (346)
 
     Initial temperature equation: 
         T ( x, 0)=τ( x ), x∈Ω   (347)
 
     Variable symbols description in thermal multi-component three-phase flow simulation analysis equations: ϕ is porosity, which is the function of average pressure  p , %; n c  is the total component number, dimensionless; λ k  is the generalized mobility of k-phase, m 2 /(Pa·s); ρ k , ρ r  are densities of k-phase and rock, kg/m 3 ; π k  is enthalpy of k phase, J/kg; U k , U r  are internal energy of k-phase and rock, J/kg; S k  is the k-phase saturation; t is time, s; t max  is the maximum of time, s; p k  is the k-phase pressure, Pa; l k  is the k-phase initial saturation distribution function; ƒ k  is the k-phase initial pressure distribution function, Pa; K iαβ  is the equilibrium constant of component i in α and β phase, dimensionless; C ik  is the mole percent of component i in k-phase, dimensionless; Z i  is the total mole percent of component i, dimensionless;   is the ratio of k-phase to the whole system, dimensionless; g k  is the boundary functions of k-phase on reservoir boundary, dimensionless; w is boundary function of temperature on reservoir boundary, dimensionless; τ is the initial temperature distribution function of reservoirs, K; p cαβ  is the capillary pressure at α-β phase interface, Pa; Ω is reservoir space; ∂Ω is reservoir boundary including internal boundary and outer boundary; c k,1  is the pressure term coefficients of k-phase on reservoir boundary, 1/Pa; c k,2  is the k-phase coefficients of derivative terms along the outer normal line on reservoir boundary condition, s/m; d 1  is the temperature term coefficient on reservoir boundary, 1/K; d 2  is temperature coefficients of directional derivative term along the normal direction outside the boundary on reservoir boundary condition, s·m 2 /J; κ is the thermal conductivity, J/(m·s·K); n ∂Ω  is outer normal direction on reservoir boundary, m; ∇ is the Hamilton operator; ∂ is the partial derivative sign. 
     Embodiment 18 
     This embodiment provides a transient temperature analysis method for single oil phase flow. 
     (1) oil single phase fluid in the reservoir, the fluid flow is a non-isothermal process,
 
(2) both fluids and rocks are slightly compressible,
 
(3) the same thickness in the reservoir,
 
(4) well is a vertical well, fully penetrating the entire formation thickness,
 
(5) there exists no heat transfer to over- and under-burden strata from the reservoir.
 
     The transient temperature analysis model for oil single-phase is created based on the assumptions above: 
     Governing equations: 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       r 
                     
                      
                     
                       
                         ∂ 
                         
                             
                         
                       
                       
                         ∂ 
                         r 
                       
                     
                      
                     
                       ( 
                       
                         r 
                          
                         
                           
                             ∂ 
                             p 
                           
                           
                             ∂ 
                             r 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         φ 
                          
                         
                             
                         
                          
                         
                           C 
                           i 
                         
                       
                       λ 
                     
                      
                     
                       
                         ∂ 
                         p 
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                 
               
               
                 
                   ( 
                   348 
                   ) 
                 
               
             
           
         
       
     
     Energy conservation equation: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         T 
                       
                       
                         ∂ 
                         t 
                       
                     
                     - 
                     
                       
                         C 
                         pR 
                       
                        
                       λ 
                        
                       
                         
                           ∂ 
                           p 
                         
                         
                           ∂ 
                           r 
                         
                       
                        
                       
                         
                           ∂ 
                           T 
                         
                         
                           ∂ 
                           r 
                         
                       
                     
                     - 
                     
                       
                         
                           α 
                           t 
                         
                         r 
                       
                        
                       
                         ∂ 
                         
                           ∂ 
                           r 
                         
                       
                        
                       
                         ( 
                         
                           r 
                            
                           
                             
                               ∂ 
                               T 
                             
                             
                               ∂ 
                               r 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       ϕ 
                       * 
                       
                         
                           ∂ 
                           p 
                         
                         
                           ∂ 
                           t 
                         
                       
                     
                     - 
                     
                       
                         ɛ 
                         JT 
                       
                        
                       
                         C 
                         pR 
                       
                        
                       λ 
                        
                       
                         
                           ∂ 
                           p 
                         
                         
                           ∂ 
                           r 
                         
                       
                        
                       
                         
                           ∂ 
                           p 
                         
                         
                           ∂ 
                           r 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   349 
                   ) 
                 
               
             
           
         
       
     
     Boundary condition equation of pressure 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               2 
                                
                               π 
                                
                               
                                   
                               
                                
                               hr 
                                
                               
                                   
                               
                                
                               λ 
                                
                               
                                 
                                   ∂ 
                                   p 
                                 
                                 
                                   ∂ 
                                   r 
                                 
                               
                             
                              
                             
                               | 
                               
                                 r 
                                 = 
                                 
                                   r 
                                   w 
                                 
                               
                             
                           
                           = 
                           
                             qB 
                             + 
                             
                               C 
                                
                               
                                 
                                   ∂ 
                                   
                                     p 
                                     w 
                                   
                                 
                                 
                                   ∂ 
                                   t 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             p 
                             w 
                           
                           = 
                           
                             
                               ( 
                               
                                 p 
                                 - 
                                 
                                   Sr 
                                    
                                   
                                     
                                       ∂ 
                                       p 
                                     
                                     
                                       ∂ 
                                       r 
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                               | 
                               
                                 r 
                                 = 
                                 
                                   r 
                                   w 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               lim 
                               
                                 r 
                                  
                                 ∞ 
                               
                             
                              
                             
                                 
                             
                              
                             p 
                           
                           = 
                           
                             p 
                             i 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   350 
                   ) 
                 
               
             
           
         
       
     
     Boundary condition equation of temperature 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               r 
                                
                               
                                 
                                   ∂ 
                                   T 
                                 
                                 
                                   ∂ 
                                   r 
                                 
                               
                             
                              
                             
                               | 
                               
                                 r 
                                 = 
                                 
                                   r 
                                   w 
                                 
                               
                             
                           
                           = 
                           0 
                         
                       
                     
                     
                       
                         
                           
                             
                               lim 
                               
                                 r 
                                  
                                 ∞ 
                               
                             
                              
                             
                                 
                             
                              
                             T 
                           
                           = 
                           
                             T 
                             i 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   351 
                   ) 
                 
               
             
           
         
       
     
     Initial pressure equation: 
         P|   t=0   =p   i   (352)
 
     Initial temperature equation: 
         T|   t=0   =T   i   (353)
 
     Model solution: 
     Defining the dimensionless pressure distribution function 
     
       
         
           
             
               
                 p 
                 D 
               
               = 
               
                 
                   2 
                    
                   πλ 
                    
                   
                       
                   
                    
                   
                     h 
                      
                     
                       ( 
                       
                         
                           p 
                           i 
                         
                         - 
                         p 
                       
                       ) 
                     
                   
                 
                 qB 
               
             
             , 
           
         
       
     
     dimensionless bottom-hole pressure function 
     
       
         
           
             
               
                 p 
                 wD 
               
               = 
               
                 
                   2 
                    
                   πλ 
                    
                   
                       
                   
                    
                   
                     h 
                      
                     
                       ( 
                       
                         
                           p 
                           i 
                         
                         - 
                         
                           p 
                           w 
                         
                       
                       ) 
                     
                   
                 
                 qB 
               
             
             , 
           
         
       
     
     dimension time 
     
       
         
           
             
               
                 t 
                 D 
               
               = 
               
                 
                   λ 
                    
                   
                       
                   
                    
                   t 
                 
                 
                   φ 
                    
                   
                       
                   
                    
                   
                     C 
                     t 
                   
                    
                   
                     r 
                     w 
                     2 
                   
                 
               
             
             , 
           
         
       
     
     dimensionless radial distance 
     
       
         
           
             
               
                 r 
                 D 
               
               = 
               
                 r 
                 
                   r 
                   w 
                 
               
             
             , 
           
         
       
     
     dimensionless wellbore storage coefficient 
     
       
         
           
             
               
                 C 
                 D 
               
               = 
               
                 C 
                 
                   2 
                    
                   
                     
                       πφ 
                        
                       C 
                     
                     t 
                   
                    
                   
                     hr 
                     w 
                     2 
                   
                 
               
             
             , 
           
         
       
     
     then convert Eqs.347, 349, 351 into dimensionless equations. 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       
                         r 
                         D 
                       
                     
                      
                     
                       ∂ 
                       
                         ∂ 
                         
                           r 
                           D 
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           r 
                           D 
                         
                          
                         
                           
                             ∂ 
                             
                               p 
                               D 
                             
                           
                           
                             ∂ 
                             
                               r 
                               D 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∂ 
                       
                         p 
                         D 
                       
                     
                     
                       ∂ 
                       
                         t 
                         D 
                       
                     
                   
                 
               
               
                 
                   ( 
                   354 
                   ) 
                 
               
             
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 r 
                                 D 
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     p 
                                     D 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     r 
                                     D 
                                   
                                 
                               
                             
                              
                             
                                
                               
                                 
                                   r 
                                   D 
                                 
                                 = 
                                 1 
                               
                             
                           
                           = 
                           
                             
                               - 
                               1 
                             
                             + 
                             
                               
                                 C 
                                 D 
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     p 
                                     wD 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     t 
                                     D 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             p 
                             wD 
                           
                           = 
                           
                             
                               ( 
                               
                                 
                                   p 
                                   D 
                                 
                                 - 
                                 
                                   
                                     Sr 
                                     D 
                                   
                                    
                                   
                                     
                                       ∂ 
                                       
                                         p 
                                         D 
                                       
                                     
                                     
                                       ∂ 
                                       
                                         r 
                                         D 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                                
                               
                                 
                                   r 
                                   D 
                                 
                                 = 
                                 1 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               lim 
                               
                                 
                                   r 
                                   D 
                                 
                                 → 
                                 ∞ 
                               
                             
                              
                             
                               p 
                               D 
                             
                           
                           = 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   355 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       p 
                       D 
                     
                      
                     
                        
                       
                         
                           t 
                           D 
                         
                         = 
                         0 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   356 
                   ) 
                 
               
             
           
         
       
     
     The Laplace space pressure distribution function can be obtained by conducting Laplace transformation to Eqs.353, 354, 355 based on to as 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       _ 
                     
                     D 
                   
                   = 
                   
                     
                       
                         K 
                         0 
                       
                        
                       
                         ( 
                         
                           
                             u 
                           
                            
                           
                             r 
                             D 
                           
                         
                         ) 
                       
                     
                     
                       u 
                        
                       
                         ( 
                         
                           
                             
                               u 
                             
                              
                             
                               C 
                               D 
                             
                              
                             
                               
                                 SuK 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   u 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               C 
                               D 
                             
                              
                             
                               
                                 uK 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   u 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               u 
                             
                              
                             
                               
                                 K 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   u 
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   357 
                   ) 
                 
               
             
           
         
       
     
     The pressure distribution function p D  in real space can be obtained through Laplace inverse transformation of Eq.356, then the pressure p D  is converted to dimensional form as 
     
       
         
           
             
               
                 
                   p 
                   = 
                   
                     
                       p 
                       i 
                     
                     - 
                     
                       
                         qB 
                         
                           2 
                            
                           
                             λ 
                              
                             h 
                           
                         
                       
                        
                       
                         p 
                         D 
                       
                     
                   
                 
               
               
                 
                   ( 
                   358 
                   ) 
                 
               
             
           
         
       
     
     Substitute Eq.357 into Eq. 348 to obtain the differential equation of temperature about time and radial distance, which cannot be solved by analytical solutions so that it need to be solved through numerical methods to obtain numerical solutions that satisfy solution conditions of Eqs.350 and 352. 
     Variable symbols description in single oil phase flow transient temperature analysis equations: ϕ is porosity, %; λ is the generalized mobility, m 2 /(Pa·s); ρ is the density, kg/m 3 ; t is time, s; t max  is the maximum of time, s; p is the pressure, Pa; T is the temperature, K; T i  is the initial temperature, K; q is he flow rate, m 3 /s; B is the volume factor, dimensionless; h is the reservoir thickness, m; p w  is the bottom-hole pressure, Pa; C t  is the total compressibility, 1/MPa; p i  is the initial pressure, Pa; ∇ is the Hamilton operator; ∂ is the partial derivative sign; C pR  is the ratio of volumetric heat capacity of a fluid to that of a saturated fluid rock, dimensionless; α t  is the thermal diffusion of saturated fluid rocks, m 2 /s; φ* is the effective heat expansion coefficient, K/Pa; ε JT  is the Joule-Thomson coefficient, K/Pa; C is the wellbore storage coefficient, m 3 /Pa; S is the skin factor, dimensionless; r w  is the wellbore radius, m; u is the Laplace variable corresponding to dimensionless time t D . 
     The embodiments given above are some typical examples of realizing this invention, but this invention is not limited to these examples. Any non-essential addition or substitution based on the technical features of the invention made by the technicians in the field all belong to the protection range of this invention.