Patent Publication Number: US-2011067857-A1

Title: Determining properties of a subterranean structure during hydraulic fracturing

Description:
BACKGROUND 
     Reservoir development is performed to produce fluids such as hydrocarbons, fresh water, and so forth, from the reservoir. Reservoir development includes drilling one or more wellbores into a subterranean formation to intersect the reservoir, and installing completion equipment in the wellbores to enable the extraction of fluids from the reservoir. Surface equipment is also provided to route or store the extracted fluids. 
     To enhance production of fluids from a subterranean reservoir, hydraulic fracturing can be employed. Hydraulic fracturing involves injecting a fluid at relatively high pressure through a wellbore into the reservoir. The injection pressure is chosen to be high enough to cause fracturing of the formation. The injection phase is followed by a shut-in phase (where the injection pressure is removed). The injection of fracturing fluids causes micro-seismic events to occur, which are also referred to as micro-earthquakes. Such micro-seismic events can be detected using seismic detectors. 
     Conventional techniques of studying reservoirs have not effectively employed available data associated with hydraulic fracturing of the reservoir to understand properties of the reservoir or characteristics of the hydraulic fracturing procedure. 
     SUMMARY 
     In general, according to an embodiment, a technique or mechanism is provided to determine, in real-time, properties of a reservoir during a hydraulic fracturing process. 
     Other or alternative features will become apparent from the following description, from the drawings, and from the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is an exemplary plot illustrating a seismic cloud produced during a hydraulic fracturing process. 
         FIGS. 2A-2B  are plots illustrating pressure build-up and pressure fall-off phases of a hydraulic fracturing process. 
         FIG. 3  is a schematic diagram of an exemplary arrangement that includes a subterranean formation and a reservoir in the subterranean formation, various sensors, and a computer that incorporates an embodiment. 
         FIG. 4  is a flow diagram of a process for performing analysis according to an embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     In the following description, numerous details are set forth to provide an understanding of the present invention. However, it will be understood by those skilled in the art that the present invention may be practiced without these details and that numerous variations or modifications from the described embodiments are possible. 
     A technique or mechanism according to some embodiments is provided to perform real-time determination of one or more properties of a subterranean structure, such as a reservoir in a subterranean formation, during a hydraulic fracturing process. Performing real-time determination of a property of a subterranean structure during a hydraulic fracturing process refers to making such determination while the hydraulic fracturing process is proceeding such that the determined property can be used to control the hydraulic fracturing process (such as to stop the hydraulic fracturing process or to change some characteristic of the hydraulic fracturing process, such as an injection rate, applied pressure, type of injected fluid, and so forth). 
     The ability to control the hydraulic fracturing process in real-time based on determination of properties of the subterranean structure allows for more efficient performance of the hydraulic fracturing process. For example, if the hydraulic fracturing has caused a reservoir to achieve target characteristics, then the hydraulic fracturing can be stopped, which would avoid unnecessary further hydraulic fracturing. 
     Hydraulic fracturing refers to application of a fluid into a wellbore at a relatively high pressure to cause the applied fluid to be communicated through perforations in the wellbore into the surrounding subterranean structure, where the applied fluid at high pressure is intended to cause fracturing of the subterranean structure. Fracturing of the subterranean structure refers to causing breaks to form in the subterranean structure, where fluid flow paths are provided as a result of the breaks to enhance flow of production fluids such as hydrocarbons, fresh water, or other fluids. Usually, the hydraulic fracturing is associated with an injection phase (where fracturing fluid is applied at high pressure, followed by a shut-in phase, where the injection of fluid is stopped and pressure is allowed to drop off). The injection phase is also referred to as a “build-up phase,” and the shut-in phase is also referred to as a “fall-off phase.” 
     Hydraulic fracturing causes micro-seismic events (also referred to as micro-earthquakes) to occur in the subterranean structure. It can be assumed that pore pressure diffusion is the primary mechanism that triggers micro-seismic events during hydraulic fracturing. Pore pressure refers to pressure of fluids within the pores of a subterranean structure. 
     Micro-seismic events are triggered only when the pore pressure field p(r,t) exceeds a threshold pressure field P T (r). Such micro-seismic events are usually exhibited in a space-time (r-t) plot, where r is the distance to the seismic event from a wellbore (the point of injection of the fracturing fluid), and t represents time. 
     An exemplary r-t plot is shown in  FIG. 1 . During the injection phase, the pore pressure field will continue to exceed the threshold pressure field, triggering a swarm of micro-seismic events resulting in the formation of a seismic cloud  100  in the r-t plot. The locus of r at the left periphery of the seismic cloud  100  is known as the triggering front  102 . The triggering front  102  depicts spatial demarcation of relaxed and unrelaxed pore pressure regions of the subterranean structure during the injection phase. Conversely, during the shut-in phase, the period of seismic quiescence, as the pore pressure field begins to recede towards the threshold pressure field, a locus of r, known as the back front  104 , will appear at the right periphery of the seismic cloud  100 . The back front  104  depicts the demarcation of relaxed and unrelaxed pore pressure regions of the formation during the shut-in phase. 
     The real-time determination of one or more properties of a subterranean structure according to some embodiments during the hydraulic fracturing process is based on various received measurement data, including data relating to micro-seismic events detected by micro-seismic detectors (e.g., geophones), pressure data, and fluid injection rate data. Pressure data can be collected by pressure sensors at the earth surface and/or downhole in a wellbore, while injection rate data can be collected by fluid rate sensors at the earth surface and/or downhole in the wellbore. 
     As examples, deduced properties regarding a subterranean structure (e.g., reservoir parameters) include storativity (ω), shape factor (α) and transmissivity (λ). Storativity is the parameter that relates fluid capacitance of the secondary (fracture) porosity to that of the combined system. Shape factor is a geometric parameter describing the distribution of a fracture network including anisotropic behavior in a heterogeneous region, and the shape factor is estimated with input from micro-seismic focal mechanism inversions. Transmissivity is the parameter governing flow between the fractures and primary matrix. 
     A technique according to an embodiment entails performing, in real-time, a constrained history matching of pressure build-up and fall-off data. The word “constrained” is used here to emphasize that during the process of history matching, reconstruction of the corresponding triggering and back fronts ( 102  and  104  in  FIG. 1 ) that envelop the evolving swarms of micro-seismic cloud is performed. The triggering front  102  is reconstructed while history matching the acquired pressure and rate data during the pressure build-up phase, and the back front  104  is reconstructed while history matching of the pressure data acquired during the pressure fall-off phase. 
       FIG. 2A  illustrates a p-r plot (plot of pressure p to distance r) corresponding to t-r plot  204  during the pressure build-up phase.  FIG. 2B  illustrates a p-r plot  206  and a corresponding t-r plot  208  during a fall-off phase. The figures show the evolution of the triggering front  210  and back front  212  during the build-up and fall-off phases, respectively. 
     Propagation of a hydraulic fracture is accompanied by creation of new fractures, where p(r,t)≧P T (r). During this process, pre-existing cracks in the reservoir are enhanced. In accordance with some embodiments, an analytic mathematical model that describes the pressure build-up during a variable rate fluid injection and the ensuing advancement of the fluid front followed by pressure fall-off during shut-in, in a dual-porosity, dual-permeability reservoir, is provided for real-time interpretation. The growth of the dominant hydraulic fracture is accounted by a time-dependent skin (a skin refers to a zone of reduced or enhanced permeability around a wellbore). The technique according to some embodiments determines key reservoir parameters that adequately describe flow behavior in a dual porosity reservoir, where the primary porosity φ m  is inter-granular and controlled by deposition and lithification, and the secondary porosity φ f  is controlled by fracturing and jointing. 
     Once the reservoir has been adequately parameterized, a semi-analytic simulator according to some embodiments is used to characterize dynamic flow behavior within the reservoir. One advantage of the simulator is that it quickly converges without gridding challenges or numerical instabilities. Other features of the simulator include one or more of the following:
         fracturing in the presence of other wells (vertical, horizontal and deviated) can be studied;   variable flow rates and bottom hole pressure (BHP) can be specified;   wellbore storage and skin can be included;   fractures (natural and induced) can be included;   non-darcy flow is possible;   closed boundary and aquifer support is provided;   multi-phase analysis is provided;   desorption is considered; and   automatic history matching is performed.       

     The analysis according to some embodiments can be performed by analysis software (e.g., analysis software  316  executable in a computer  314  as shown in  FIG. 3 ). As further shown in  FIG. 3 , the computer  314  includes a processor  318  on which the analysis software  316  is executable. The processor  318  is connected to storage media  320 , which can be implemented when one or more disk-based storage devices and/or one or more integrated circuit or semiconductor storage devices. The storage media  320  contains measurement data  322  collected by various sensors  308  and  310 . The storage media  320  also stores a model  324  that is used by techniques according to some embodiments. 
     The sensors  308  shown in  FIG. 3  are sensors deployed downhole in wellbores  306  that are drilled into a subterranean formation  302 . The sensors  310  are earth surface sensors deployed at the earth surface, such as part of wellhead equipment  312 . In other implementations, earth surface sensors  310  or downhole sensors  308  may be omitted. 
     The wellbores  306  intersect a reservoir  304  in the subterranean formation  302 . One of the wellbores  306  can be used to produce fluids from the reservoir  304 , while another one of the wellbores  306  can be used to inject fluids into the reservoir  304 , such as fracturing fluids used for fracturing the reservoir  304  as part of the hydraulic fracturing process. 
       FIG. 4  illustrates a workflow procedure according to an embodiment. A reservoir model is initialized (at  402 ). The reservoir model is initialized with approximations of various model parameters derived from nearby wells, where such approximations of model parameters are used as initial estimates that are input into the model. The model that is considered according to some embodiments is a model of a subterranean formation that includes at least one wellbore that is located in an infinite homogeneous isotropic medium of uniform thickness. In the model, it is assumed that the formation and fluid properties are independent of pressure, the fluids are of relatively small compressibility, and that gravity effects are negligible. 
     Reservoir parameters  401 A of the model that are initialized include dual porosity parameters including the shape factor (α), transmissivity (λ), and storativity (ω)). In addition, flow parameters  401 B for the model that are initialized include the reservoir permeability and skin. Fixed parameters  401 C for the model include reservoir thickness (h), porosity (φ), and pressure, volume, and temperature. The initial estimates for the various model parameters can be obtained from one or more of the following: well logs, formation micro-imager (FMI) data, sonic scanner data, nearby micro-seismic data, and so forth. 
     Next, after initializing (at  402 ) the model, during a pressure build-up phase of a hydraulic fracturing process in which fracturing fluid is injected, the pressure as a function of time and position, p(r,t), is computed (at  404 ). To compute p(r,t), real-time injection rate measurement data is acquired (at  403 ) at the treatment wellbore (the wellbore used to inject fluid) and used as an input. The injection rate is the rate of injection of the fracturing fluid for the hydraulic fracturing operation. The computed pressure includes a fracture pressure p fi  (pressure in fractures) and matrix pressure p mi , (pressure in the reservoir containing the fractures) that are calculated according to Eqs. 21 and 22 (below) during the early stages of the hydraulic fracturing process. The fracture pressure p fi  and matrix pressure p mi  are calculated according to Eqs. 10 and 11 (below) in subsequent stages of the hydraulic fracturing process. 
     The computed pressure is uncorrected for skin. The skin refers to a zone of reduced permeability around a wellbore. Skin can be caused by particles clogging up pores in the reservoir. Real-time wellbore pressure measurement data is acquired (at  405 ), and a skin calculator (which is part of the analysis software  316  of  FIG. 3 ) is used to calculate (at  406 ) the skin (according to Eq. 41 below) of the reservoir in real-time at predetermined intervals. 
     If the rate of skin decrease falls below a predetermined threshold, as determined at  408 , a notification can be provided to a well operator to allow the well operator to stop the fracturing job (at  410 ). Otherwise, the procedure proceeds back to re-perform tasks  404  and  406 . The rate of skin decrease falling below the predetermined threshold indicates that the hydraulic fracturing has assisted in increasing the permeability of the reservoir to an extent such that any further hydraulic fracturing may not substantially or effectively enhance further reduced skin. 
     Once the fracturing fluid injection is stopped (at  410 ), the pressure fall-off phase is started. The pressure fall-off phase is continued for some predetermined time, during which all real-time data measurement acquisitions are continued. The fall-off phase is ended (at  412 ) after the predetermined time period. At that point, recording of real-time measurement data is stopped (at  414 ), and historical data is stored (at  416 ), where the stored historical data includes measurement data collected during the pressure build-up phase and fall-off phase of the hydraulic fracturing process. 
     After ending of the pressure fall-off phase, the model is updated (at  418 ), which includes setting the improved permeability in the invaded zone to account for skin improvement due to hydraulic fracturing. Effectively, the updated model contains the effect of the hydraulic fracturing process that has been performed. 
     The pressure p(r,t) is then computed (at  420 ) again, by computing the fracture pressure p fi  according to Eq. 33 and the matrix pressure p mi  according to Eq. 34. The computation of the fracture pressure and matrix pressure uses the stored historical information ( 416 ) and the updated model. The re-computed fracture pressure p fi  and the matrix pressure p mi , now reflect the improved skin effect resulting from the hydraulic fracturing process. 
     Also, at  420 , while the pressure p(r,t) (including the fracture pressure p fi  and the matrix pressure p mi ) is being computed, the triggering front  102  is reconstructed as the calculated p(r,t) exceeds a micro-seismic event activation pressure threshold P T (r) that is essentially the upper bound on a pseudo-random pore pressure function. Similarly, the back front  104  is reconstructed based on declines of treatment well pressures below P T (r). 
     A parabolic expression, r tf =√{square root over (4πη ap t)}, for the triggering front  102 , is derived by considering the pore pressure perturbations induced by a point source, where η ap  is an apparent hydraulic diffusion coefficient associated with the fracturing process. This expression for the triggering front  102  is then used in conjunction with a volumetric balance to estimate fracture geometry parameters such as fracture width, lateral and vertical extent and fluid loss coefficient, a parameter associated with estimation of fluid loss from the fracture into the surrounding matrix. An expression, 
     
       
         
           
             
               
                 r 
                 bf 
               
               = 
               
                 
                   2 
                    
                   
                     η 
                      
                     
                       ( 
                       
                         
                           t 
                           
                             t 
                             0 
                           
                         
                         - 
                         1 
                       
                       ) 
                     
                   
                    
                   
                     ln 
                      
                     
                       ( 
                       
                         t 
                         
                           t 
                           - 
                           
                             t 
                             0 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             , 
           
         
       
     
     is derived for the back front  104  that develops during pore pressure relaxation after shutdown (during the pressure fall-off phase), where η is a pore pressure diffusivity constant associated with the established fracture-matrix system. 
     Non-linear regression is performed (at  422 ) to update the dual porosity and flow parameters, including transmissivity (λ), storativity (ω), shape factor (α), permeability, and skin. The updated parameters further characterize the updated reservoir model. The volumetric distribution of micro-seismic activity and the volume of fluid injected are used to update the storativity. Also, the algorithm aims to minimize the objective function that incorporates both wellbore pressure and the triggering and back front positions. 
     In performing the non-linear regression, the reconstructed triggering front and back front are matched to the trigger front and back front derived (at  426 ) based on real-time micro-seismic event locations ( 428 ). The trigger and back fronts derived based on the real-time micro-seismic event locations are considered the measured trigger and back fronts. The micro-seismic event locations ( 428 ) are based on seismic data acquired by seismic sensors that are able to measure micro-seismic events induced by the hydraulic fracturing process. 
     Next, in accordance with some embodiments, performance of a well and the reservoir can be predicted (at  430 ) using the updated reservoir model. This is accomplished by running a simulator (which can be part of the analysis software  316  of  FIG. 3 ) that uses the updated reservoir model (depicted as  324  in  FIG. 3 ). 
     The following provides further details regarding various parameters and calculations of pressure, skin, and other variables. 
     As noted above, storativity (w) is a parameter relating fluid capacitance of the secondary (fracture) porosity to that of the combined system. Classically it is defined as: 
     
       
         
           
             
               
                 
                   
                     ω 
                     = 
                     
                       
                         
                           φ 
                           f 
                         
                          
                         
                           c 
                           f 
                         
                       
                       
                         
                           
                             φ 
                             m 
                           
                            
                           
                             c 
                             m 
                           
                         
                         + 
                         
                           
                             φ 
                             f 
                           
                            
                           
                             c 
                             f 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
     where φ m =primary porosity (matrix), φ f =secondary porosity (e.g., due to fractures), and c m  and c f  are total compressibilities within the matrix and fracture, respectively. 
     The primary porosity φ m  will typically be determined from laboratory core analysis (analysis of core samples retrieved from the subterranean formation). Porosity due to fractures and joints φ f  can be estimated in the simplest case using the total injected fluid volume distributed in the reservoir and micro-seismic density, taking into account leakoff to the primary matrix. This is particularly easy when the primary matrix has a very low permeability (e.g., a gas shale), the injected fluid is incompressible (e.g., water), and leakoff from fractures to the matrix during injection can be considered negligible. Consider a blocked region full of detected and located micro-seismic events, then in each region of the blocked reservoir i, 
     
       
         
           
             
               
                 
                   
                     
                       φ 
                       fi 
                     
                     ≅ 
                     
                       
                         
                           n 
                           i 
                         
                          
                         
                           
                             ∫ 
                             0 
                             t 
                           
                            
                           
                             
                               q 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                              
                             
                                 
                             
                              
                             
                                
                               t 
                             
                           
                         
                       
                       
                         NV 
                         i 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
     where η i  is the number of micro-seismic events in block i, V i  is the volume of block i, N is the total number of micro-seismic events, and q(t) is the injection rate. It is also assumed that 
     
       
         
           
             
               
                 
                   
                     
                       c 
                       m 
                     
                     = 
                     
                       
                         c 
                         0 
                       
                       + 
                       
                         
                           
                             c 
                             p 
                           
                           + 
                           
                             
                               S 
                               wi 
                             
                              
                             
                               c 
                               w 
                             
                           
                         
                         
                           1 
                           - 
                           
                             S 
                             wi 
                           
                         
                       
                     
                   
                   , 
                   and 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     3 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       c 
                       f 
                     
                     = 
                     
                       c 
                       0 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
     where c 0  is the compressibility of flowing liquid, c p  is the effective pore compressibility, c w  is the compressibility of connate water, and S wi  represents the connate water saturation. 
     Alternatively, the micro-seismic waveforms may be used to define discrete planar fracture surfaces, and then the frequency content of the waveforms can be analyzed to estimate the equivalent radius of the fracture plane. The contributions of individual micro-seismic events to porosity can then be weighted according to the derived fracture area associated with each event. Alternatively, induced porosity may be derived from micro-seismic density weighted by seismic moment determined from the frequency analysis. 
     Waveforms may be inverted for the moment tensor associated with micro-seismic events. When a sufficient observation network (of sensors) is available, more reliable moment tensor solutions can be obtained with components representing double couple (pure shear), tension and compensated linear vector dipole (CLVD). In these cases, the relative amount of tensile to double couple can be used to further specify the opening of cracks, and the associated induced porosity. 
     Shape factor (α) is the geometric parameter describing the distribution of a fracture network including anisotropic behavior in a heterogeneous region. The shape factor reflects the geometry of the matrix elements and the shape factor controls the flow between the two porous regions. It generally allows specification of variable fracture spacing and/or width in different directions so it can be used to indicate the proper degree of anisotropy. 
     As the fracture network or mesh is formed, an interaction with natural fractures causes alternating jogs between almost pure shear fracture along pre-existing natural fractures and induced fractures oriented parallel to the direction of maximum horizontal stress. A large difference in angle between the natural fractures and max stress direction or low stress anisotropy may create more complex fracture networks with less preferential flow direction. One of three methods may be used to characterize the fracture network and estimate the shape factor from micro-seismic data. 
     The simplest estimation of shape factor may come from knowledge (length versus width) of the overall frac geometry coupled with other knowledge of preferred fluid propagation direction such as from stress anisotropy interpretation. 
     Another method that involves composite fault phase solutions exploits the observation that just a few characteristic fracture plane orientations typically exist within a hydraulic fracture network. In addition, the mechanism for most of the fracturing events appears to be almost purely double couple allowing the corresponding solutions to be fit to the aggregate data. The method involves extraction of the amplitudes and first motion polarities of the P, S-H and S-V arrivals, and then fitting the rations (e.g., S-H/P) to theoretical double couple solutions as a function of arrival angle (at the receivers). The final step in this method is to assign a characteristic fracture plane orientation to each of the micro-seismic events and compute an overall shape factor utilizing the event locations and orientations. The fracture plane areas, derived from conventional earthquake spectral analysis methods, may also be used in the shape factor calculation as it contributes to the characterization of the fracture network geometry. 
     Alternatively, a more advanced shape factor estimation may be derived from full moment sensor solutions for individual micro-seismic events when they are reliable (when the sensor network provides sufficient focal sphere coverage). This method then assigns a unique fracture plane location, orientation and area to each micro-seismic event in the characterization of the fracture network and calculation of shape factor. 
     Transmissivity (λ) is the parameter governing flow between the fractures and the primary matrix defined as, 
     
       
         
           
             
               
                 
                   
                     λ 
                     = 
                     
                       
                         α 
                          
                         
                             
                         
                          
                         
                           k 
                           m 
                         
                          
                         
                           a 
                           2 
                         
                       
                       
                         k 
                         e 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     5 
                   
                   ) 
                 
               
             
           
         
       
     
     where α is the shape factor (geometric parameter for heterogeneous region), k m  is the permeability of the primary matrix, a is the radius of the well, and k e  is the effective permeability of anisotropic medium. Transmissivity is deduced from the constrained history matching of the pressure build-up and fall-off data. 
     The physical model considered in this analysis includes a wellbore located in an infinite homogeneous isotropic medium of uniform thickness. The formation and fluid properties are independent of pressure, the fluids are of relatively small compressibility, and gravity effects are negligible. Quantities of fluid q(t) are continuously injected at r=a over the entire thickness of the reservoir and displaces the in-situ fluids in a piston-like manner, such that a uniform, immobile in-situ fluid saturation exists behind the advancing fluid front. The resulting pressure disturbance is left to diffuse through a semi-infinite homogeneous porous medium. After a specified period, the injection is terminated and the pressure is allowed to recede. The pressure from the diffusivity equation in a naturally fractured formation is described as follows: 
     For the invaded region (a&lt;r&lt;r f (t)), where r f (t) represents the advancing fluid front over time t of injected fluids: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           φ 
                           f 
                         
                          
                         
                           c 
                           f 
                         
                          
                         
                           
                             ∂ 
                             
                               p 
                               fi 
                             
                           
                           
                             ∂ 
                             t 
                           
                         
                       
                       + 
                       
                         
                           φ 
                           m 
                         
                          
                         
                           c 
                           m 
                         
                          
                         
                           
                             ∂ 
                             
                               p 
                               mi 
                             
                           
                           
                             ∂ 
                             t 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           k 
                           fi 
                         
                         
                           μ 
                           w 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 ∂ 
                                 2 
                               
                                
                               
                                 p 
                                 fi 
                               
                             
                             
                               ∂ 
                               
                                 r 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               1 
                               r 
                             
                              
                             
                               
                                 ∂ 
                                 
                                   p 
                                   fi 
                                 
                               
                               
                                 ∂ 
                                 r 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     6 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       φ 
                       m 
                     
                      
                     
                       c 
                       m 
                     
                      
                     
                       
                         ∂ 
                         
                           p 
                           mi 
                         
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                   = 
                   
                     α 
                      
                     
                       
                         k 
                         mi 
                       
                       
                         μ 
                         w 
                       
                     
                      
                     
                       
                         ( 
                         
                           
                             p 
                             fi 
                           
                           - 
                           
                             p 
                             mi 
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
     The invaded region is the region in which injected fluid extends. For the uninvaded region (r&gt;r f  (t)): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           φ 
                           f 
                         
                          
                         
                           c 
                           f 
                         
                          
                         
                           
                             ∂ 
                             
                               p 
                               fu 
                             
                           
                           
                             ∂ 
                             t 
                           
                         
                       
                       + 
                       
                         
                           φ 
                           m 
                         
                          
                         
                           c 
                           m 
                         
                          
                         
                           
                             ∂ 
                             
                               p 
                               mu 
                             
                           
                           
                             ∂ 
                             t 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           k 
                           fu 
                         
                         
                           μ 
                           o 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 ∂ 
                                 2 
                               
                                
                               
                                 p 
                                 fu 
                               
                             
                             
                               ∂ 
                               
                                 r 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               1 
                               r 
                             
                              
                             
                               
                                 ∂ 
                                 
                                   p 
                                   fu 
                                 
                               
                               
                                 ∂ 
                                 r 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     8 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       φ 
                       m 
                     
                      
                     
                       c 
                       m 
                     
                      
                     
                       
                         ∂ 
                         
                           p 
                           mu 
                         
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                   = 
                   
                     α 
                      
                     
                       
                         k 
                         mu 
                       
                       
                         μ 
                         o 
                       
                     
                      
                     
                       
                         ( 
                         
                           
                             p 
                             fu 
                           
                           - 
                           
                             p 
                             mu 
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
     The uninvaded region is the region of the reservoir that the injected fluid does not reach. In these equations, p fi  and p mi  represent the fracture and matrix pressures in the flooded zones whereas, p fu  and p mu , denote the fracture and matrix pressures in the oil zone. Here, matrix diffusivities of the invaded and uninvaded regions are defined, respectively, by 
     
       
         
           
             
               
                 η 
                 mi 
               
               = 
               
                 
                   
                     k 
                     mi 
                   
                   
                     
                       μ 
                       w 
                     
                      
                     
                       φ 
                       m 
                     
                      
                     
                       c 
                       m 
                     
                   
                 
                 = 
                 
                   
                     
                       k 
                       m 
                       α 
                     
                      
                     
                       
                         k 
                         rw 
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             S 
                             or 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       μ 
                       w 
                     
                      
                     
                       φ 
                       m 
                     
                      
                     
                       c 
                       m 
                     
                   
                 
               
             
             , 
             
                 
             
              
             and 
           
         
       
       
         
           
             
                 
             
              
             
               
                 η 
                 mu 
               
               = 
               
                 
                   
                     k 
                     mu 
                   
                   
                     
                       μ 
                       o 
                     
                      
                     
                       φ 
                       m 
                     
                      
                     
                       c 
                       m 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         k 
                         m 
                         α 
                       
                        
                       
                         
                           k 
                           ro 
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               S 
                               
                                 ω 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         μ 
                         o 
                       
                        
                       
                         φ 
                         m 
                       
                        
                       
                         c 
                         m 
                       
                     
                   
                   . 
                 
               
             
           
         
       
     
     Similarly, η fi  and η fu  denote the fracture diffusivities of the invaded and uninvaded regions defined respectively by 
     
       
         
           
             
               η 
               fi 
             
             = 
             
               
                 
                   k 
                   
                     f 
                      
                     i 
                   
                 
                 
                   
                     μ 
                     w 
                   
                    
                   
                     φ 
                     f 
                   
                    
                   
                     c 
                     f 
                   
                 
               
               = 
               
                 
                   
                     
                       k 
                       fi 
                       α 
                     
                      
                     
                       
                         k 
                         rw 
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             S 
                             or 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       μ 
                       w 
                     
                      
                     
                       φ 
                       f 
                     
                      
                     
                       c 
                       f 
                     
                   
                 
                  
                 
                     
                 
                  
                 and 
               
             
           
         
       
       
         
           
             
               η 
               fu 
             
             = 
             
               
                 
                   k 
                   fu 
                 
                 
                   
                     μ 
                     o 
                   
                    
                   
                     φ 
                     f 
                   
                    
                   
                     c 
                     f 
                   
                 
               
               = 
               
                 
                   
                     
                       k 
                       fu 
                       α 
                     
                      
                     
                       
                         k 
                         ro 
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             S 
                             
                               ω 
                                
                               
                                   
                               
                                
                               i 
                             
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       μ 
                       o 
                     
                      
                     
                       φ 
                       f 
                     
                      
                     
                       c 
                       f 
                     
                   
                 
                 . 
               
             
           
         
       
     
     Note that the fracture absolute permeability in the invaded zone, k fi   a , is different from that of the uninvaded zone, k fu   a , in order to model the variable mechanical skin. 
     The boundary conditions are at 
     
       
         
           
             
               r 
               = 
               a 
             
             , 
             
               
                 
                   ∂ 
                   
                     
                       p 
                       f 
                     
                      
                     
                       ( 
                       
                         a 
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 
                   ∂ 
                   r 
                 
               
               = 
               
                 
                   - 
                   
                     ( 
                     
                       
                         u 
                         w 
                       
                       
                         k 
                         ft 
                       
                     
                     ) 
                   
                 
                  
                 
                   q 
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
               
             
             , 
           
         
       
     
     and at the moving interface r=r f (t), p fi {r f (t),t}=p fu {r f (t),t}, p mi {r f (t),t}=p mu {r f (t),t} and 
     
       
         
           
             
               
                 [ 
                 
                   
                     
                       ∂ 
                       
                         p 
                         fi 
                       
                     
                      
                     
                       { 
                       
                         
                           
                             r 
                             f 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                         , 
                         t 
                       
                       } 
                     
                   
                   
                     ∂ 
                     r 
                   
                 
                 ] 
               
                
               
                 ( 
                 
                   
                     μ 
                     w 
                   
                   
                     k 
                     fi 
                   
                 
                 ) 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       ∂ 
                       
                         p 
                         fu 
                       
                     
                      
                     
                       { 
                       
                         
                           
                             r 
                             f 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                         , 
                         t 
                       
                       } 
                     
                   
                   
                     ∂ 
                     r 
                   
                 
                 ] 
               
                
               
                 
                   ( 
                   
                     
                       u 
                       o 
                     
                     
                       k 
                       fu 
                     
                   
                   ) 
                 
                 . 
               
             
           
         
       
     
     Here i and u denote the invaded and uninvaded regions. The initial condition p f (r,0)=p m (r,0)=φ(r)=p I . The above solves the pressure build-up phase of the problem during injection. Fluid injection is terminated at t=t 0 . The radial pressure profile at t=t 0 , obtained from the pressure build up solution is used as the initial condition to solve the pressure fall-off problem. The invaded and uninvaded region solutions for the pressure build-up and fall-off phases are given below. 
     The following describes solutions during the pressure build-up phase for the invaded region. The general solutions for the fracture and matrix pressure, with the exception of very early times, are given by 
     
       
         
           
             
               
                 
                   
                     p 
                     fi 
                   
                   = 
                   
                     
                       2 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 f 
                                 2 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             - 
                             
                               a 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 
                                   k 
                                   mi 
                                 
                                  
                                 
                                   η 
                                   fi 
                                 
                               
                               
                                 
                                   k 
                                   fi 
                                 
                                  
                                 
                                   η 
                                   mi 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         ∫ 
                         0 
                         t 
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 A 
                                 f 
                               
                                
                               
                                 ( 
                                 τ 
                                 ) 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     k 
                                     mi 
                                   
                                    
                                   
                                     η 
                                     fi 
                                   
                                 
                                 
                                   
                                     k 
                                     fi 
                                   
                                    
                                   
                                     η 
                                     mi 
                                   
                                 
                               
                                
                               
                                 
                                   A 
                                   m 
                                 
                                  
                                 
                                   ( 
                                   τ 
                                   ) 
                                 
                               
                             
                              
                             
                                 
                             
                             + 
                             
                               
                                 
                                   
                                     k 
                                     mi 
                                   
                                    
                                   
                                     η 
                                     fi 
                                   
                                 
                                 
                                   
                                     k 
                                     fi 
                                   
                                    
                                   
                                     η 
                                     mi 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       A 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       τ 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       A 
                                       m 
                                     
                                      
                                     
                                       ( 
                                       τ 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     
                                       
                                         αη 
                                         mi 
                                       
                                        
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           
                                             
                                               
                                                 k 
                                                 mi 
                                               
                                                
                                               
                                                 η 
                                                 fi 
                                               
                                             
                                             
                                               
                                                 k 
                                                 fi 
                                               
                                                
                                               
                                                 η 
                                                 mi 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       τ 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             τ 
                             ++ 
                           
                         
                          
                         
                           
                             π 
                             2 
                           
                           2 
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 
                                   ξ 
                                   n 
                                   2 
                                 
                                  
                                 
                                   J 
                                   1 
                                   2 
                                 
                                  
                                 
                                   { 
                                   
                                     
                                       ξ 
                                       n 
                                     
                                      
                                     
                                       
                                         r 
                                         f 
                                       
                                        
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                   
                                   } 
                                 
                                  
                                 
                                   
                                     κ 
                                     n 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         ξ 
                                         n 
                                       
                                        
                                       r 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 [ 
                                 
                                   
                                     
                                       J 
                                       1 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         a 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       J 
                                       1 
                                       2 
                                     
                                      
                                     
                                       { 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             t 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                 
                                 ] 
                               
                             
                              
                             
                                
                               
                                 - 
                                 
                                   
                                     ϑ 
                                     nn 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                             
                              
                             
                                 
                               
                                 [ 
                                 
                                   
                                     ∫ 
                                     0 
                                     t 
                                   
                                    
                                   
                                     
                                        
                                       
                                         - 
                                         
                                           
                                             ϑ 
                                             nn 
                                           
                                            
                                           
                                             ( 
                                             τ 
                                             ) 
                                           
                                         
                                       
                                     
                                      
                                     
                                         
                                       
                                           
                                       
                                        
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 B 
                                                 f 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ξ 
                                                     n 
                                                   
                                                   , 
                                                   τ 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               
                                                 cosh 
                                                  
                                                 
                                                   ( 
                                                   
                                                     α 
                                                      
                                                     
                                                       
                                                         
                                                           
                                                             k 
                                                             mi 
                                                           
                                                           
                                                             k 
                                                             fi 
                                                           
                                                         
                                                          
                                                         
                                                           η 
                                                           mi 
                                                         
                                                          
                                                         
                                                           η 
                                                           fi 
                                                         
                                                       
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         t 
                                                         - 
                                                         τ 
                                                       
                                                       ) 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               ++ 
                                             
                                              
                                             
                                               
                                                 
                                                   
                                                     k 
                                                     mi 
                                                   
                                                    
                                                   
                                                     η 
                                                     fi 
                                                   
                                                 
                                                 
                                                   
                                                     k 
                                                     fi 
                                                   
                                                    
                                                   
                                                     η 
                                                     mi 
                                                   
                                                 
                                               
                                             
                                              
                                             
                                               sinh 
                                                
                                               
                                                 ( 
                                                 
                                                   α 
                                                    
                                                   
                                                     
                                                       
                                                         
                                                           k 
                                                           mi 
                                                         
                                                         
                                                           k 
                                                           fi 
                                                         
                                                       
                                                        
                                                       
                                                         η 
                                                         mi 
                                                       
                                                        
                                                       
                                                         η 
                                                         fi 
                                                       
                                                     
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       t 
                                                       - 
                                                       τ 
                                                     
                                                     ) 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                                 
                                               
                                                 [ 
                                                 
                                                   
                                                     
                                                       B 
                                                       m 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           ξ 
                                                           n 
                                                         
                                                         , 
                                                         τ 
                                                       
                                                       ) 
                                                     
                                                   
                                                   + 
                                                   
                                                     
                                                       ( 
                                                       
                                                         
                                                           α 
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             ξ 
                                                             n 
                                                             2 
                                                             
                                                             + 
                                                             
                                                             α 
                                                              
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         - 
                                                         
                                                           αη 
                                                           mi 
                                                         
                                                       
                                                       ) 
                                                     
                                                      
                                                     
                                                       
                                                         
                                                           p 
                                                           _ 
                                                         
                                                         mi 
                                                         
                                                           ( 
                                                           
                                                             k 
                                                             - 
                                                             1 
                                                           
                                                           ) 
                                                         
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           
                                                             ξ 
                                                             n 
                                                           
                                                           , 
                                                           τ 
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                 
                                                 ] 
                                               
                                               ) 
                                             
                                              
                                             
                                                
                                               τ 
                                             
                                           
                                           ] 
                                         
                                         + 
                                         
                                           pI 
                                            
                                           
                                             
 
                                           
                                            
                                           and 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     10 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     mi 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       2 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 f 
                                 2 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             - 
                             
                               a 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 
                                   k 
                                   mi 
                                 
                                  
                                 
                                   η 
                                   fi 
                                 
                               
                               
                                 
                                   k 
                                   fi 
                                 
                                  
                                 
                                   η 
                                   mi 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         ∫ 
                         0 
                         t 
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 A 
                                 f 
                               
                                
                               
                                 ( 
                                 τ 
                                 ) 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     k 
                                     mi 
                                   
                                    
                                   
                                     η 
                                     fi 
                                   
                                 
                                 
                                   
                                     k 
                                     fi 
                                   
                                    
                                   
                                     η 
                                     mi 
                                   
                                 
                               
                                
                               
                                 
                                   A 
                                   m 
                                 
                                  
                                 
                                   ( 
                                   τ 
                                   ) 
                                 
                               
                             
                              
                             
                                 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       A 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       τ 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       A 
                                       m 
                                     
                                      
                                     
                                       ( 
                                       τ 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     
                                       
                                         αη 
                                         mi 
                                       
                                        
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           
                                             
                                               
                                                 k 
                                                 mi 
                                               
                                                
                                               
                                                 η 
                                                 fi 
                                               
                                             
                                             
                                               
                                                 k 
                                                 fi 
                                               
                                                
                                               
                                                 η 
                                                 mi 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       τ 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             τ 
                             ++ 
                           
                         
                          
                         
                           
                             π 
                             2 
                           
                           2 
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 
                                   ξ 
                                   n 
                                   2 
                                 
                                  
                                 
                                   J 
                                   1 
                                   2 
                                 
                                  
                                 
                                   { 
                                   
                                     
                                       ξ 
                                       n 
                                     
                                      
                                     
                                       
                                         r 
                                         f 
                                       
                                        
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                   
                                   } 
                                 
                                  
                                 
                                   
                                     κ 
                                     n 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         ξ 
                                         n 
                                       
                                        
                                       r 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 [ 
                                 
                                   
                                     
                                       J 
                                       1 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         a 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       J 
                                       1 
                                       2 
                                     
                                      
                                     
                                       { 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             t 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                 
                                 ] 
                               
                             
                              
                             
                                
                               
                                 - 
                                 
                                   
                                     ϑ 
                                     nn 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                             
                              
                             
                                 
                               
                                 [ 
                                 
                                   
                                     ∫ 
                                     0 
                                     t 
                                   
                                    
                                   
                                     
                                        
                                       
                                         - 
                                         
                                           
                                             ϑ 
                                             nn 
                                           
                                            
                                           
                                             ( 
                                             τ 
                                             ) 
                                           
                                         
                                       
                                     
                                      
                                     
                                         
                                       
                                           
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     k 
                                                     fi 
                                                   
                                                    
                                                   
                                                     η 
                                                     mi 
                                                   
                                                 
                                                 
                                                   
                                                     k 
                                                     mi 
                                                   
                                                    
                                                   
                                                     η 
                                                     fi 
                                                   
                                                 
                                               
                                             
                                              
                                             
                                               
                                                 B 
                                                 f 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ξ 
                                                     n 
                                                   
                                                   , 
                                                   τ 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               sinh 
                                               ( 
                                               
                                                 α 
                                                  
                                                 
                                                   
                                                     
                                                       
                                                         k 
                                                         mi 
                                                       
                                                       
                                                         k 
                                                         fi 
                                                       
                                                     
                                                      
                                                     
                                                       η 
                                                       mi 
                                                     
                                                      
                                                     
                                                       η 
                                                       fi 
                                                     
                                                   
                                                 
                                                  
                                                 
                                                   
                                                       
                                                     
                                                       ( 
                                                       
                                                         t 
                                                         - 
                                                         τ 
                                                       
                                                       ) 
                                                     
                                                     ) 
                                                   
                                                   ++ 
                                                 
                                                  
                                                 
                                                   cosh 
                                                    
                                                   
                                                     ( 
                                                     
                                                       α 
                                                        
                                                       
                                                         
                                                           
                                                             
                                                             k 
                                                             mi 
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                           
                                                            
                                                           
                                                             η 
                                                             mi 
                                                           
                                                            
                                                           
                                                             η 
                                                             fi 
                                                           
                                                         
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           t 
                                                           - 
                                                           τ 
                                                         
                                                         ) 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                  
                                                 
                                                     
                                                   
                                                     [ 
                                                     
                                                       
                                                         
                                                           B 
                                                           m 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             ξ 
                                                             n 
                                                             
                                                             , 
                                                             τ 
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                       
                                                         
                                                           ( 
                                                           
                                                             
                                                             α 
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             ξ 
                                                             n 
                                                             2 
                                                             
                                                             + 
                                                             
                                                             α 
                                                              
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             αη 
                                                             mi 
                                                             
                                                           
                                                           ) 
                                                         
                                                          
                                                         
                                                           
                                                             
                                                             p 
                                                             _ 
                                                             
                                                             mi 
                                                             
                                                             ( 
                                                             
                                                             k 
                                                             - 
                                                             1 
                                                             
                                                             ) 
                                                             
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             ξ 
                                                             n 
                                                             
                                                             , 
                                                             τ 
                                                             
                                                             ) 
                                                           
                                                         
                                                       
                                                     
                                                     ] 
                                                   
                                                   ) 
                                                 
                                                  
                                                 
                                                    
                                                   τ 
                                                 
                                               
                                               ] 
                                             
                                           
                                           + 
                                           pI 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     11 
                   
                   ) 
                 
               
             
           
         
       
     
     where the subscript i denotes the invaded region a≦r≦r f (t) and 
     
       
         
           
             
               
                 
                   
                     
                       
                         A 
                         f 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             q 
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                             p 
                             f 
                           
                            
                           
                             { 
                             
                               
                                 r 
                                 f 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             } 
                           
                         
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           h 
                            
                           
                               
                           
                            
                           
                             φ 
                              
                             
                               ( 
                               
                                 
                                   S 
                                   w 
                                 
                                 - 
                                 
                                   S 
                                   wi 
                                 
                               
                               ) 
                             
                           
                         
                       
                       + 
                       
                         
                           1 
                           
                             
                               φ 
                               f 
                             
                              
                             
                               c 
                               f 
                             
                           
                         
                          
                         
                           { 
                           
                             
                               aq 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             - 
                             
                               
                                 
                                   r 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                                
                               
                                 
                                   ψ 
                                   
                                     r 
                                     f 
                                   
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     12 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         A 
                         m 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           q 
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                          
                         
                           p 
                           m 
                         
                          
                         
                           { 
                           
                             
                               r 
                               f 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           } 
                         
                       
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         h 
                          
                         
                             
                         
                          
                         
                           φ 
                            
                           
                             ( 
                             
                               
                                 S 
                                 w 
                               
                               - 
                               
                                 S 
                                 wi 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     13 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         B 
                         f 
                       
                        
                       
                         ( 
                         
                           
                             ξ 
                             n 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             q 
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                             κ 
                             n 
                           
                            
                           
                             { 
                             
                               
                                 ξ 
                                 n 
                               
                                
                               
                                 
                                   r 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             } 
                           
                            
                           
                             p 
                             f 
                           
                            
                           
                             { 
                             
                               
                                 r 
                                 f 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             } 
                           
                         
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           h 
                            
                           
                               
                           
                            
                           
                             φ 
                              
                             
                               ( 
                               
                                 
                                   S 
                                   w 
                                 
                                 - 
                                 
                                   S 
                                   wi 
                                 
                               
                               ) 
                             
                           
                         
                       
                       + 
                       
                         
                           
                             2 
                             q 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                         
                           
                             πφ 
                             f 
                           
                            
                           
                             c 
                             f 
                           
                            
                           
                             ξ 
                             n 
                           
                         
                       
                       - 
                       
                         
                           2 
                            
                           
                               
                           
                            
                           
                             
                               J 
                               1 
                             
                              
                             
                               ( 
                               
                                 
                                   ξ 
                                   n 
                                 
                                  
                                 a 
                               
                               ) 
                             
                           
                            
                           
                             
                               ψ 
                               
                                 r 
                                 f 
                               
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         
                           
                             πφ 
                             f 
                           
                            
                           
                             c 
                             f 
                           
                            
                           
                             ξ 
                             n 
                           
                            
                           
                             J 
                             1 
                           
                            
                           
                             { 
                             
                               
                                 ξ 
                                 n 
                               
                                
                               
                                 
                                   r 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     14 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         B 
                         m 
                       
                        
                       
                         ( 
                         
                           
                             ξ 
                             n 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           q 
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                          
                         
                           κ 
                           n 
                         
                          
                         
                           { 
                           
                             
                               ξ 
                               n 
                             
                              
                             
                               
                                 r 
                                 f 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           } 
                         
                          
                         
                           p 
                           m 
                         
                          
                         
                           { 
                           
                             
                               r 
                               f 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           } 
                         
                       
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         h 
                          
                         
                             
                         
                          
                         
                           φ 
                            
                           
                             ( 
                             
                               
                                 S 
                                 w 
                               
                               - 
                               
                                 S 
                                 wi 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     15 
                   
                   ) 
                 
               
             
           
         
       
     
     where K n (ξ n r)=Y 0 (ξ n r)J 1 (ξ n a)−J 0 (ξ n r)Y 1 (ξ n a). The set of eigenvalues ξ n , which are time dependent, are the positive roots of the transcendental equation J 1 (ξ n a)Y 1 {ξ n r f (t)}−Y 1 (ξ n a)J 1 {ξ n r f (t)}=0, n=1,2 . . . : 
     
       
         
           
             
               
                 
                   
                     
                       
                         ϑ 
                         nn 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         0 
                         t 
                       
                        
                       
                         
                           
                             ϖ 
                             nn 
                             f 
                           
                            
                           
                             ( 
                             τ 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           τ 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     16 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ϖ 
                         pn 
                         f 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             η 
                             f 
                           
                            
                           
                             ( 
                             
                               
                                 ξ 
                                 n 
                                 2 
                               
                               + 
                               
                                 α 
                                  
                                 
                                   
                                     k 
                                     mi 
                                   
                                   
                                     k 
                                     fi 
                                   
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           δ 
                           p 
                           n 
                         
                       
                       - 
                       
                         Ω 
                          
                         
                           ( 
                           
                             
                               ξ 
                               n 
                             
                             , 
                             
                               ξ 
                               p 
                             
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     17 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ϖ 
                         pn 
                         m 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           αη 
                           mi 
                         
                          
                         
                           δ 
                           p 
                           n 
                         
                       
                       - 
                       
                         Ω 
                          
                         
                           ( 
                           
                             
                               ξ 
                               n 
                             
                             , 
                             
                               ξ 
                               p 
                             
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       where 
                        
                       
                           
                       
                        
                       
                         δ 
                         p 
                         n 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             0 
                           
                           
                             
                               p 
                               ≠ 
                               n 
                             
                           
                         
                         
                           
                             1 
                           
                           
                             
                               p 
                               = 
                               n 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     18 
                   
                   ) 
                 
               
             
           
         
       
     
     is the Kronecker delta function. Also, 
     
       
         
           
             
               
                 
                   
                     Ω 
                      
                     
                       ( 
                       
                         
                           ξ 
                           n 
                         
                         , 
                         
                           ξ 
                           p 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           π 
                           2 
                         
                          
                         
                           ξ 
                           p 
                           2 
                         
                          
                         
                           J 
                           1 
                           2 
                         
                          
                         
                           { 
                           
                             
                               ξ 
                               p 
                             
                              
                             
                               
                                 r 
                                 f 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           } 
                         
                       
                       
                         2 
                          
                         
                           { 
                           
                             
                               
                                 J 
                                 1 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ξ 
                                     p 
                                   
                                    
                                   a 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 J 
                                 1 
                                 2 
                               
                                
                               
                                 { 
                                 
                                   
                                     ξ 
                                     p 
                                   
                                    
                                   
                                     
                                       r 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                 
                                 } 
                               
                             
                           
                           } 
                         
                       
                     
                      
                     
                       
                         ∫ 
                         a 
                         
                           
                             r 
                             f 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                        
                       
                         r 
                          
                         
                             
                         
                          
                         
                           
                             κ 
                             p 
                           
                            
                           
                             ( 
                             
                               
                                 ξ 
                                 p 
                               
                                
                               r 
                             
                             ) 
                           
                         
                          
                         
                           
                             ∂ 
                             
                               
                                 κ 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ξ 
                                     n 
                                   
                                    
                                   r 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             t 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                              
                             r 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     19 
                   
                   ) 
                 
               
             
           
         
       
     
     The advancing fluid front is given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         r 
                         f 
                         2 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         a 
                         2 
                       
                       + 
                       
                         
                           
                             ∫ 
                             0 
                             t 
                           
                            
                           
                             
                               q 
                                
                               
                                 ( 
                                 τ 
                                 ) 
                               
                             
                              
                             
                                
                               τ 
                             
                           
                         
                         
                           π 
                            
                           
                               
                           
                            
                           h 
                            
                           
                               
                           
                            
                           
                             φ 
                              
                             
                               ( 
                               
                                 
                                   S 
                                   w 
                                 
                                 - 
                                 
                                   S 
                                   wi 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     20 
                   
                   ) 
                 
               
             
           
         
       
     
     where h is the thickness of the reservoir, S w  is the water saturation and S ωi  the initial water saturation. Note that an iterative procedure is performed to evaluate the fracture and matrix pressures, where k represents the iteration counter. 
     During very early times when the fluid front of the injected fluids is still close to the wellbore a good iterative approximation of the fractures and matrix pressures p fi  and p mi  can be made. The solutions are given by 
     
       
         
           
             
               
                 
                   
                     p 
                     fi 
                     
                       ( 
                       j 
                       ) 
                     
                   
                   = 
                   
                     
                       2 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 f 
                                 2 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             - 
                             
                               a 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 
                                   k 
                                   m 
                                 
                                 , 
                                 
                                   η 
                                   fi 
                                 
                               
                               
                                 
                                   k 
                                   f 
                                 
                                 , 
                                 
                                   η 
                                   mi 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     × 
                     × 
                     
                       
                         ∫ 
                         0 
                         t 
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 A 
                                 f 
                               
                                
                               
                                 ( 
                                 τ 
                                 ) 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     k 
                                     mi 
                                   
                                    
                                   
                                     η 
                                     fi 
                                   
                                 
                                 
                                   
                                     k 
                                     fi 
                                   
                                    
                                   
                                     η 
                                     mi 
                                   
                                 
                               
                                
                               
                                 
                                   A 
                                   m 
                                 
                                  
                                 
                                   ( 
                                   τ 
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     k 
                                     mi 
                                   
                                    
                                   
                                     η 
                                     fi 
                                   
                                 
                                 
                                   
                                     k 
                                     fi 
                                   
                                    
                                   
                                     η 
                                     mi 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       A 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       τ 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       A 
                                       m 
                                     
                                      
                                     
                                       ( 
                                       τ 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                  
                                 
                                   
                                     
                                       αη 
                                       mi 
                                     
                                      
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           
                                             
                                               k 
                                               mi 
                                             
                                              
                                             
                                               η 
                                               fi 
                                             
                                           
                                           
                                             
                                               k 
                                               fi 
                                             
                                              
                                             
                                               η 
                                               mi 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       l 
                                       - 
                                       τ 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             τ 
                             ++ 
                           
                         
                          
                         
                           
                             π 
                             2 
                           
                           2 
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                             
                               
                                 
                                   ξ 
                                   n 
                                   2 
                                 
                                  
                                 
                                   J 
                                   1 
                                   2 
                                 
                                  
                                 
                                   { 
                                   
                                     
                                       ξ 
                                       n 
                                     
                                      
                                     
                                       
                                         r 
                                         f 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                   } 
                                 
                                  
                                 
                                   
                                      
                                     n 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         ξ 
                                         n 
                                       
                                        
                                       r 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 [ 
                                 
                                   
                                     
                                       J 
                                       1 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         a 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       J 
                                       1 
                                       2 
                                     
                                      
                                     
                                       { 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                 
                                 ] 
                               
                             
                              
                             
                                 
                               
                                 [ 
                                 
                                   
                                     ∫ 
                                     0 
                                     t 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         cosh 
                                          
                                         
                                           ( 
                                           
                                             α 
                                              
                                             
                                               
                                                 
                                                   
                                                     k 
                                                     mi 
                                                   
                                                   
                                                     k 
                                                     fi 
                                                   
                                                 
                                                  
                                                 
                                                   η 
                                                   mi 
                                                 
                                                  
                                                 
                                                   η 
                                                   fi 
                                                 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 - 
                                                 τ 
                                               
                                               ) 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                           
                                         
                                           
                                             
                                               [ 
                                               
                                                 
                                                   
                                                     B 
                                                     f 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         ξ 
                                                         n 
                                                       
                                                       , 
                                                       τ 
                                                     
                                                     ) 
                                                   
                                                 
                                                 - 
                                                 
                                                   
                                                     C 
                                                     f 
                                                     
                                                       ( 
                                                       
                                                         j 
                                                         - 
                                                         1 
                                                       
                                                       ) 
                                                     
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         ξ 
                                                         n 
                                                       
                                                       , 
                                                       τ 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ] 
                                             
                                             ++ 
                                           
                                            
                                           
                                             
                                               
                                                 
                                                   k 
                                                   mi 
                                                 
                                                  
                                                 
                                                   η 
                                                   fi 
                                                 
                                               
                                               
                                                 
                                                   k 
                                                   fi 
                                                 
                                                  
                                                 
                                                   η 
                                                   mi 
                                                 
                                               
                                             
                                           
                                            
                                           sinh 
                                            
                                           
                                               
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       α 
                                                        
                                                       
                                                         
                                                           
                                                             
                                                             k 
                                                             mi 
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                           
                                                            
                                                           
                                                             η 
                                                             mi 
                                                           
                                                            
                                                           
                                                             η 
                                                             fi 
                                                           
                                                         
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           t 
                                                           - 
                                                           τ 
                                                         
                                                         ) 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   
                                                       
                                                     
                                                       
                                                         [ 
                                                         
                                                           
                                                             
                                                             B 
                                                             m 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             ξ 
                                                             n 
                                                             
                                                             , 
                                                             τ 
                                                             
                                                             ) 
                                                             
                                                           
                                                           - 
                                                           
                                                             
                                                             C 
                                                             m 
                                                             
                                                             
                                                             ( 
                                                             
                                                             k 
                                                             - 
                                                             1 
                                                             
                                                             ) 
                                                             
                                                             / 
                                                             A 
                                                             
                                                             
                                                              
                                                             
                                                               
                                                             
                                                             
                                                             
                                                             ( 
                                                             
                                                             
                                                             ξ 
                                                             n 
                                                             
                                                             , 
                                                             τ 
                                                             
                                                             ) 
                                                             
                                                             ++ 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             α 
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             ξ 
                                                             n 
                                                             2 
                                                             
                                                             + 
                                                             
                                                             α 
                                                              
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             αη 
                                                             mi 
                                                             
                                                             
                                                             ) 
                                                             
                                                              
                                                             
                                                             
                                                             
                                                             p 
                                                             _ 
                                                             
                                                             mi 
                                                             
                                                             ( 
                                                             
                                                             k 
                                                             - 
                                                             1 
                                                             
                                                             ) 
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             ξ 
                                                             n 
                                                             
                                                             , 
                                                             τ 
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             ] 
                                                             
                                                           
                                                         
                                                         ) 
                                                       
                                                        
                                                       
                                                          
                                                         
                                                           - 
                                                           
                                                             
                                                             ϑ 
                                                             
                                                             n 
                                                             , 
                                                             n 
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             t 
                                                             ) 
                                                             
                                                           
                                                         
                                                       
                                                        
                                                       
                                                          
                                                         τ 
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 + 
                                                 
                                                   p 
                                                   I 
                                                 
                                               
                                               , 
                                               
                                                 
 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               and 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     21 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     mi 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       2 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 f 
                                 2 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             - 
                             
                               a 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 
                                   k 
                                   mi 
                                 
                                  
                                 
                                   η 
                                   fi 
                                 
                               
                               
                                 
                                   k 
                                   fi 
                                 
                                  
                                 
                                   η 
                                   mi 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         ∫ 
                         0 
                         t 
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 A 
                                 f 
                               
                                
                               
                                 ( 
                                 τ 
                                 ) 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     k 
                                     mi 
                                   
                                    
                                   
                                     η 
                                     fi 
                                   
                                 
                                 
                                   
                                     k 
                                     fi 
                                   
                                    
                                   
                                     η 
                                     mi 
                                   
                                 
                               
                                
                               
                                 
                                   A 
                                   m 
                                 
                                  
                                 
                                   ( 
                                   τ 
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       A 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       τ 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       A 
                                       m 
                                     
                                      
                                     
                                       ( 
                                       τ 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     
                                       
                                         αη 
                                         mi 
                                       
                                        
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           
                                             
                                               
                                                 k 
                                                 mi 
                                               
                                                
                                               
                                                 η 
                                                 fi 
                                               
                                             
                                             
                                               
                                                 k 
                                                 fi 
                                               
                                                
                                               
                                                 η 
                                                 mi 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       r 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             τ 
                             ++ 
                           
                         
                          
                         
                           
                             π 
                             2 
                           
                           2 
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                             
                               
                                 
                                   ξ 
                                   n 
                                   2 
                                 
                                  
                                 
                                   J 
                                   1 
                                   2 
                                 
                                  
                                 
                                   { 
                                   
                                     
                                       ξ 
                                       n 
                                     
                                      
                                     
                                       
                                         r 
                                         f 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                   } 
                                 
                                  
                                 
                                   
                                      
                                     n 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         ξ 
                                         n 
                                       
                                        
                                       r 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 [ 
                                 
                                   
                                     
                                       J 
                                       1 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         a 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       J 
                                       1 
                                       2 
                                     
                                      
                                     
                                       { 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                 
                                 ] 
                               
                             
                             × 
                             × 
                             
                                 
                               
                                 [ 
                                 
                                   
                                     ∫ 
                                     0 
                                     t 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   k 
                                                   fi 
                                                 
                                                  
                                                 
                                                   η 
                                                   mi 
                                                 
                                               
                                               
                                                 
                                                   k 
                                                   mi 
                                                 
                                                  
                                                 
                                                   η 
                                                   fi 
                                                 
                                               
                                             
                                           
                                            
                                           
                                             sinh 
                                              
                                             
                                               ( 
                                               
                                                 α 
                                                  
                                                 
                                                   
                                                     
                                                       
                                                         k 
                                                         mi 
                                                       
                                                       
                                                         k 
                                                         fi 
                                                       
                                                     
                                                      
                                                     
                                                       η 
                                                       mi 
                                                     
                                                      
                                                     
                                                       η 
                                                       fi 
                                                     
                                                   
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     - 
                                                     τ 
                                                   
                                                   ) 
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                               
                                             
                                               
                                                 
                                                   [ 
                                                   
                                                     
                                                       
                                                         B 
                                                         f 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           
                                                             ξ 
                                                             n 
                                                           
                                                           , 
                                                           τ 
                                                         
                                                         ) 
                                                       
                                                     
                                                     - 
                                                     
                                                       
                                                         C 
                                                         f 
                                                         
                                                           ( 
                                                           
                                                             j 
                                                             - 
                                                             1 
                                                           
                                                           ) 
                                                         
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           
                                                             ξ 
                                                             n 
                                                           
                                                           , 
                                                           τ 
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                   ] 
                                                 
                                                 ++ 
                                               
                                                
                                               
                                                 cosh 
                                                  
                                                 
                                                   ( 
                                                   
                                                     α 
                                                      
                                                     
                                                       
                                                         
                                                           
                                                             k 
                                                             mi 
                                                           
                                                           
                                                             k 
                                                             fi 
                                                           
                                                         
                                                          
                                                         
                                                           η 
                                                           mi 
                                                         
                                                          
                                                         
                                                           η 
                                                           fi 
                                                         
                                                       
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         t 
                                                         - 
                                                         τ 
                                                       
                                                       ) 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                                
                                               
                                                   
                                                 
                                                   [ 
                                                   
                                                     
                                                       
                                                         B 
                                                         m 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           
                                                             ξ 
                                                             n 
                                                           
                                                           , 
                                                           τ 
                                                         
                                                         ) 
                                                       
                                                     
                                                     - 
                                                     
                                                       
                                                         
                                                           
                                                             C 
                                                             m 
                                                             
                                                             ( 
                                                             
                                                             k 
                                                             - 
                                                             1 
                                                             
                                                             ) 
                                                             
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             ξ 
                                                             n 
                                                             
                                                             , 
                                                             τ 
                                                             
                                                             ) 
                                                           
                                                         
                                                         ++ 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           
                                                             α 
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             ξ 
                                                             n 
                                                             2 
                                                             
                                                             + 
                                                             
                                                             α 
                                                              
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                           
                                                           - 
                                                           
                                                             αη 
                                                             mi 
                                                           
                                                         
                                                         ) 
                                                       
                                                        
                                                       
                                                         
                                                           
                                                             p 
                                                             _ 
                                                           
                                                           mi 
                                                           
                                                             ( 
                                                             
                                                             k 
                                                             - 
                                                             1 
                                                             
                                                             ) 
                                                           
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             ξ 
                                                             n 
                                                             
                                                             , 
                                                             τ 
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                   ] 
                                                 
                                                 ) 
                                               
                                                
                                               
                                                  
                                                 
                                                   - 
                                                   
                                                     
                                                       ϑ 
                                                       
                                                         n 
                                                         , 
                                                         n 
                                                       
                                                     
                                                      
                                                     
                                                       ( 
                                                       t 
                                                       ) 
                                                     
                                                   
                                                 
                                               
                                                
                                               
                                                  
                                                 τ 
                                               
                                             
                                             ] 
                                           
                                         
                                         + 
                                         
                                           p 
                                           I 
                                         
                                       
                                       , 
                                       
                                         
 
                                       
                                        
                                       
                                           
                                       
                                        
                                       where 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     22 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       C 
                       f 
                       
                         ( 
                         
                           j 
                           - 
                           1 
                         
                         ) 
                       
                     
                      
                     
                       ( 
                       
                         
                           ξ 
                           n 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             0 
                           
                           
                             
                               j 
                               = 
                               0 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     p 
                                     = 
                                     1 
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                     
                                       ϖ 
                                       pn 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       
                                         p 
                                         _ 
                                       
                                       fi 
                                       
                                         ( 
                                         
                                           j 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ξ 
                                           p 
                                         
                                         , 
                                         t 
                                       
                                       ) 
                                     
                                   
                                    
                                   p 
                                 
                               
                               ≠ 
                               n 
                             
                           
                           
                             
                               
                                 j 
                                 = 
                                 1 
                               
                               , 
                               2 
                               , 
                               … 
                                
                               
                                   
                               
                               , 
                             
                           
                         
                       
                        
                       
                         
 
                       
                        
                       
                           
                       
                        
                       and 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     23 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       C 
                       m 
                       
                         ( 
                         
                           k 
                           - 
                           1 
                         
                         ) 
                       
                     
                      
                     
                       ( 
                       
                         
                           ξ 
                           n 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           0 
                         
                         
                           
                             k 
                             = 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   p 
                                   = 
                                   1 
                                 
                                 ∞ 
                               
                                
                               
                                 
                                   
                                     ϖ 
                                     pn 
                                     m 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     
                                       p 
                                       _ 
                                     
                                     mi 
                                     
                                       ( 
                                       
                                         k 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         ξ 
                                         p 
                                       
                                       , 
                                       t 
                                     
                                     ) 
                                   
                                 
                                  
                                 p 
                               
                             
                             ≠ 
                             n 
                           
                         
                         
                           
                             
                               k 
                               = 
                               1 
                             
                             , 
                             2 
                             , 
                             … 
                              
                             
                                 
                             
                             , 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     24 
                   
                   ) 
                 
               
             
           
         
       
     
     Here, j and k are the iteration counters. 
     The following describes the solutions during pressure build-up for the uninvaded region. The general solutions for the fracture pressure and the matrix pressure are given respectively by: 
     
       
         
           
             
               
                 
                   
                     p 
                     fu 
                     
                       ( 
                       j 
                       ) 
                     
                   
                   = 
                   
                     
                       
                          
                         
                           
                             - 
                             
                               αη 
                               mu 
                             
                           
                            
                           t 
                         
                       
                        
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                          
                         
                           
                             
                               
                                 ϛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   ϛ 
                                    
                                   
                                       
                                   
                                    
                                   r 
                                 
                                 ) 
                               
                             
                             
                               [ 
                               
                                 
                                   
                                     J 
                                     1 
                                     2 
                                   
                                    
                                   
                                     { 
                                     
                                       ϛ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 + 
                                 
                                   
                                     Y 
                                     1 
                                     2 
                                   
                                    
                                   
                                     { 
                                     
                                       ϛ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                             
                               
                                 
                                   λ 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   ϛ 
                                   ) 
                                 
                               
                                
                               
                                 
                                   λ 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   ϛ 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   λ 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   ϛ 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   λ 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   ϛ 
                                   ) 
                                 
                               
                             
                           
                           × 
                           × 
                           
                             
                               ∫ 
                               0 
                               t 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                              
                                             
                                               
                                                 
                                                   λ 
                                                   2 
                                                 
                                                  
                                                 
                                                   ( 
                                                   ϛ 
                                                   ) 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   r 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           
                                             
                                               λ 
                                               1 
                                             
                                              
                                             
                                               ( 
                                               ϛ 
                                               ) 
                                             
                                           
                                         
                                         - 
                                         
                                           
                                              
                                             
                                               
                                                 
                                                   λ 
                                                   1 
                                                 
                                                  
                                                 
                                                   ( 
                                                   ϛ 
                                                   ) 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   τ 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           
                                             
                                               λ 
                                               2 
                                             
                                              
                                             
                                               ( 
                                               ϛ 
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             C 
                                             f 
                                             
                                               ( 
                                               j 
                                               ) 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               ϛ 
                                               , 
                                               τ 
                                             
                                             ) 
                                           
                                         
                                         + 
                                         
                                           
                                             Ξ 
                                             f 
                                             
                                               ( 
                                               
                                                 j 
                                                 - 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               ϛ 
                                               , 
                                               τ 
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     
                                       1 
                                       
                                         αη 
                                         mu 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                            
                                           
                                             
                                               
                                                 λ 
                                                 1 
                                               
                                                
                                               
                                                 ( 
                                                 ϛ 
                                                 ) 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 - 
                                                 τ 
                                               
                                               ) 
                                             
                                           
                                         
                                         - 
                                         
                                            
                                           
                                             
                                               
                                                 λ 
                                                 2 
                                               
                                                
                                               
                                                 ( 
                                                 ϛ 
                                                 ) 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 - 
                                                 τ 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             C 
                                             m 
                                             
                                               ( 
                                               k 
                                               ) 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               ϛ 
                                               , 
                                               τ 
                                             
                                             ) 
                                           
                                         
                                         + 
                                         
                                           
                                             Ξ 
                                             m 
                                             
                                               ( 
                                               
                                                 k 
                                                 - 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               ϛ 
                                               , 
                                               τ 
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                  
                                 τ 
                               
                                
                               
                                  
                                 ϛ 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       p 
                       I 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     25 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       p 
                       mu 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       
                         
                            
                           
                             
                               - 
                               
                                 αη 
                                 mu 
                               
                             
                              
                             t 
                           
                         
                          
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                            
                           
                             
                               
                                 
                                   ϛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     ϛ 
                                      
                                     
                                         
                                     
                                      
                                     r 
                                   
                                   ) 
                                 
                               
                               
                                 [ 
                                 
                                   
                                     
                                       J 
                                       1 
                                       2 
                                     
                                      
                                     
                                       { 
                                       
                                         ϛ 
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             t 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                   + 
                                   
                                     
                                       Y 
                                       1 
                                       2 
                                     
                                      
                                     
                                       { 
                                       
                                         ϛ 
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             t 
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                 
                                 ] 
                               
                             
                              
                             
                               
                                 
                                   
                                     λ 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     ϛ 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     λ 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     ϛ 
                                     ) 
                                   
                                 
                               
                               
                                 
                                   
                                     λ 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     ϛ 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     λ 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     ϛ 
                                     ) 
                                   
                                 
                               
                             
                             × 
                             × 
                             
                               
                                 ∫ 
                                 0 
                                 t 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                                
                                               
                                                 
                                                   
                                                     λ 
                                                     1 
                                                   
                                                    
                                                   
                                                     ( 
                                                     ϛ 
                                                     ) 
                                                   
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     - 
                                                     r 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             
                                               
                                                 λ 
                                                 1 
                                               
                                                
                                               
                                                 ( 
                                                 ϛ 
                                                 ) 
                                               
                                             
                                           
                                           - 
                                           
                                             
                                                
                                               
                                                 
                                                   
                                                     λ 
                                                     2 
                                                   
                                                    
                                                   
                                                     ( 
                                                     ϛ 
                                                     ) 
                                                   
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     - 
                                                     τ 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             
                                               
                                                 λ 
                                                 2 
                                               
                                                
                                               
                                                 ( 
                                                 ϛ 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               C 
                                               m 
                                               
                                                 ( 
                                                 k 
                                                 ) 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 ϛ 
                                                 , 
                                                 τ 
                                               
                                               ) 
                                             
                                           
                                           + 
                                           
                                             
                                               Ξ 
                                               m 
                                               
                                                 ( 
                                                 
                                                   k 
                                                   - 
                                                   1 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 ϛ 
                                                 , 
                                                 τ 
                                               
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     , 
                                     
                                       
                                         + 
                                         
                                           
                                             αη 
                                             mu 
                                           
                                           
                                             
                                               
                                                 λ 
                                                 1 
                                               
                                                
                                               
                                                 ( 
                                                 ϛ 
                                                 ) 
                                               
                                             
                                              
                                             
                                               
                                                 λ 
                                                 2 
                                               
                                                
                                               
                                                 ( 
                                                 ϛ 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                              
                                             
                                               
                                                 
                                                   λ 
                                                   2 
                                                 
                                                  
                                                 
                                                   ( 
                                                   ϛ 
                                                   ) 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   τ 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           - 
                                           
                                              
                                             
                                               
                                                 
                                                   λ 
                                                   1 
                                                 
                                                  
                                                 
                                                   ( 
                                                   ϛ 
                                                   ) 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   τ 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               C 
                                               f 
                                               
                                                 ( 
                                                 j 
                                                 ) 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 ϛ 
                                                 , 
                                                 τ 
                                               
                                               ) 
                                             
                                           
                                           + 
                                           
                                             
                                               Ξ 
                                               f 
                                               
                                                 ( 
                                                 
                                                   j 
                                                   - 
                                                   1 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 ϛ 
                                                 , 
                                                 τ 
                                               
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                    
                                   τ 
                                 
                                  
                                 
                                    
                                   ϛ 
                                 
                               
                             
                           
                         
                       
                       + 
                       
                         p 
                         I 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     26 
                   
                   ) 
                 
               
             
           
         
       
     
     where the subscript u denotes the uninvaded region, r f (t)≦r≦∞, and the functions C f (ç,t), C m (ç,t), λ 1 (ç) and λ 2 (ç) are defined by the following equations: 
     
       
         
           
             
               
                 
                   
                     
                       
                         C 
                         f 
                       
                        
                       
                         ( 
                         
                           ϛ 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               2 
                                
                               
                                 
                                   ψ 
                                   rj 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             
                               
                                 πφ 
                                 f 
                               
                                
                               
                                 c 
                                 f 
                               
                                
                               ϛ 
                             
                           
                           - 
                           
                             
                               
                                 q 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                                
                               
                                  
                                 0 
                               
                                
                               
                                 { 
                                 
                                   ϛ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       r 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                 
                                 } 
                               
                                
                               
                                 p 
                                 f 
                               
                                
                               
                                 { 
                                 
                                   
                                     r 
                                     f 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 } 
                               
                             
                             
                               2 
                                
                               π 
                                
                               
                                   
                               
                                
                               h 
                                
                               
                                   
                               
                                
                               
                                 φ 
                                  
                                 
                                   ( 
                                   
                                     
                                       S 
                                       w 
                                     
                                     - 
                                     
                                       S 
                                       wi 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                          
                         
                           
                             αη 
                             mu 
                           
                            
                           t 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     27 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         C 
                         m 
                       
                        
                       
                         ( 
                         
                           ϛ 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             
                               q 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                              
                             
                                
                               0 
                             
                              
                             
                               { 
                               
                                 ϛ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     r 
                                     f 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                               } 
                             
                              
                             
                               p 
                               m 
                             
                              
                             
                               { 
                               
                                 
                                   r 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               } 
                             
                           
                           
                             2 
                              
                             π 
                              
                             
                                 
                             
                              
                             h 
                              
                             
                                 
                             
                              
                             
                               φ 
                                
                               
                                 ( 
                                 
                                   
                                     S 
                                     w 
                                   
                                   - 
                                   
                                     S 
                                     wi 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                        
                       
                          
                         
                           
                             αη 
                             mu 
                           
                            
                           t 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     28 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         Ξ 
                         f 
                         
                           ( 
                           
                             j 
                             - 
                             1 
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           ϛ 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                          
                         
                           
                             αη 
                             mu 
                           
                            
                           t 
                         
                       
                        
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                          
                         
                           
                             
                               ζ 
                                
                               
                                   
                               
                                
                               
                                 
                                   
                                     p 
                                     _ 
                                   
                                   fu 
                                   
                                     ( 
                                     
                                       j 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     ζ 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             
                               [ 
                               
                                 
                                   
                                     J 
                                     1 
                                     2 
                                   
                                    
                                   
                                     { 
                                     
                                       ζ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 + 
                                 
                                   
                                     Y 
                                     1 
                                     2 
                                   
                                    
                                   
                                     { 
                                     
                                       ζ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                             
                               ∫ 
                               
                                 
                                   r 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               ∞ 
                             
                              
                             
                               r 
                                
                               
                                   
                               
                                
                               
                                 
                                    
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     ζ 
                                      
                                     
                                         
                                     
                                      
                                     r 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     
                                        
                                       0 
                                     
                                      
                                     
                                       ( 
                                       
                                         ϛ 
                                          
                                         
                                             
                                         
                                          
                                         r 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   ∂ 
                                   t 
                                 
                               
                                
                               
                                  
                                 r 
                               
                                
                               
                                  
                                 ζ 
                               
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     29 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         Ξ 
                         m 
                         
                           ( 
                           
                             k 
                             - 
                             1 
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           ϛ 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                          
                         
                           
                             αη 
                             mu 
                           
                            
                           t 
                         
                       
                        
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                          
                         
                           
                             
                               ζ 
                                
                               
                                   
                               
                                
                               
                                 
                                   
                                     p 
                                     _ 
                                   
                                   mu 
                                   
                                     ( 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     ζ 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             
                               [ 
                               
                                 
                                   
                                     J 
                                     1 
                                     2 
                                   
                                    
                                   
                                     { 
                                     
                                       ζ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 + 
                                 
                                   
                                     Y 
                                     1 
                                     2 
                                   
                                    
                                   
                                     { 
                                     
                                       ζ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                             
                               ∫ 
                               
                                 
                                   r 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               ∞ 
                             
                              
                             
                               r 
                                
                               
                                   
                               
                                
                               
                                 
                                    
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     ζ 
                                      
                                     
                                         
                                     
                                      
                                     r 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   ∂ 
                                   
                                     
                                        
                                       0 
                                     
                                      
                                     
                                       ( 
                                       
                                         ϛ 
                                          
                                         
                                             
                                         
                                          
                                         r 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   ∂ 
                                   t 
                                 
                               
                                
                               
                                  
                                 r 
                               
                                
                               
                                  
                                 ζ 
                               
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     30 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       λ 
                       1 
                     
                      
                     
                       ( 
                       ϛ 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       
                         ( 
                         
                           
                             αη 
                             mu 
                           
                           - 
                           
                             
                               η 
                               fu 
                             
                              
                             
                               ( 
                               
                                 
                                   ϛ 
                                   2 
                                 
                                 + 
                                 
                                   α 
                                    
                                   
                                     
                                       k 
                                       mu 
                                     
                                     
                                       k 
                                       fu 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         η 
                                         fu 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             ϛ 
                                             2 
                                           
                                           + 
                                           
                                             α 
                                              
                                             
                                               
                                                 k 
                                                 mu 
                                               
                                               
                                                 k 
                                                 fu 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       αη 
                                       mu 
                                     
                                   
                                   ] 
                                 
                                 2 
                               
                               + 
                               
                                 4 
                                  
                                 
                                   α 
                                   2 
                                 
                                  
                                 
                                   
                                     k 
                                     mu 
                                   
                                   
                                     k 
                                     fu 
                                   
                                 
                                  
                                 
                                   η 
                                   fu 
                                 
                                  
                                 
                                   η 
                                   mu 
                                 
                               
                             
                           
                         
                         ) 
                       
                       . 
                       
                         
 
                       
                        
                       
                           
                       
                        
                       and 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     31 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       λ 
                       2 
                     
                      
                     
                       ( 
                       ϛ 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       
                         ( 
                         
                           
                             αη 
                             mu 
                           
                           - 
                           
                             
                               η 
                               fu 
                             
                              
                             
                               ( 
                               
                                 
                                   ϛ 
                                   2 
                                 
                                 + 
                                 
                                   α 
                                    
                                   
                                     
                                       k 
                                       mu 
                                     
                                     
                                       k 
                                       fu 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         η 
                                         fu 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             ϛ 
                                             2 
                                           
                                           + 
                                           
                                             α 
                                              
                                             
                                               
                                                 k 
                                                 mu 
                                               
                                               
                                                 k 
                                                 fu 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       αη 
                                       mu 
                                     
                                   
                                   ] 
                                 
                                 2 
                               
                               + 
                               
                                 4 
                                  
                                 
                                   α 
                                   2 
                                 
                                  
                                 
                                   
                                     k 
                                     mu 
                                   
                                   
                                     k 
                                     fu 
                                   
                                 
                                  
                                 
                                   η 
                                   fu 
                                 
                                  
                                 
                                   η 
                                   mu 
                                 
                               
                             
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     32 
                   
                   ) 
                 
               
             
           
         
       
     
     Note the fracture and matrix pressure solutions are to be used in an iterative scheme. To start the iteration at j=k=1, we assume Ξ f   (0) (ç, t)=Ξ m   (0) (ç, t)=0 and for subsequent iterations, Ξ f   (j−1)  and Ξ m   k−1  are given by Eqs. 29 and 30, respectively. At the interface r=r f (t), between the invaded and uninvaded regions, matching the fracture and matrix pressure solutions of the invaded and uninvaded regions, four integral equations with three unknowns are obtained: the fracture pressure p f {r f (t), t}, the matrix pressure p m {r f (t), t} and the flux ψr f (t). The parameters p f {,r f (t), t}, p m {r f (t), t} and ψ rf (t) deduced from these equations can then be used in the general solutions to obtain the fracture and matrix pressures as a function of r and t. 
     The following describes solutions during the pressure fall-off phase. 
     Fluid injection is terminated at t=t 0  and the interface between the invaded and uninvaded regions at r=r f (t 0 ) is static and is obtained from 
     
       
         
           
             
               
                 r 
                 f 
                 2 
               
                
               
                 ( 
                 
                   t 
                   0 
                 
                 ) 
               
             
             = 
             
               
                 a 
                 2 
               
               + 
               
                 
                   
                     
                       ∫ 
                       0 
                       
                         t 
                         0 
                       
                     
                      
                     
                       
                         q 
                          
                         
                           ( 
                           τ 
                           ) 
                         
                       
                        
                       
                          
                         r 
                       
                     
                   
                   
                     π 
                      
                     
                         
                     
                      
                     h 
                      
                     
                         
                     
                      
                     
                       φ 
                        
                       
                         ( 
                         
                           
                             S 
                             2 
                           
                           - 
                           
                             S 
                             wi 
                           
                         
                         ) 
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
     The boundary conditions at the interface are p fi {r f (t 0 ),t}=p fu {f f (t 0 ), t},p mi {r f (t 0 ), t} and 
     
       
         
           
             
               
                 [ 
                 
                   
                     
                       ∂ 
                       
                         p 
                         fi 
                       
                     
                      
                     
                       { 
                       
                         
                           
                             r 
                             f 
                           
                            
                           
                             ( 
                             
                               t 
                               o 
                             
                             ) 
                           
                         
                         , 
                         t 
                       
                       } 
                     
                   
                   
                     ∂ 
                     r 
                   
                 
                 ] 
               
                
               
                 ( 
                 
                   
                     u 
                     w 
                   
                   
                     k 
                     fi 
                   
                 
                 ) 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       ∂ 
                       
                         p 
                         fu 
                       
                     
                      
                     
                       { 
                       
                         
                           
                             r 
                             f 
                           
                            
                           
                             ( 
                             
                               t 
                               o 
                             
                             ) 
                           
                         
                         , 
                         t 
                       
                       } 
                     
                   
                   
                     ∂ 
                     r 
                   
                 
                 ] 
               
                
               
                 
                   ( 
                   
                     
                       u 
                       o 
                     
                     
                       k 
                       fu 
                     
                   
                   ) 
                 
                 . 
               
             
           
         
       
     
     The initial condition at t=t 0 , start time of the pressure fall-off phase, is obtained from the pressure build-up solutions, which are: 
     
       
         
           
             
               
                 p 
                 f 
               
                
               
                 ( 
                 
                   r 
                   , 
                   
                     t 
                     0 
                   
                 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             p 
                             fi 
                           
                            
                           
                             ( 
                             
                               r 
                               , 
                               
                                 t 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           a 
                           ≤ 
                           r 
                           ≤ 
                           
                             
                               r 
                               f 
                             
                              
                             
                               ( 
                               
                                 t 
                                 0 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             p 
                             fu 
                           
                            
                           
                             ( 
                             
                               r 
                               , 
                               
                                 t 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             r 
                             ≥ 
                             
                               
                                 r 
                                 f 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                   0 
                                 
                                 ) 
                               
                             
                           
                           , 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                    
                   
                     
 
                   
                    
                   
                     
                       p 
                       m 
                     
                      
                     
                       ( 
                       
                         r 
                         , 
                         
                           t 
                           0 
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             p 
                             mi 
                           
                            
                           
                             ( 
                             
                               r 
                               , 
                               
                                 t 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           a 
                           ≤ 
                           r 
                           ≤ 
                           
                             
                               r 
                               f 
                             
                              
                             
                               ( 
                               
                                 t 
                                 0 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             p 
                             mu 
                           
                            
                           
                             ( 
                             
                               r 
                               , 
                               
                                 t 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           r 
                           ≥ 
                           
                             
                               
                                 r 
                                 f 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                   0 
                                 
                                 ) 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     For the invaded region, the solutions in Laplace domain are: 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       _ 
                     
                     fi 
                   
                   = 
                   
                     
                       2 
                       
                         ( 
                         
                           
                             
                               r 
                               f 
                               2 
                             
                              
                             
                               ( 
                               
                                 t 
                                 0 
                               
                               ) 
                             
                           
                           - 
                           
                             a 
                             2 
                           
                         
                         ) 
                       
                     
                     × 
                     × 
                     
                         
                       
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     α 
                                      
                                     
                                       
                                         k 
                                         mi 
                                       
                                       
                                         k 
                                         fi 
                                       
                                     
                                      
                                     
                                       η 
                                       fi 
                                     
                                      
                                     
                                       
                                         ∫ 
                                         a 
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         
                                           
                                             up 
                                             mi 
                                           
                                            
                                           
                                             ( 
                                             
                                               u 
                                               , 
                                               
                                                 t 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                            
                                           u 
                                         
                                       
                                     
                                   
                                   
                                     s 
                                      
                                     
                                       ( 
                                       
                                         s 
                                         + 
                                         
                                           
                                             αη 
                                             mi 
                                           
                                            
                                           
                                             ( 
                                             
                                               1 
                                               + 
                                               
                                                 
                                                   
                                                     k 
                                                     mi 
                                                   
                                                    
                                                   
                                                     η 
                                                     fi 
                                                   
                                                 
                                                 
                                                   
                                                     k 
                                                     fi 
                                                   
                                                    
                                                   
                                                     η 
                                                     mi 
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       ( 
                                       
                                         s 
                                         + 
                                         
                                           αη 
                                           mi 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       
                                         ∫ 
                                         a 
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         
                                           
                                             up 
                                             fi 
                                           
                                            
                                           
                                             ( 
                                             
                                               u 
                                               , 
                                               
                                                 t 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                            
                                           u 
                                         
                                       
                                     
                                   
                                   
                                     s 
                                      
                                     
                                       ( 
                                       
                                         s 
                                         + 
                                         
                                           
                                             αη 
                                             mi 
                                           
                                            
                                           
                                             ( 
                                             
                                               1 
                                               + 
                                               
                                                 
                                                   
                                                     k 
                                                     mi 
                                                   
                                                    
                                                   
                                                     η 
                                                     fi 
                                                   
                                                 
                                                 
                                                   
                                                     k 
                                                     fi 
                                                   
                                                    
                                                   
                                                     η 
                                                     mi 
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       
                                         r 
                                         f 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           0 
                                         
                                         ) 
                                       
                                     
                                     
                                       
                                         φ 
                                         f 
                                       
                                        
                                       
                                         c 
                                         f 
                                       
                                     
                                   
                                    
                                   
                                     
                                       
                                         ( 
                                         
                                           s 
                                           + 
                                           
                                             αη 
                                             mi 
                                           
                                         
                                         ) 
                                       
                                        
                                       
                                         
                                           
                                             ψ 
                                             _ 
                                           
                                           
                                             r 
                                             f 
                                           
                                         
                                          
                                         
                                           ( 
                                           s 
                                           ) 
                                         
                                       
                                     
                                     
                                       s 
                                        
                                       
                                         ( 
                                         
                                           s 
                                           + 
                                           
                                             
                                               αη 
                                               mi 
                                             
                                              
                                             
                                               ( 
                                               
                                                 1 
                                                 + 
                                                 
                                                   
                                                     
                                                       k 
                                                       mi 
                                                     
                                                      
                                                     
                                                       η 
                                                       fi 
                                                     
                                                   
                                                   
                                                     
                                                       k 
                                                       fi 
                                                     
                                                      
                                                     
                                                       η 
                                                       mi 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             ++ 
                           
                            
                           
                             
                               π 
                               2 
                             
                             2 
                           
                            
                           α 
                            
                           
                             
                               k 
                               mi 
                             
                             
                               k 
                               fi 
                             
                           
                            
                           
                             η 
                             fi 
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 1 
                               
                               ∞ 
                             
                              
                             
                               
                                 
                                   
                                     ξ 
                                     n 
                                     2 
                                   
                                    
                                   
                                     J 
                                     1 
                                     2 
                                   
                                    
                                   
                                     { 
                                     
                                       
                                         ξ 
                                         n 
                                       
                                        
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                    
                                   
                                     
                                        
                                       n 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         r 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   [ 
                                   
                                     
                                       
                                         J 
                                         1 
                                         2 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             ξ 
                                             n 
                                           
                                            
                                           a 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         J 
                                         1 
                                         2 
                                       
                                        
                                       
                                         { 
                                         
                                           
                                             ξ 
                                             n 
                                           
                                            
                                           
                                             
                                               r 
                                               f 
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                         
                                         } 
                                       
                                     
                                   
                                   ] 
                                 
                               
                                
                               
                                 
                                   
                                     
                                       ∫ 
                                       α 
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                     
                                       
                                         
                                           p 
                                           mi 
                                         
                                          
                                         
                                           ( 
                                           
                                             u 
                                             , 
                                             
                                               t 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       u 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                            
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               ξ 
                                               n 
                                             
                                              
                                             u 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                          
                                         u 
                                       
                                     
                                   
                                   
                                     
                                       s 
                                        
                                       
                                         [ 
                                         
                                           s 
                                           + 
                                           
                                             
                                               αη 
                                               mi 
                                             
                                              
                                             
                                               ( 
                                               
                                                 1 
                                                 + 
                                                 
                                                   
                                                     
                                                       k 
                                                       mi 
                                                     
                                                      
                                                     
                                                       η 
                                                       fi 
                                                     
                                                   
                                                   
                                                     
                                                       k 
                                                       fi 
                                                     
                                                      
                                                     
                                                       η 
                                                       mi 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                     + 
                                     
                                       
                                         η 
                                         fi 
                                       
                                        
                                       
                                         
                                           ξ 
                                           n 
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             s 
                                             + 
                                             
                                               αη 
                                               mi 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ++ 
                               
                                
                               
                                 
                                   π 
                                   2 
                                 
                                 2 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                     
                                       
                                         ξ 
                                         n 
                                         2 
                                       
                                        
                                       
                                         J 
                                         1 
                                         2 
                                       
                                        
                                       
                                         { 
                                         
                                           
                                             ξ 
                                             n 
                                           
                                            
                                           
                                             
                                               r 
                                               f 
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                         
                                         } 
                                       
                                        
                                       
                                         
                                            
                                           n 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               ξ 
                                               n 
                                             
                                              
                                             r 
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       [ 
                                       
                                         
                                           
                                             J 
                                             1 
                                             2 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 ξ 
                                                 n 
                                               
                                                
                                               a 
                                             
                                             ) 
                                           
                                         
                                         - 
                                         
                                           
                                             J 
                                             1 
                                             2 
                                           
                                            
                                           
                                             { 
                                             
                                               
                                                 ξ 
                                                 n 
                                               
                                                
                                               
                                                 
                                                   r 
                                                   f 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     0 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             } 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                    
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             s 
                                             + 
                                             
                                               αη 
                                               mi 
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                           
                                             ∫ 
                                             α 
                                             
                                               
                                                 r 
                                                 f 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   0 
                                                 
                                                 ) 
                                               
                                             
                                           
                                            
                                           
                                             
                                               
                                                 p 
                                                 fi 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   u 
                                                   , 
                                                   
                                                     t 
                                                     0 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             u 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               
                                                  
                                                 0 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ξ 
                                                     n 
                                                   
                                                    
                                                   u 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                                
                                               u 
                                             
                                           
                                         
                                       
                                       
                                         
                                           s 
                                            
                                           
                                             [ 
                                             
                                               s 
                                               + 
                                               
                                                 
                                                   αη 
                                                   mi 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     1 
                                                     + 
                                                     
                                                       
                                                         
                                                           k 
                                                           mi 
                                                         
                                                          
                                                         
                                                           η 
                                                           fi 
                                                         
                                                       
                                                       
                                                         
                                                           k 
                                                           fi 
                                                         
                                                          
                                                         
                                                           η 
                                                           mi 
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                         + 
                                         
                                           
                                             η 
                                             fi 
                                           
                                            
                                           
                                             
                                               ξ 
                                               n 
                                               2 
                                             
                                              
                                             
                                               ( 
                                               
                                                 s 
                                                 + 
                                                 
                                                   αη 
                                                   mi 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                     -- 
                                   
                                    
                                   
                                     π 
                                     
                                       
                                         φ 
                                         f 
                                       
                                        
                                       
                                         c 
                                         f 
                                       
                                     
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         1 
                                       
                                       ∞ 
                                     
                                      
                                     
                                       
                                         
                                           
                                             ξ 
                                             n 
                                           
                                            
                                           
                                             J 
                                             1 
                                           
                                            
                                           
                                             { 
                                             
                                               
                                                 ξ 
                                                 n 
                                               
                                                
                                               
                                                 
                                                   r 
                                                   f 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     0 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             } 
                                           
                                            
                                           
                                             
                                               J 
                                               1 
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   ξ 
                                                   n 
                                                 
                                                  
                                                 a 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             
                                                
                                               n 
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   ξ 
                                                   n 
                                                 
                                                  
                                                 r 
                                               
                                               ) 
                                             
                                           
                                         
                                         
                                           [ 
                                           
                                             
                                               
                                                 J 
                                                 1 
                                                 2 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ξ 
                                                     n 
                                                   
                                                    
                                                   a 
                                                 
                                                 ) 
                                               
                                             
                                             - 
                                             
                                               
                                                 J 
                                                 1 
                                                 2 
                                               
                                                
                                               
                                                 { 
                                                 
                                                   
                                                     ξ 
                                                     n 
                                                   
                                                    
                                                   
                                                     
                                                       r 
                                                       f 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         t 
                                                         0 
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 } 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                        
                                       
                                         
                                           
                                             ( 
                                             
                                               s 
                                               + 
                                               
                                                 αη 
                                                 mi 
                                               
                                             
                                             ) 
                                           
                                            
                                           
                                             
                                               
                                                 ψ 
                                                 _ 
                                               
                                               
                                                 r 
                                                 f 
                                               
                                             
                                              
                                             
                                               ( 
                                               s 
                                               ) 
                                             
                                           
                                         
                                         
                                           
                                             s 
                                              
                                             
                                               [ 
                                               
                                                 s 
                                                 + 
                                                 
                                                   
                                                     αη 
                                                     mi 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       1 
                                                       + 
                                                       
                                                         
                                                           
                                                             k 
                                                             mi 
                                                           
                                                            
                                                           
                                                             η 
                                                             fi 
                                                           
                                                         
                                                         
                                                           
                                                             k 
                                                             fi 
                                                           
                                                            
                                                           
                                                             η 
                                                             mi 
                                                           
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ] 
                                             
                                           
                                           + 
                                           
                                             
                                               η 
                                               fi 
                                             
                                              
                                             
                                               
                                                 ξ 
                                                 n 
                                                 2 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   s 
                                                   + 
                                                   
                                                     αη 
                                                     mi 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         , 
                         
                           
 
                         
                          
                         and 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     33 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       p 
                       _ 
                     
                     mi 
                   
                   = 
                   
                     
                       
                         
                           p 
                           mi 
                         
                          
                         
                           ( 
                           
                             r 
                             , 
                             
                               t 
                               0 
                             
                           
                           ) 
                         
                       
                       
                         s 
                         + 
                         
                           αη 
                           mi 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             
                               2 
                                
                               
                                 αη 
                                 mi 
                               
                             
                             
                               ( 
                               
                                 
                                   
                                     r 
                                     f 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       t 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   a 
                                   2 
                                 
                               
                               ) 
                             
                           
                            
                           
                             [ 
                             
                               
                                 
                                   α 
                                    
                                   
                                     
                                       k 
                                       mi 
                                     
                                     
                                       k 
                                       fi 
                                     
                                   
                                    
                                   
                                     η 
                                     fi 
                                   
                                    
                                   
                                     
                                       ∫ 
                                       a 
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                     
                                       
                                         
                                           up 
                                           mi 
                                         
                                          
                                         
                                           ( 
                                           
                                             u 
                                             , 
                                             
                                               t 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                          
                                         u 
                                       
                                     
                                   
                                 
                                 
                                   
                                     s 
                                      
                                     
                                       ( 
                                       
                                         s 
                                         + 
                                         
                                           αη 
                                           mi 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       s 
                                       + 
                                       
                                         
                                           αη 
                                           mi 
                                         
                                          
                                         
                                           ( 
                                           
                                             1 
                                             + 
                                             
                                               
                                                 
                                                   k 
                                                   mi 
                                                 
                                                  
                                                 
                                                   η 
                                                   fi 
                                                 
                                               
                                               
                                                 
                                                   k 
                                                   fi 
                                                 
                                                  
                                                 
                                                   η 
                                                   mi 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     
                                       
                                         ∫ 
                                         a 
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         
                                           
                                             up 
                                             fi 
                                           
                                            
                                           
                                             ( 
                                             
                                               u 
                                               , 
                                               
                                                 t 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                            
                                           u 
                                         
                                       
                                     
                                     
                                       s 
                                        
                                       
                                         ( 
                                         
                                           s 
                                           + 
                                           
                                             
                                               αη 
                                               mi 
                                             
                                              
                                             
                                               ( 
                                               
                                                 1 
                                                 + 
                                                 
                                                   
                                                     
                                                       k 
                                                       mi 
                                                     
                                                      
                                                     
                                                       η 
                                                       fi 
                                                     
                                                   
                                                   
                                                     
                                                       k 
                                                       fi 
                                                     
                                                      
                                                     
                                                       η 
                                                       mi 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   -- 
                                 
                                  
                                 
                                   
                                     
                                       r 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         0 
                                       
                                       ) 
                                     
                                   
                                   
                                     
                                       φ 
                                       f 
                                     
                                      
                                     
                                       c 
                                       f 
                                     
                                   
                                 
                                  
                                 
                                   
                                     
                                       
                                         ψ 
                                         _ 
                                       
                                       
                                         r 
                                         f 
                                       
                                     
                                      
                                     
                                       ( 
                                       s 
                                       ) 
                                     
                                   
                                   
                                     s 
                                      
                                     
                                       ( 
                                       
                                         s 
                                         + 
                                         
                                           
                                             αη 
                                             mi 
                                           
                                            
                                           
                                             ( 
                                             
                                               1 
                                               + 
                                               
                                                 
                                                   
                                                     k 
                                                     mi 
                                                   
                                                    
                                                   
                                                     η 
                                                     fi 
                                                   
                                                 
                                                 
                                                   
                                                     k 
                                                     fi 
                                                   
                                                    
                                                   
                                                     η 
                                                     mi 
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         ++ 
                       
                        
                       
                         
                           π 
                           2 
                         
                         2 
                       
                        
                       
                         αη 
                         mi 
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           ∞ 
                         
                          
                         
                           
                             
                               
                                 ξ 
                                 n 
                                 2 
                               
                                
                               
                                 J 
                                 1 
                                 2 
                               
                                
                               
                                 { 
                                 
                                   
                                     ξ 
                                     n 
                                   
                                    
                                   
                                     
                                       r 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         0 
                                       
                                       ) 
                                     
                                   
                                 
                                 } 
                               
                                
                               
                                 
                                    
                                   n 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       ξ 
                                       n 
                                     
                                      
                                     r 
                                   
                                   ) 
                                 
                               
                             
                             
                               [ 
                               
                                 
                                   
                                     J 
                                     1 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         ξ 
                                         n 
                                       
                                        
                                       a 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     J 
                                     1 
                                     2 
                                   
                                    
                                   
                                     { 
                                     
                                       
                                         ξ 
                                         n 
                                       
                                        
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                             
                               
                                 
                                   ∫ 
                                   α 
                                   
                                     
                                       r 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         0 
                                       
                                       ) 
                                     
                                   
                                 
                                  
                                 
                                   
                                     
                                       p 
                                       fi 
                                     
                                      
                                     
                                       ( 
                                       
                                         u 
                                         , 
                                         
                                           t 
                                           0 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   u 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                        
                                       0 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         u 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                      
                                     u 
                                   
                                 
                               
                               
                                 
                                   s 
                                    
                                   
                                     [ 
                                     
                                       s 
                                       + 
                                       
                                         
                                           αη 
                                           mi 
                                         
                                          
                                         
                                           ( 
                                           
                                             1 
                                             + 
                                             
                                               
                                                 
                                                   k 
                                                   mi 
                                                 
                                                  
                                                 
                                                   η 
                                                   fi 
                                                 
                                               
                                               
                                                 
                                                   k 
                                                   fi 
                                                 
                                                  
                                                 
                                                   η 
                                                   mi 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                 + 
                                 
                                   
                                     η 
                                     fi 
                                   
                                    
                                   
                                     
                                       ξ 
                                       n 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         s 
                                         + 
                                         
                                           αη 
                                           mi 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ++ 
                           
                            
                           
                             
                               π 
                               2 
                             
                             2 
                           
                            
                           α 
                            
                           
                             
                               k 
                               mi 
                             
                             
                               k 
                               fi 
                             
                           
                            
                           
                             η 
                             fi 
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 1 
                               
                               ∞ 
                             
                              
                             
                               
                                 
                                   
                                     ξ 
                                     n 
                                     2 
                                   
                                    
                                   
                                     J 
                                     1 
                                     2 
                                   
                                    
                                   
                                     { 
                                     
                                       
                                         ξ 
                                         n 
                                       
                                        
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                    
                                   
                                     
                                        
                                       n 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         r 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   [ 
                                   
                                     
                                       
                                         J 
                                         1 
                                         2 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             ξ 
                                             n 
                                           
                                            
                                           a 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         J 
                                         1 
                                         2 
                                       
                                        
                                       
                                         { 
                                         
                                           
                                             ξ 
                                             n 
                                           
                                            
                                           
                                             
                                               r 
                                               f 
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                         
                                         } 
                                       
                                     
                                   
                                   ] 
                                 
                               
                                
                               
                                 
                                   
                                     
                                       ∫ 
                                       α 
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                     
                                       
                                         
                                           p 
                                           mi 
                                         
                                          
                                         
                                           ( 
                                           
                                             u 
                                             , 
                                             
                                               t 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       u 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                            
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               ξ 
                                               n 
                                             
                                              
                                             u 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                          
                                         u 
                                       
                                     
                                   
                                   
                                     
                                       ( 
                                       
                                         s 
                                         + 
                                         
                                           αη 
                                           mi 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           s 
                                            
                                           
                                             [ 
                                             
                                               s 
                                               + 
                                               
                                                 
                                                   αη 
                                                   mi 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     1 
                                                     + 
                                                     
                                                       
                                                         
                                                           k 
                                                           mi 
                                                         
                                                          
                                                         
                                                           η 
                                                           fi 
                                                         
                                                       
                                                       
                                                         
                                                           k 
                                                           fi 
                                                         
                                                          
                                                         
                                                           η 
                                                           mi 
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                         + 
                                         
                                           
                                             η 
                                             fi 
                                           
                                            
                                           
                                             
                                               ξ 
                                               n 
                                               2 
                                             
                                              
                                             
                                               ( 
                                               
                                                 s 
                                                 + 
                                                 
                                                   αη 
                                                   mi 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 -- 
                               
                                
                               
                                 
                                   παη 
                                   mi 
                                 
                                 
                                   
                                     φ 
                                     f 
                                   
                                    
                                   
                                     c 
                                     f 
                                   
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                     
                                       
                                         ξ 
                                         n 
                                       
                                        
                                       
                                         J 
                                         1 
                                       
                                        
                                       
                                         { 
                                         
                                           
                                             ξ 
                                             n 
                                           
                                            
                                           
                                             
                                               r 
                                               f 
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                         
                                         } 
                                       
                                        
                                       
                                         
                                           J 
                                           1 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               ξ 
                                               n 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                            
                                           n 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               ξ 
                                               n 
                                             
                                              
                                             r 
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       [ 
                                       
                                         
                                           
                                             J 
                                             1 
                                             2 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 ξ 
                                                 n 
                                               
                                                
                                               a 
                                             
                                             ) 
                                           
                                         
                                         - 
                                         
                                           
                                             J 
                                             1 
                                             2 
                                           
                                            
                                           
                                             { 
                                             
                                               
                                                 ξ 
                                                 n 
                                               
                                                
                                               
                                                 
                                                   r 
                                                   f 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     0 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             } 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                    
                                   
                                     
                                       
                                         
                                           ψ 
                                           _ 
                                         
                                         
                                           r 
                                           f 
                                         
                                       
                                        
                                       
                                         ( 
                                         s 
                                         ) 
                                       
                                     
                                     
                                       
                                         s 
                                          
                                         
                                           [ 
                                           
                                             s 
                                             + 
                                             
                                               
                                                 αη 
                                                 mi 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   1 
                                                   + 
                                                   
                                                     
                                                       
                                                         k 
                                                         mi 
                                                       
                                                        
                                                       
                                                         η 
                                                         fi 
                                                       
                                                     
                                                     
                                                       
                                                         k 
                                                         fi 
                                                       
                                                        
                                                       
                                                         η 
                                                         mi 
                                                       
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       + 
                                       
                                         
                                           η 
                                           fi 
                                         
                                          
                                         
                                           
                                             ξ 
                                             n 
                                             2 
                                           
                                            
                                           
                                             ( 
                                             
                                               s 
                                               + 
                                               
                                                 αη 
                                                 mi 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     34 
                   
                   ) 
                 
               
             
           
         
       
     
     where  ψ   rf (s)=∫ 0   t−t     0   ψ rf (τ)e −sτ dτ, and K n (ξ n r)=Y 0 (ξ n r)J 1 (ξ n a)−J 0 (ξ n r)Y 1 (ξ n a). The corresponding eigenvalues are ξ 0 =0 and ξ n . The set of eigenvalues are the positive roots of the transcendental equation J 1 (ξ n a)Y 1 {ξ n r f (t 0 )}−Y 1 (ξ n a)J 1 {ξ n r f (t 0)}= 0, n=1, 2, . . . . 
     The solutions in time domain of the fracture pressure p fi  and matrix pressure p mi ; during the fall-off phase are: 
     
       
         
           
             
               
                 
                   
                     p 
                     fi 
                   
                   = 
                   
                     
                       2 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 f 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                   0 
                                 
                                 ) 
                               
                             
                             - 
                             
                               a 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 
                                   k 
                                   mi 
                                 
                                  
                                 
                                   η 
                                   fi 
                                 
                               
                               
                                 
                                   k 
                                   fi 
                                 
                                  
                                 
                                   η 
                                   mi 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                         
                       
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       k 
                                       mi 
                                     
                                      
                                     
                                       η 
                                       fi 
                                     
                                   
                                   
                                     
                                       k 
                                       fi 
                                     
                                      
                                     
                                       η 
                                       mi 
                                     
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                           
                                             
                                               αη 
                                               mi 
                                             
                                              
                                             
                                               ( 
                                               
                                                 1 
                                                 + 
                                                 
                                                   
                                                     
                                                       k 
                                                       mi 
                                                     
                                                      
                                                     
                                                       η 
                                                       fi 
                                                     
                                                   
                                                   
                                                     
                                                       k 
                                                       fi 
                                                     
                                                      
                                                     
                                                       η 
                                                       mi 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             - 
                                             
                                               t 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   
                                     ∫ 
                                     a 
                                     
                                       
                                         r 
                                         f 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     
                                       
                                         up 
                                         mi 
                                       
                                        
                                       
                                         ( 
                                         
                                           u 
                                           , 
                                           
                                             t 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                        
                                       
                                         u 
                                         ++ 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           
                                             
                                               
                                                 k 
                                                 mi 
                                               
                                                
                                               
                                                 η 
                                                 fi 
                                               
                                             
                                             
                                               k 
                                                
                                               
                                                 ? 
                                               
                                                
                                               
                                                 η 
                                                 mi 
                                               
                                             
                                           
                                            
                                           
                                              
                                             
                                               
                                                 - 
                                                 
                                                   
                                                     αη 
                                                     mi 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       1 
                                                       + 
                                                       
                                                         
                                                           
                                                             k 
                                                             mi 
                                                           
                                                            
                                                           
                                                             η 
                                                             fi 
                                                           
                                                         
                                                         
                                                           
                                                             k 
                                                             fi 
                                                           
                                                            
                                                           
                                                             η 
                                                             mi 
                                                           
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   
                                                     t 
                                                     0 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       
                                         ∫ 
                                         
                                           ? 
                                         
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         
                                           
                                             up 
                                             fi 
                                           
                                            
                                           
                                             ( 
                                             
                                               u 
                                               , 
                                               
                                                 t 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                            
                                           
                                             u 
                                             -- 
                                           
                                         
                                          
                                         
                                           
                                             
                                               r 
                                               f 
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                           
                                             
                                               φ 
                                               f 
                                             
                                              
                                             
                                               c 
                                               f 
                                             
                                           
                                         
                                          
                                         
                                           
                                             ∫ 
                                             0 
                                             
                                               t 
                                               - 
                                               
                                                 t 
                                                 0 
                                               
                                             
                                           
                                            
                                           
                                             
                                               
                                                 ψ 
                                                 
                                                   r 
                                                   f 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   
                                                     t 
                                                     0 
                                                   
                                                   - 
                                                   τ 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 1 
                                                 + 
                                                 
                                                   
                                                     
                                                       
                                                         k 
                                                         mi 
                                                       
                                                        
                                                       
                                                         η 
                                                         fi 
                                                       
                                                     
                                                     
                                                       
                                                         k 
                                                         fi 
                                                       
                                                        
                                                       
                                                         η 
                                                         mi 
                                                       
                                                     
                                                   
                                                    
                                                   
                                                      
                                                     
                                                       
                                                         - 
                                                         
                                                           
                                                             αη 
                                                             mi 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             1 
                                                             + 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                       
                                                        
                                                       τ 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                              
                                             
                                                
                                               τ 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             ++ 
                           
                            
                           
                             
                               π 
                               2 
                             
                             2 
                           
                            
                           α 
                            
                           
                             
                               k 
                               mi 
                             
                             
                               k 
                               fi 
                             
                           
                            
                           
                             η 
                             fi 
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 1 
                               
                               ∞ 
                             
                              
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           ξ 
                                           n 
                                           2 
                                         
                                          
                                         
                                           J 
                                           1 
                                           2 
                                         
                                          
                                         
                                           { 
                                           
                                             
                                               ξ 
                                               n 
                                             
                                              
                                             
                                               
                                                 r 
                                                 f 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   0 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           } 
                                         
                                          
                                         
                                           
                                              
                                             n 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 ξ 
                                                 n 
                                               
                                                
                                               r 
                                             
                                             ) 
                                           
                                         
                                       
                                       
                                         [ 
                                         
                                           
                                             
                                               J 
                                               1 
                                               2 
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   ξ 
                                                   n 
                                                 
                                                  
                                                 a 
                                               
                                               ) 
                                             
                                           
                                           - 
                                           
                                             
                                               J 
                                               1 
                                               2 
                                             
                                              
                                             
                                               { 
                                               
                                                 
                                                   ξ 
                                                   n 
                                                 
                                                  
                                                 
                                                   
                                                     r 
                                                     f 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       t 
                                                       0 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               } 
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                      
                                     
                                       
                                         ∫ 
                                         α 
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         
                                           
                                             p 
                                             mi 
                                           
                                            
                                           
                                             ( 
                                             
                                               u 
                                               , 
                                               
                                                 t 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         u 
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                              
                                             0 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 ξ 
                                                 n 
                                               
                                                
                                               u 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                            
                                           u 
                                         
                                         × 
                                         × 
                                         
                                           ( 
                                           
                                             
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                     
                                                       1 
                                                       2 
                                                     
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         
                                                           αη 
                                                           mi 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             1 
                                                             + 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                       
                                                         σ 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       t 
                                                       - 
                                                       
                                                         t 
                                                         0 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               
                                                 
                                                   
                                                     η 
                                                     fi 
                                                   
                                                    
                                                   
                                                     ξ 
                                                     n 
                                                     2 
                                                   
                                                 
                                                 - 
                                                 
                                                   σ 
                                                 
                                               
                                             
                                             + 
                                             
                                               
                                                  
                                                 
                                                   
                                                     1 
                                                     2 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         - 
                                                         
                                                           
                                                             αη 
                                                             mi 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             1 
                                                             + 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                       
                                                       + 
                                                       
                                                         σ 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       t 
                                                       - 
                                                       
                                                         t 
                                                         0 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               
                                                 
                                                   
                                                     η 
                                                     fi 
                                                   
                                                    
                                                   
                                                     ξ 
                                                     n 
                                                     2 
                                                   
                                                 
                                                 + 
                                                 
                                                   σ 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 ++ 
                               
                                
                               
                                 
                                   π 
                                   2 
                                 
                                 4 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               ξ 
                                               n 
                                               2 
                                             
                                              
                                             
                                               J 
                                               1 
                                               2 
                                             
                                              
                                             
                                               { 
                                               
                                                 
                                                   ξ 
                                                   n 
                                                 
                                                  
                                                 
                                                   
                                                     r 
                                                     f 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       t 
                                                       0 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               } 
                                             
                                              
                                             
                                               
                                                  
                                                 n 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ξ 
                                                     n 
                                                   
                                                    
                                                   r 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           
                                             [ 
                                             
                                               
                                                 
                                                   J 
                                                   1 
                                                   2 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       ξ 
                                                       n 
                                                     
                                                      
                                                     a 
                                                   
                                                   ) 
                                                 
                                               
                                               - 
                                               
                                                 
                                                   J 
                                                   1 
                                                   2 
                                                 
                                                  
                                                 
                                                   { 
                                                   
                                                     
                                                       ξ 
                                                       n 
                                                     
                                                      
                                                     
                                                       
                                                         r 
                                                         f 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           t 
                                                           0 
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                   } 
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                          
                                         
                                           
                                             ∫ 
                                             α 
                                             
                                               
                                                 r 
                                                 f 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   0 
                                                 
                                                 ) 
                                               
                                             
                                           
                                            
                                           
                                             
                                               
                                                 p 
                                                 fi 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   u 
                                                   , 
                                                   
                                                     t 
                                                     0 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             u 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               
                                                  
                                                 0 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ξ 
                                                     n 
                                                   
                                                    
                                                   u 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                                
                                               u 
                                             
                                             × 
                                             × 
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     
                                                       ( 
                                                       
                                                         
                                                           
                                                             αη 
                                                             mi 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             1 
                                                             - 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         - 
                                                         
                                                           σ 
                                                         
                                                       
                                                       ) 
                                                     
                                                      
                                                     
                                                        
                                                       
                                                         
                                                           - 
                                                           
                                                             1 
                                                             2 
                                                           
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             
                                                             αη 
                                                             mi 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             1 
                                                             + 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             σ 
                                                             
                                                           
                                                           ) 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             t 
                                                             - 
                                                             
                                                             t 
                                                             0 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         η 
                                                         fi 
                                                       
                                                        
                                                       
                                                         ξ 
                                                         n 
                                                         2 
                                                       
                                                     
                                                     - 
                                                     
                                                       σ 
                                                     
                                                   
                                                 
                                                 + 
                                                 
                                                   
                                                     
                                                       ( 
                                                       
                                                         
                                                           
                                                             αη 
                                                             mi 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             1 
                                                             - 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                         
                                                           σ 
                                                         
                                                       
                                                       ) 
                                                     
                                                      
                                                     
                                                        
                                                       
                                                         
                                                           1 
                                                           2 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             - 
                                                             
                                                             
                                                             αη 
                                                             mi 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             1 
                                                             + 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             + 
                                                             
                                                             σ 
                                                             
                                                           
                                                           ) 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             t 
                                                             - 
                                                             
                                                             t 
                                                             0 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         η 
                                                         fi 
                                                       
                                                        
                                                       
                                                         ξ 
                                                         n 
                                                         2 
                                                       
                                                     
                                                     + 
                                                     
                                                       σ 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                     -- 
                                   
                                    
                                   
                                     π 
                                     
                                       2 
                                        
                                       
                                         φ 
                                         f 
                                       
                                        
                                       
                                         c 
                                         f 
                                       
                                     
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         1 
                                       
                                       ∞ 
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               ξ 
                                               n 
                                             
                                              
                                             
                                               J 
                                               1 
                                             
                                              
                                             
                                               { 
                                               
                                                 
                                                   ξ 
                                                   n 
                                                 
                                                  
                                                 
                                                   
                                                     r 
                                                     f 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       t 
                                                       0 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               } 
                                             
                                              
                                             
                                               
                                                 J 
                                                 1 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ξ 
                                                     n 
                                                   
                                                    
                                                   a 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               
                                                  
                                                 n 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ξ 
                                                     n 
                                                   
                                                    
                                                   r 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           
                                             [ 
                                             
                                               
                                                 
                                                   J 
                                                   1 
                                                   2 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       ξ 
                                                       n 
                                                     
                                                      
                                                     a 
                                                   
                                                   ) 
                                                 
                                               
                                               - 
                                               
                                                 
                                                   J 
                                                   1 
                                                   2 
                                                 
                                                  
                                                 
                                                   { 
                                                   
                                                     
                                                       ξ 
                                                       n 
                                                     
                                                      
                                                     
                                                       
                                                         r 
                                                         f 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           t 
                                                           0 
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                   } 
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                          
                                         
                                           
                                             ∫ 
                                             0 
                                             
                                               t 
                                               - 
                                               
                                                 t 
                                                 0 
                                               
                                             
                                           
                                            
                                           
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     
                                                       
                                                         ( 
                                                         
                                                           
                                                             
                                                             αη 
                                                             mi 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             1 
                                                             - 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                           
                                                           - 
                                                           
                                                             σ 
                                                           
                                                         
                                                         ) 
                                                       
                                                        
                                                       
                                                          
                                                         
                                                           
                                                             - 
                                                             
                                                             1 
                                                             2 
                                                             
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             
                                                             αη 
                                                             mi 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             1 
                                                             + 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             σ 
                                                             
                                                             
                                                             ) 
                                                           
                                                            
                                                           τ 
                                                         
                                                       
                                                     
                                                     
                                                       
                                                         
                                                           η 
                                                           fi 
                                                         
                                                          
                                                         
                                                           ξ 
                                                           n 
                                                           2 
                                                         
                                                       
                                                       - 
                                                       
                                                         σ 
                                                       
                                                     
                                                   
                                                   ++ 
                                                 
                                                  
                                                 
                                                   
                                                     
                                                       ( 
                                                       
                                                         
                                                           
                                                             αη 
                                                             mi 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             1 
                                                             - 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                         
                                                           σ 
                                                         
                                                       
                                                       ) 
                                                     
                                                      
                                                     
                                                        
                                                       
                                                         
                                                           1 
                                                           2 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             - 
                                                             
                                                             
                                                             αη 
                                                             mi 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             1 
                                                             + 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             + 
                                                             
                                                             σ 
                                                             
                                                           
                                                           ) 
                                                         
                                                          
                                                         τ 
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         η 
                                                         fi 
                                                       
                                                        
                                                       
                                                         ξ 
                                                         n 
                                                         2 
                                                       
                                                     
                                                     + 
                                                     
                                                       σ 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                              
                                             
                                               
                                                 ψ 
                                                 
                                                   r 
                                                   f 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   
                                                     t 
                                                     0 
                                                   
                                                   - 
                                                   τ 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                                
                                               τ 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                           
                         
                         , 
                         
                           
 
                         
                          
                         
                             
                         
                          
                         and 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     35 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       p 
                       mi 
                     
                     = 
                     
                       
                         
                           
                             p 
                             mi 
                           
                            
                           
                             ( 
                             
                               r 
                               , 
                               
                                 t 
                                 0 
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             
                               - 
                               
                                 
                                   αη 
                                   mi 
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       t 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                           
                         
                       
                       + 
                       
                         
                           2 
                           
                             
                               ( 
                               
                                 
                                   
                                     r 
                                     f 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       t 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   a 
                                   2 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     
                                       k 
                                       mi 
                                     
                                      
                                     
                                       η 
                                       fi 
                                     
                                   
                                   
                                     
                                       k 
                                       fi 
                                     
                                      
                                     
                                       η 
                                       mi 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                          
                         
                           
                               
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           k 
                                           mi 
                                         
                                          
                                         
                                           η 
                                           fi 
                                         
                                       
                                       
                                         
                                           k 
                                           fi 
                                         
                                          
                                         
                                           η 
                                           mi 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           
                                             
                                               
                                                 k 
                                                 mi 
                                               
                                                
                                               
                                                 η 
                                                 fi 
                                               
                                             
                                             
                                               
                                                 k 
                                                 fi 
                                               
                                                
                                               
                                                 η 
                                                 mi 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                        
                                       
                                          
                                         
                                           - 
                                           
                                             
                                               αη 
                                               mi 
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 - 
                                                 
                                                   t 
                                                   0 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                     + 
                                     
                                        
                                       
                                         
                                           - 
                                           
                                             
                                               αη 
                                               mi 
                                             
                                              
                                             
                                               ( 
                                               
                                                 1 
                                                 + 
                                                 
                                                   
                                                     
                                                       k 
                                                       mi 
                                                     
                                                      
                                                     
                                                       η 
                                                       fi 
                                                     
                                                   
                                                   
                                                     
                                                       k 
                                                       fi 
                                                     
                                                      
                                                     
                                                       η 
                                                       mi 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             - 
                                             
                                               t 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 × 
                                 × 
                                 
                                   
                                     ∫ 
                                     a 
                                     
                                       
                                         r 
                                         f 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     
                                       
                                         up 
                                         mi 
                                       
                                        
                                       
                                         ( 
                                         
                                           u 
                                           , 
                                           
                                             t 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                        
                                       u 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                           
                                             
                                               αη 
                                               mi 
                                             
                                              
                                             
                                               ( 
                                               
                                                 1 
                                                 + 
                                                 
                                                   
                                                     
                                                       k 
                                                       mi 
                                                     
                                                      
                                                     
                                                       η 
                                                       fi 
                                                     
                                                   
                                                   
                                                     
                                                       k 
                                                       fi 
                                                     
                                                      
                                                     
                                                       η 
                                                       mi 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             - 
                                             
                                               t 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   
                                     ∫ 
                                     
                                       ? 
                                     
                                     
                                       
                                         r 
                                         f 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     
                                       
                                         up 
                                         fi 
                                       
                                        
                                       
                                         ( 
                                         
                                           u 
                                           , 
                                           
                                             t 
                                             0 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                        
                                       
                                         u 
                                         -- 
                                       
                                     
                                      
                                     
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             0 
                                           
                                           ) 
                                         
                                       
                                       
                                         
                                           φ 
                                           f 
                                         
                                          
                                         
                                           c 
                                           f 
                                         
                                       
                                     
                                      
                                     
                                       
                                         ∫ 
                                         0 
                                         
                                           t 
                                           - 
                                           
                                             t 
                                             0 
                                           
                                         
                                       
                                        
                                       
                                         
                                           
                                             ψ 
                                             
                                               r 
                                               f 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               - 
                                               
                                                 t 
                                                 0 
                                               
                                               - 
                                               τ 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             1 
                                             - 
                                             
                                                
                                               
                                                 
                                                   - 
                                                   
                                                     
                                                       αη 
                                                       mi 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         1 
                                                         + 
                                                         
                                                           
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                           
                                                           
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                  
                                                 τ 
                                               
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                            
                                           τ 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ++ 
                         
                          
                         
                           
                             π 
                             2 
                           
                           2 
                         
                          
                         
                           α 
                           2 
                         
                          
                         
                           
                             k 
                             mi 
                           
                           
                             k 
                             fi 
                           
                         
                          
                         
                           η 
                           fi 
                         
                          
                         
                           η 
                           mi 
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     ξ 
                                     n 
                                     2 
                                   
                                    
                                   
                                     J 
                                     1 
                                     2 
                                   
                                    
                                   
                                     { 
                                     
                                       
                                         ξ 
                                         n 
                                       
                                        
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                    
                                   
                                     
                                        
                                       n 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         r 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   [ 
                                   
                                     
                                       
                                         J 
                                         1 
                                         2 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             ξ 
                                             n 
                                           
                                            
                                           a 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         J 
                                         1 
                                         2 
                                       
                                        
                                       
                                         { 
                                         
                                           
                                             ξ 
                                             n 
                                           
                                            
                                           
                                             
                                               r 
                                               f 
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                         
                                         } 
                                       
                                     
                                   
                                   ] 
                                 
                               
                                
                               
                                 
                                   ∫ 
                                   α 
                                   
                                     
                                       r 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         0 
                                       
                                       ) 
                                     
                                   
                                 
                                  
                                 
                                   
                                     
                                       p 
                                       mi 
                                     
                                      
                                     
                                       ( 
                                       
                                         u 
                                         , 
                                         
                                           t 
                                           0 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   u 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                        
                                       0 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           ξ 
                                           n 
                                         
                                          
                                         u 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                      
                                     u 
                                   
                                   × 
                                   × 
                                   
                                     ( 
                                     
                                       
                                         
                                           2 
                                            
                                           
                                              
                                             
                                               
                                                 - 
                                                 
                                                   1 
                                                   2 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       αη 
                                                       mi 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         1 
                                                         + 
                                                         
                                                           
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                           
                                                           
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   + 
                                                   
                                                     σ 
                                                   
                                                 
                                                 ) 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   
                                                     t 
                                                     0 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         
                                           
                                             ( 
                                             
                                               
                                                 
                                                   αη 
                                                   mi 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     1 
                                                     - 
                                                     
                                                       
                                                         
                                                           k 
                                                           mi 
                                                         
                                                          
                                                         
                                                           η 
                                                           fi 
                                                         
                                                       
                                                       
                                                         
                                                           k 
                                                           fi 
                                                         
                                                          
                                                         
                                                           η 
                                                           mi 
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               - 
                                               
                                                 σ 
                                               
                                             
                                             ) 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   η 
                                                   fi 
                                                 
                                                  
                                                 
                                                   ξ 
                                                   n 
                                                   2 
                                                 
                                               
                                               - 
                                               
                                                 σ 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           2 
                                            
                                           
                                              
                                             
                                               
                                                 - 
                                                 
                                                   1 
                                                   2 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       αη 
                                                       mi 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         1 
                                                         + 
                                                         
                                                           
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                           
                                                           
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   + 
                                                   
                                                     σ 
                                                   
                                                 
                                                 ) 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   
                                                     t 
                                                     0 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         
                                           
                                             ( 
                                             
                                               
                                                 
                                                   αη 
                                                   mi 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     1 
                                                     - 
                                                     
                                                       
                                                         
                                                           k 
                                                           mi 
                                                         
                                                          
                                                         
                                                           η 
                                                           fi 
                                                         
                                                       
                                                       
                                                         
                                                           k 
                                                           fi 
                                                         
                                                          
                                                         
                                                           η 
                                                           mi 
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               + 
                                               
                                                 σ 
                                               
                                             
                                             ) 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   η 
                                                   fi 
                                                 
                                                  
                                                 
                                                   ξ 
                                                   n 
                                                   2 
                                                 
                                               
                                               + 
                                               
                                                 σ 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                            
                                           
                                             - 
                                             
                                               
                                                 αη 
                                                 mi 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   
                                                     t 
                                                     0 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         
                                           
                                             αη 
                                             mi 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   αη 
                                                   mi 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     2 
                                                     + 
                                                     
                                                       
                                                         
                                                           k 
                                                           mi 
                                                         
                                                          
                                                         
                                                           η 
                                                           fi 
                                                         
                                                       
                                                       
                                                         
                                                           k 
                                                           fi 
                                                         
                                                          
                                                         
                                                           η 
                                                           mi 
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               + 
                                               
                                                 2 
                                                  
                                                 
                                                   η 
                                                   fi 
                                                 
                                                  
                                                 
                                                   ξ 
                                                   n 
                                                   2 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                       + 
                       
                         
                           
                             π 
                             2 
                           
                           2 
                         
                          
                         
                           αη 
                           mi 
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         ξ 
                                         n 
                                         2 
                                       
                                        
                                       
                                         J 
                                         1 
                                         2 
                                       
                                        
                                       
                                         { 
                                         
                                           
                                             ξ 
                                             n 
                                           
                                            
                                           
                                             
                                               r 
                                               f 
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                         
                                         } 
                                       
                                        
                                       
                                         
                                            
                                           n 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               ξ 
                                               n 
                                             
                                              
                                             r 
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       [ 
                                       
                                         
                                           
                                             J 
                                             1 
                                             2 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 ξ 
                                                 n 
                                               
                                                
                                               a 
                                             
                                             ) 
                                           
                                         
                                         - 
                                         
                                           
                                             J 
                                             1 
                                             2 
                                           
                                            
                                           
                                             { 
                                             
                                               
                                                 ξ 
                                                 n 
                                               
                                                
                                               
                                                 
                                                   r 
                                                   f 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     0 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             } 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                    
                                   
                                     
                                       ∫ 
                                       α 
                                       
                                         
                                           r 
                                           f 
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             0 
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                     
                                       
                                         
                                           p 
                                           fi 
                                         
                                          
                                         
                                           ( 
                                           
                                             u 
                                             , 
                                             
                                               t 
                                               0 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       u 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                            
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               ξ 
                                               n 
                                             
                                              
                                             u 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                          
                                         u 
                                       
                                       × 
                                       × 
                                       
                                         ( 
                                         
                                           
                                             
                                                
                                               
                                                 
                                                   - 
                                                   
                                                     1 
                                                     2 
                                                   
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       
                                                         αη 
                                                         mi 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           1 
                                                           + 
                                                           
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                     + 
                                                     
                                                       σ 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     - 
                                                     
                                                       t 
                                                       0 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   η 
                                                   fi 
                                                 
                                                  
                                                 
                                                   ξ 
                                                   n 
                                                   2 
                                                 
                                               
                                               - 
                                               
                                                 σ 
                                               
                                             
                                           
                                           + 
                                           
                                             
                                                
                                               
                                                 
                                                   1 
                                                   2 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       - 
                                                       
                                                         
                                                           αη 
                                                           mi 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             1 
                                                             + 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                     + 
                                                     
                                                       σ 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     - 
                                                     
                                                       t 
                                                       0 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   η 
                                                   fi 
                                                 
                                                  
                                                 
                                                   ξ 
                                                   n 
                                                   2 
                                                 
                                               
                                               + 
                                               
                                                 σ 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               -- 
                             
                              
                             
                               
                                 παη 
                                 mi 
                               
                               
                                 
                                   φ 
                                   f 
                                 
                                  
                                 
                                   c 
                                   f 
                                 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   1 
                                 
                                 ∞ 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         ξ 
                                         n 
                                       
                                        
                                       
                                         J 
                                         1 
                                       
                                        
                                       
                                         { 
                                         
                                           
                                             ξ 
                                             n 
                                           
                                            
                                           
                                             
                                               r 
                                               f 
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 0 
                                               
                                               ) 
                                             
                                           
                                         
                                         } 
                                       
                                        
                                       
                                         
                                           J 
                                           1 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               ξ 
                                               n 
                                             
                                              
                                             a 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                            
                                           n 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               ξ 
                                               n 
                                             
                                              
                                             r 
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       [ 
                                       
                                         
                                           
                                             J 
                                             1 
                                             2 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 ξ 
                                                 n 
                                               
                                                
                                               a 
                                             
                                             ) 
                                           
                                         
                                         - 
                                         
                                           
                                             J 
                                             1 
                                             2 
                                           
                                            
                                           
                                             { 
                                             
                                               
                                                 ξ 
                                                 n 
                                               
                                                
                                               
                                                 
                                                   r 
                                                   f 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     0 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             } 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                    
                                   
                                     
                                       ∫ 
                                       0 
                                       
                                         t 
                                         - 
                                         
                                           t 
                                           0 
                                         
                                       
                                     
                                      
                                     
                                       
                                         
                                           ψ 
                                           
                                             r 
                                             f 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             - 
                                             
                                               t 
                                               0 
                                             
                                             - 
                                             τ 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                     
                                                       1 
                                                       2 
                                                     
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         
                                                           αη 
                                                           mi 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             1 
                                                             + 
                                                             
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                       
                                                         σ 
                                                       
                                                     
                                                     ) 
                                                   
                                                    
                                                   τ 
                                                 
                                               
                                               
                                                 
                                                   
                                                     η 
                                                     fi 
                                                   
                                                    
                                                   
                                                     ξ 
                                                     n 
                                                     2 
                                                   
                                                 
                                                 - 
                                                 
                                                   σ 
                                                 
                                               
                                             
                                             ++ 
                                           
                                            
                                           
                                             
                                                
                                               
                                                 
                                                   - 
                                                   
                                                     1 
                                                     2 
                                                   
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       
                                                         αη 
                                                         mi 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           1 
                                                           + 
                                                           
                                                             
                                                             
                                                             k 
                                                             mi 
                                                             
                                                              
                                                             
                                                             η 
                                                             fi 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fi 
                                                             
                                                              
                                                             
                                                             η 
                                                             mi 
                                                             
                                                             
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                     + 
                                                     
                                                       σ 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 τ 
                                               
                                             
                                             
                                               
                                                 
                                                   η 
                                                   fi 
                                                 
                                                  
                                                 
                                                   ξ 
                                                   n 
                                                   2 
                                                 
                                               
                                               + 
                                               
                                                 σ 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                        
                                       
                                          
                                         τ 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                       
                   
                    
                   with 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     36 
                   
                   ) 
                 
               
             
             
               
                 
                   σ 
                   = 
                   
                     
                       
                         α 
                         2 
                       
                        
                       
                         
                           
                             η 
                             mi 
                             2 
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   
                                     k 
                                     mi 
                                   
                                    
                                   
                                     η 
                                     fi 
                                   
                                 
                                 
                                   
                                     k 
                                     fi 
                                   
                                    
                                   
                                     η 
                                     mi 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         2 
                       
                     
                     + 
                     
                       
                         η 
                         fi 
                         2 
                       
                        
                       
                         ξ 
                         n 
                         4 
                       
                     
                     + 
                     
                       2 
                        
                       
                         αξ 
                         n 
                         2 
                       
                        
                       
                         η 
                         mi 
                       
                        
                       
                         
                           
                             η 
                             fi 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     k 
                                     mi 
                                   
                                    
                                   
                                     η 
                                     fi 
                                   
                                 
                                 
                                   
                                     k 
                                     fi 
                                   
                                    
                                   
                                     η 
                                     mi 
                                   
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                         
                         . 
                         
                           
 
                         
                          
                         
                             
                         
                          
                         
                           ? 
                         
                       
                        
                       
                         indicates text missing or illegible when filed 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     37 
                   
                   ) 
                 
               
             
           
         
       
     
     For the uninvaded region, the solutions in Laplace domain are: 
     
       
         
           
             
               
                 
                   
                     
                       
                         p 
                         _ 
                       
                       fu 
                     
                     = 
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                        
                       
                         
                           
                             
                               ϛ 
                               0 
                             
                              
                             
                               ( 
                               
                                 ϛ 
                                  
                                 
                                     
                                 
                                  
                                 r 
                               
                               ) 
                             
                           
                           
                             [ 
                             
                               
                                 
                                   J 
                                   1 
                                   2 
                                 
                                  
                                 
                                   { 
                                   
                                     ϛ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         r 
                                         f 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                   } 
                                 
                               
                               + 
                               
                                 
                                   Y 
                                   1 
                                   2 
                                 
                                  
                                 
                                   { 
                                   
                                     ϛ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         r 
                                         f 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                   } 
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           
                             
                               ( 
                               
                                 s 
                                 + 
                                 
                                   αη 
                                   mu 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 ∫ 
                                 
                                   
                                     r 
                                     f 
                                   
                                    
                                   
                                     ( 
                                     
                                       t 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 ∞ 
                               
                                
                               
                                 u 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                      
                                     0 
                                   
                                    
                                   
                                     ( 
                                     
                                       ϛ 
                                        
                                       
                                           
                                       
                                        
                                       u 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     p 
                                     fu 
                                   
                                    
                                   
                                     ( 
                                     
                                       u 
                                       , 
                                       
                                         t 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   u 
                                 
                               
                             
                           
                           
                             ( 
                             
                               
                                 s 
                                  
                                 
                                   [ 
                                   
                                     s 
                                     + 
                                     
                                       
                                         αη 
                                         mu 
                                       
                                        
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           
                                             
                                               
                                                 k 
                                                 mu 
                                               
                                                
                                               
                                                 η 
                                                 fu 
                                               
                                             
                                             
                                               
                                                 k 
                                                 fu 
                                               
                                                
                                               
                                                 η 
                                                 mu 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               + 
                               
                                 
                                   η 
                                   fu 
                                 
                                  
                                 
                                   
                                     ϛ 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       s 
                                       + 
                                       
                                         αη 
                                         mu 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             ϛ 
                             ++ 
                           
                         
                          
                         α 
                          
                         
                           
                             k 
                             mu 
                           
                           
                             k 
                             fu 
                           
                         
                          
                         
                           η 
                           fu 
                         
                          
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                            
                           
                             
                               
                                 
                                   ϛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     ϛ 
                                      
                                     
                                         
                                     
                                      
                                     r 
                                   
                                   ) 
                                 
                               
                               
                                 [ 
                                 
                                   
                                     
                                       J 
                                       1 
                                       2 
                                     
                                      
                                     
                                       { 
                                       
                                         ϛ 
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                   + 
                                   
                                     
                                       Y 
                                       1 
                                       2 
                                     
                                      
                                     
                                       { 
                                       
                                         ϛ 
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                 
                                 ] 
                               
                             
                              
                             
                               
                                 
                                   ∫ 
                                   
                                     
                                       r 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         0 
                                       
                                       ) 
                                     
                                   
                                   ∞ 
                                 
                                  
                                 
                                   u 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                        
                                       0 
                                     
                                      
                                     
                                       ( 
                                       
                                         ϛ 
                                          
                                         
                                             
                                         
                                          
                                         u 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       p 
                                       fu 
                                     
                                      
                                     
                                       ( 
                                       
                                         u 
                                         , 
                                         
                                           t 
                                           0 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                      
                                     u 
                                   
                                 
                               
                               
                                 ( 
                                 
                                   
                                     s 
                                      
                                     
                                       [ 
                                       
                                         s 
                                         + 
                                         
                                           
                                             αη 
                                             mu 
                                           
                                            
                                           
                                             ( 
                                             
                                               1 
                                               + 
                                               
                                                 
                                                   
                                                     k 
                                                     mu 
                                                   
                                                    
                                                   
                                                     η 
                                                     fu 
                                                   
                                                 
                                                 
                                                   
                                                     k 
                                                     fu 
                                                   
                                                    
                                                   
                                                     η 
                                                     mu 
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   + 
                                   
                                     
                                       η 
                                       fu 
                                     
                                      
                                     
                                       
                                         ϛ 
                                         2 
                                       
                                        
                                       
                                         ( 
                                         
                                           s 
                                           + 
                                           
                                             αη 
                                             mu 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 ϛ 
                                 ++ 
                               
                             
                              
                             
                               
                                 2 
                                  
                                 
                                   
                                     
                                       φ 
                                       _ 
                                     
                                     
                                       r 
                                       f 
                                     
                                   
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                               
                                 
                                   πφ 
                                   f 
                                 
                                  
                                 
                                   c 
                                   f 
                                 
                               
                             
                              
                             
                               
                                 ∫ 
                                 0 
                                 ∞ 
                               
                                
                               
                                 
                                   
                                     
                                        
                                       0 
                                     
                                      
                                     
                                       ( 
                                       
                                         ϛ 
                                          
                                         
                                             
                                         
                                          
                                         r 
                                       
                                       ) 
                                     
                                   
                                   
                                     [ 
                                     
                                       
                                         
                                           J 
                                           1 
                                           2 
                                         
                                          
                                         
                                           { 
                                           
                                             ϛ 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               
                                                 r 
                                                 f 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   0 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           } 
                                         
                                       
                                       + 
                                       
                                         
                                           Y 
                                           1 
                                           2 
                                         
                                          
                                         
                                           { 
                                           
                                             ϛ 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               
                                                 r 
                                                 f 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   0 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           } 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       s 
                                       + 
                                       
                                         αη 
                                         mu 
                                       
                                     
                                     ) 
                                   
                                   
                                     ( 
                                     
                                       
                                         s 
                                          
                                         
                                           [ 
                                           
                                             s 
                                             + 
                                             
                                               
                                                 αη 
                                                 mu 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   1 
                                                   + 
                                                   
                                                     
                                                       
                                                         k 
                                                         mu 
                                                       
                                                        
                                                       
                                                         η 
                                                         fu 
                                                       
                                                     
                                                     
                                                       
                                                         k 
                                                         fu 
                                                       
                                                        
                                                       
                                                         η 
                                                         mu 
                                                       
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       + 
                                       
                                         
                                           η 
                                           fu 
                                         
                                          
                                         
                                           
                                             ϛ 
                                             2 
                                           
                                            
                                           
                                             ( 
                                             
                                               s 
                                               + 
                                               
                                                 αη 
                                                 mu 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   ϛ 
                                 
                               
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                       
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     38 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         p 
                         _ 
                       
                       mu 
                     
                     = 
                     
                       
                         
                           
                             p 
                             mu 
                           
                            
                           
                             ( 
                             
                               r 
                               , 
                               
                                 t 
                                 0 
                               
                             
                             ) 
                           
                         
                         
                           s 
                           + 
                           
                             αη 
                             mu 
                           
                         
                       
                       + 
                       
                         
                           αη 
                           mu 
                         
                          
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                            
                           
                             
                               
                                 
                                   ϛ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     ϛ 
                                      
                                     
                                         
                                     
                                      
                                     r 
                                   
                                   ) 
                                 
                               
                               
                                 [ 
                                 
                                   
                                     
                                       J 
                                       1 
                                       2 
                                     
                                      
                                     
                                       { 
                                       
                                         ϛ 
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                   + 
                                   
                                     
                                       Y 
                                       1 
                                       2 
                                     
                                      
                                     
                                       { 
                                       
                                         ϛ 
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                       } 
                                     
                                   
                                 
                                 ] 
                               
                             
                              
                             
                               
                                 
                                   ∫ 
                                   
                                     
                                       r 
                                       f 
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         0 
                                       
                                       ) 
                                     
                                   
                                   ∞ 
                                 
                                  
                                 
                                   u 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                        
                                       0 
                                     
                                      
                                     
                                       ( 
                                       
                                         ϛ 
                                          
                                         
                                             
                                         
                                          
                                         u 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       p 
                                       fu 
                                     
                                      
                                     
                                       ( 
                                       
                                         u 
                                         , 
                                         
                                           t 
                                           0 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                      
                                     u 
                                   
                                 
                               
                               
                                 ( 
                                 
                                   
                                     s 
                                      
                                     
                                       [ 
                                       
                                         s 
                                         + 
                                         
                                           
                                             αη 
                                             mu 
                                           
                                            
                                           
                                             ( 
                                             
                                               1 
                                               + 
                                               
                                                 
                                                   
                                                     k 
                                                     mu 
                                                   
                                                    
                                                   
                                                     η 
                                                     fu 
                                                   
                                                 
                                                 
                                                   
                                                     k 
                                                     fu 
                                                   
                                                    
                                                   
                                                     η 
                                                     mu 
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   + 
                                   
                                     
                                       η 
                                       fu 
                                     
                                      
                                     
                                       
                                         ϛ 
                                         2 
                                       
                                        
                                       
                                         ( 
                                         
                                           s 
                                           + 
                                           
                                             αη 
                                             mu 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 ϛ 
                                 ++ 
                               
                             
                              
                             
                               α 
                               2 
                             
                              
                             
                               
                                 k 
                                 mu 
                               
                               
                                 k 
                                 fu 
                               
                             
                              
                             
                               η 
                               fu 
                             
                              
                             
                               η 
                               mu 
                             
                              
                             
                               
                                 ∫ 
                                 0 
                                 ∞ 
                               
                                
                               
                                 
                                   
                                     
                                       ϛ 
                                       0 
                                     
                                      
                                     
                                       ( 
                                       
                                         ϛ 
                                          
                                         
                                             
                                         
                                          
                                         r 
                                       
                                       ) 
                                     
                                   
                                   
                                     [ 
                                     
                                       
                                         
                                           J 
                                           1 
                                           2 
                                         
                                          
                                         
                                           { 
                                           
                                             ϛ 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               
                                                 r 
                                                 f 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   0 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           } 
                                         
                                       
                                       + 
                                       
                                         
                                           Y 
                                           1 
                                           2 
                                         
                                          
                                         
                                           { 
                                           
                                             ϛ 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               
                                                 r 
                                                 f 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   0 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           } 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       
                                         ∫ 
                                         
                                           
                                             r 
                                             f 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               0 
                                             
                                             ) 
                                           
                                         
                                         ∞ 
                                       
                                        
                                       
                                         u 
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                              
                                             0 
                                           
                                            
                                           
                                             ( 
                                             
                                               ϛ 
                                                
                                               
                                                   
                                               
                                                
                                               u 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           
                                             p 
                                             mu 
                                           
                                            
                                           
                                             ( 
                                             
                                               u 
                                               , 
                                               
                                                 t 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                            
                                           u 
                                         
                                       
                                     
                                     ) 
                                   
                                   
                                     
                                       ( 
                                       
                                         s 
                                         + 
                                         
                                           αη 
                                           mu 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           s 
                                            
                                           
                                             [ 
                                             
                                               s 
                                               + 
                                               
                                                 
                                                   αη 
                                                   mu 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     1 
                                                     + 
                                                     
                                                       
                                                         
                                                           k 
                                                           mu 
                                                         
                                                          
                                                         
                                                           η 
                                                           fu 
                                                         
                                                       
                                                       
                                                         
                                                           k 
                                                           fu 
                                                         
                                                          
                                                         
                                                           η 
                                                           mu 
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                         + 
                                         
                                           
                                             η 
                                             fu 
                                           
                                            
                                           
                                             
                                               ϛ 
                                               2 
                                             
                                              
                                             
                                               ( 
                                               
                                                 s 
                                                 + 
                                                 
                                                   αη 
                                                   mu 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                  
                                 
                                    
                                   
                                     ϛ 
                                     ++ 
                                   
                                 
                                  
                                 
                                   
                                     2 
                                      
                                     
                                       αη 
                                       mu 
                                     
                                      
                                     
                                       
                                         
                                           φ 
                                           _ 
                                         
                                         
                                           r 
                                           f 
                                         
                                       
                                        
                                       
                                         ( 
                                         s 
                                         ) 
                                       
                                     
                                   
                                   
                                     
                                       πφ 
                                       f 
                                     
                                      
                                     
                                       c 
                                       f 
                                     
                                   
                                 
                                  
                                 
                                   
                                     ∫ 
                                     0 
                                     ∞ 
                                   
                                    
                                   
                                     
                                       
                                         
                                            
                                           0 
                                         
                                          
                                         
                                           ( 
                                           
                                             ϛ 
                                              
                                             
                                                 
                                             
                                              
                                             r 
                                           
                                           ) 
                                         
                                       
                                       
                                         [ 
                                         
                                           
                                             
                                               J 
                                               1 
                                               2 
                                             
                                              
                                             
                                               { 
                                               
                                                 ϛ 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   
                                                     r 
                                                     f 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       t 
                                                       0 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               } 
                                             
                                           
                                           + 
                                           
                                             
                                               Y 
                                               1 
                                               2 
                                             
                                              
                                             
                                               { 
                                               
                                                 ϛ 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   
                                                     r 
                                                     f 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       t 
                                                       0 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               } 
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                      
                                     
                                       
                                          
                                         ϛ 
                                       
                                       
                                         ( 
                                         
                                           
                                             s 
                                              
                                             
                                               [ 
                                               
                                                 s 
                                                 + 
                                                 
                                                   
                                                     αη 
                                                     mu 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       1 
                                                       + 
                                                       
                                                         
                                                           
                                                             k 
                                                             mu 
                                                           
                                                            
                                                           
                                                             η 
                                                             fu 
                                                           
                                                         
                                                         
                                                           
                                                             k 
                                                             fu 
                                                           
                                                            
                                                           
                                                             η 
                                                             mu 
                                                           
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ] 
                                             
                                           
                                           + 
                                           
                                             
                                               η 
                                               fu 
                                             
                                              
                                             
                                               
                                                 ϛ 
                                                 2 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   s 
                                                   + 
                                                   
                                                     αη 
                                                     mu 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     39 
                   
                   ) 
                 
               
             
           
         
       
     
     The solutions in time domain are: 
     
       
         
           
             
               
                 
                   
                     p 
                     fu 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                        
                       
                         
                           
                             
                               
                                 ϛ 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   ϛ 
                                    
                                   
                                       
                                   
                                    
                                   r 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 ∫ 
                                 
                                   
                                     r 
                                     f 
                                   
                                    
                                   
                                     ( 
                                     
                                       t 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 ∞ 
                               
                                
                               
                                 u 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                      
                                     0 
                                   
                                    
                                   
                                     ( 
                                     
                                       ϛ 
                                        
                                       
                                           
                                       
                                        
                                       u 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     p 
                                     fu 
                                   
                                    
                                   
                                     ( 
                                     
                                       u 
                                       , 
                                       
                                         t 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   u 
                                 
                               
                             
                           
                           
                             [ 
                             
                               
                                 
                                   J 
                                   1 
                                   2 
                                 
                                  
                                 
                                   { 
                                   
                                     ϛ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         r 
                                         f 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                   } 
                                 
                               
                               + 
                               
                                 
                                   Y 
                                   1 
                                   2 
                                 
                                  
                                 
                                   { 
                                   
                                     ϛ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         r 
                                         f 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           0 
                                         
                                         ) 
                                       
                                     
                                   
                                   } 
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                             
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               αη 
                                               mu 
                                             
                                              
                                             
                                               ( 
                                               
                                                 1 
                                                 - 
                                                 
                                                   
                                                     
                                                       k 
                                                       mu 
                                                     
                                                      
                                                     
                                                       η 
                                                       fu 
                                                     
                                                   
                                                   
                                                     
                                                       k 
                                                       fu 
                                                     
                                                      
                                                     
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                                                           k 
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                                                           k 
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                                                   ) 
                                                 
                                               
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                                               - 
                                               
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                                                 0 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     
                                       
                                         
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                                                           k 
                                                           fu 
                                                         
                                                          
                                                         
                                                           η 
                                                           mu 
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             + 
                                             
                                               γ 
                                             
                                           
                                           ) 
                                         
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                                           ( 
                                           
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                                               t 
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                                           ) 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         η 
                                         fu 
                                       
                                        
                                       
                                         ϛ 
                                         2 
                                       
                                     
                                     + 
                                     
                                       γ 
                                     
                                   
                                 
                               
                               ] 
                             
                              
                             
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                                 ϛ 
                                 ++ 
                               
                             
                              
                             α 
                              
                             
                               
                                 k 
                                 mu 
                               
                               
                                 k 
                                 fu 
                               
                             
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                                 0 
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                                           r 
                                         
                                         ) 
                                       
                                     
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                                             ) 
                                           
                                         
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                                         u 
                                          
                                         
                                             
                                         
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                                           1 
                                           2 
                                         
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                                                 ( 
                                                 
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                                                   0 
                                                 
                                                 ) 
                                               
                                             
                                           
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                                       + 
                                       
                                         
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                                           { 
                                           
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                                                   0 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           } 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                  
                                 
                                     
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             
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                                                             fu 
                                                             
                                                             
                                                             
                                                             
                                                             k 
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                                                             η 
                                                             mu 
                                                             
                                                             
                                                           
                                                         
                                                         ) 
                                                       
                                                     
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                                                   ) 
                                                 
                                               
                                             
                                             
                                               
                                                 
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                                                   2 
                                                 
                                               
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                                                 ( 
                                                 
                                                   
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                                                         mu 
                                                       
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                                                             fu 
                                                             
                                                             
                                                             
                                                             
                                                             k 
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                                                         ) 
                                                       
                                                     
                                                   
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                                                 ) 
                                               
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                                                 ( 
                                                 
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                                       ] 
                                     
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                                                 r 
                                               
                                               ) 
                                             
                                           
                                           
                                             [ 
                                             
                                               
                                                 
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                                                   1 
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                                                   { 
                                                   
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                                                           t 
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                                                   { 
                                                   
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                                                         r 
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                                                   } 
                                                 
                                               
                                             
                                             ] 
                                           
                                         
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                                             0 
                                             
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                                            
                                           
                                               
                                             
                                                 
                                             
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                                               [ 
                                               
                                                 
                                                   
                                                       
                                                     
                                                       
                                                         
                                                           
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                                                             ) 
                                                             
                                                             
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                                                             ( 
                                                             
                                                             
                                                             
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                                                             2 
                                                             
                                                             
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                                                             ) 
                                                           
                                                         
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                                                        
                                                       
                                                         
                                                           ( 
                                                           
                                                             
                                                             ( 
                                                             
                                                             
                                                             
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                                                             mu 
                                                             
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                                                             η 
                                                             fu 
                                                             
                                                             
                                                             
                                                             
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                                                             fu 
                                                             
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                                                             η 
                                                             mu 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
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                                                             γ 
                                                             
                                                             
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                                                           ( 
                                                           
                                                             
                                                             
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                                                         r 
                                                         f 
                                                       
                                                     
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                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
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                   ) 
                 
               
             
             
               
                 
                   
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                                     1 
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                                           ( 
                                           
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                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                 
                               
                               ] 
                             
                           
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                                 [ 
                                 
                                   
                                     
                                       
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                                               1 
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                                                     0 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             } 
                                           
                                         
                                         + 
                                         
                                           
                                             Y 
                                             1 
                                             2 
                                           
                                            
                                           
                                             { 
                                             
                                               ϛ 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 
                                                   r 
                                                   f 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     0 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             } 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                    
                                   
                                       
                                     
                                       
                                         
                                           [ 
                                           
                                             
                                               
                                                 
                                                   
                                                     2 
                                                      
                                                     
                                                        
                                                       
                                                         
                                                           - 
                                                           
                                                             1 
                                                             2 
                                                           
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             
                                                             αη 
                                                             mu 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             1 
                                                             + 
                                                             
                                                             
                                                             
                                                             k 
                                                             mu 
                                                             
                                                              
                                                             
                                                             η 
                                                             fu 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fu 
                                                             
                                                              
                                                             
                                                             η 
                                                             mu 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             γ 
                                                             
                                                           
                                                           ) 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             t 
                                                             - 
                                                             
                                                             t 
                                                             0 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       ( 
                                                       
                                                         
                                                           
                                                             αη 
                                                             mu 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             1 
                                                             - 
                                                             
                                                             
                                                             
                                                             k 
                                                             mu 
                                                             
                                                              
                                                             
                                                             η 
                                                             fu 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fu 
                                                             
                                                              
                                                             
                                                             η 
                                                             mu 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                         
                                                           γ 
                                                         
                                                       
                                                       ) 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           
                                                             η 
                                                             fu 
                                                           
                                                            
                                                           
                                                             ϛ 
                                                             2 
                                                           
                                                         
                                                         - 
                                                         
                                                           γ 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 ++ 
                                               
                                                
                                               
                                                 
                                                   2 
                                                    
                                                   
                                                      
                                                     
                                                       
                                                         1 
                                                         2 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           
                                                             - 
                                                             
                                                             
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                                                             mu 
                                                             
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                                                             ( 
                                                             
                                                             1 
                                                             + 
                                                             
                                                             
                                                             
                                                             k 
                                                             mu 
                                                             
                                                              
                                                             
                                                             η 
                                                             fu 
                                                             
                                                             
                                                             
                                                             
                                                             k 
                                                             fu 
                                                             
                                                              
                                                             
                                                             η 
                                                             mu 
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                           
                                                           + 
                                                           
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                                                         ) 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           t 
                                                           - 
                                                           
                                                             t 
                                                             0 
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         
                                                           αη 
                                                           mu 
                                                         
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                                                             1 
                                                             - 
                                                             
                                                             
                                                             
                                                             k 
                                                             mu 
                                                             
                                                              
                                                             
                                                             η 
                                                             fu 
                                                             
                                                             
                                                             
                                                             
                                                             k 
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                                                             mu 
                                                             
                                                             
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                       
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                                                     ) 
                                                   
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                                                     ( 
                                                     
                                                       
                                                         
                                                           η 
                                                           fu 
                                                         
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                                                           ϛ 
                                                           2 
                                                         
                                                       
                                                       + 
                                                       
                                                         γ 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                             + 
                                             
                                               
                                                  
                                                 
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                                                       αη 
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                                                           t 
                                                           0 
                                                         
                                                       
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                                                     mu 
                                                   
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                                                     ( 
                                                     
                                                       2 
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                                                             k 
                                                             mu 
                                                           
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                                                             k 
                                                             fu 
                                                           
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                                                             η 
                                                             mu 
                                                           
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                 + 
                                                 
                                                   2 
                                                    
                                                   
                                                     η 
                                                     fu 
                                                   
                                                    
                                                   
                                                     ϛ 
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                                           ] 
                                         
                                          
                                         
                                            
                                           
                                             ϛ 
                                             ++ 
                                           
                                         
                                          
                                         
                                           
                                             2 
                                              
                                             
                                               αη 
                                               mu 
                                             
                                           
                                           
                                             
                                               πφ 
                                               f 
                                             
                                              
                                             
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                                               f 
                                             
                                           
                                         
                                          
                                         
                                           
                                             ∫ 
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                                             ∞ 
                                           
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                                                   0 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
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                                                      
                                                     r 
                                                   
                                                   ) 
                                                 
                                               
                                               
                                                 [ 
                                                 
                                                   
                                                     
                                                       J 
                                                       1 
                                                       2 
                                                     
                                                      
                                                     
                                                       { 
                                                       
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                                                             ( 
                                                             
                                                             t 
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                                                             ) 
                                                           
                                                         
                                                       
                                                       } 
                                                     
                                                   
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                                                             r 
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                                                             ( 
                                                             
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                                                             ) 
                                                           
                                                         
                                                       
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                                                 ] 
                                               
                                             
                                              
                                             
                                               
                                                 ∫ 
                                                 0 
                                                 
                                                   t 
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                                                     t 
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                                                            
                                                           
                                                             
                                                             - 
                                                             
                                                             1 
                                                             2 
                                                             
                                                             
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                                                             + 
                                                             
                                                             γ 
                                                             
                                                             
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                                                           τ 
                                                         
                                                       
                                                       
                                                         
                                                           
                                                             η 
                                                             fu 
                                                           
                                                            
                                                           
                                                             ϛ 
                                                             2 
                                                           
                                                         
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                                                   ] 
                                                 
                                                  
                                                 
                                                   
                                                     ψ 
                                                     
                                                       r 
                                                       f 
                                                     
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       t 
                                                       - 
                                                       
                                                         t 
                                                         0 
                                                       
                                                       - 
                                                       τ 
                                                     
                                                     ) 
                                                   
                                                 
                                                  
                                                 
                                                    
                                                   τ 
                                                 
                                                  
                                                 
                                                    
                                                   ϛ 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       , 
                                       
                                         
 
                                       
                                        
                                       
                                           
                                       
                                        
                                       with 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     41 
                   
                   ) 
                 
               
             
             
               
                 
                   γ 
                   = 
                   
                     
                       
                         α 
                         2 
                       
                        
                       
                         
                           
                             η 
                             mu 
                             2 
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   
                                     k 
                                     mu 
                                   
                                    
                                   
                                     η 
                                     fu 
                                   
                                 
                                 
                                   
                                     k 
                                     fu 
                                   
                                    
                                   
                                     η 
                                     mu 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         2 
                       
                     
                     + 
                     
                       
                         η 
                         fu 
                         2 
                       
                        
                       
                         γ 
                         4 
                       
                     
                     + 
                     
                       2 
                        
                       
                         αγ 
                         2 
                       
                        
                       
                         η 
                         mu 
                       
                        
                       
                         
                           
                             η 
                             fu 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     k 
                                     mu 
                                   
                                    
                                   
                                     η 
                                     fu 
                                   
                                 
                                 
                                   
                                     k 
                                     fu 
                                   
                                    
                                   
                                     η 
                                     mu 
                                   
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                         
                         . 
                         
                           
 
                         
                          
                         
                             
                         
                          
                         
                           ? 
                         
                       
                        
                       
                         indicates text missing or illegible when filed 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     42 
                   
                   ) 
                 
               
             
           
         
       
     
     At the interface r=r f (t 0 ), matching the fracture and matrix pressure solutions of the invaded and uninvaded regions, four integral equations with three unknowns are obtained: the fracture pressure p f {r f (t),t}, the matrix pressure p m {r f (t),t} and the flux ψr f (t). The p f {r f (t), t}, p m {r f (t),t} and ψr f (t) deduced from these equations can then be used in the general solutions to obtain the fracture and matrix pressures as a function of r and t. 
     The following describes the time-dependent skin computations for longitudinal fracture growth. During fracturing fluid injection, fluid rate and downhole pressure can change significantly with time. To compute the skin evolution of a variable rate well, the flow rate normalization technique can be employed. The time dependent skin is given by 
     
       
         
           
             
               
                 
                   
                     s 
                     = 
                     
                       
                         
                           ( 
                           
                             p 
                             
                               D 
                                
                               
                                   
                               
                                
                               C 
                             
                           
                           ) 
                         
                         
                           rD 
                           = 
                           1 
                         
                       
                       - 
                       
                         
                           p 
                           wD 
                         
                         
                           q 
                           wD 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     43 
                   
                   ) 
                 
               
             
           
         
       
     
     where p ωD  is the measured bottom hole dimensionless pressure in the wellbore, and q ωD  is the measured dimensionless rate defined respectively by the following equations: 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       wD 
                     
                     = 
                     
                       
                         
                           k 
                           fi 
                         
                          
                         
                           h 
                            
                           
                             ( 
                             
                               
                                 p 
                                 i 
                               
                               - 
                               
                                 p 
                                 wf 
                               
                             
                             ) 
                           
                         
                       
                       
                         141.2 
                          
                         q 
                          
                         
                             
                         
                          
                         
                           μ 
                           w 
                         
                          
                         
                           B 
                           w 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     44 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       q 
                       wD 
                     
                     = 
                     
                       
                         q 
                         w 
                       
                       q 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     45 
                   
                   ) 
                 
               
             
           
         
       
     
     In Eq. 39, (P DC ) rD=1  is the dimensionless pressure at the sandface for a constant rate well with no skin defined by 
     
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           p 
                           DC 
                         
                         ) 
                       
                       
                         rD 
                         = 
                         1 
                       
                     
                     = 
                     
                       
                         
                           k 
                           fi 
                         
                          
                         
                           h 
                            
                           
                             ( 
                             
                               
                                 p 
                                 i 
                               
                               - 
                               
                                 
                                   p 
                                   fi 
                                 
                                  
                                 
                                   ( 
                                   
                                     a 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         141.2 
                          
                         q 
                          
                         
                             
                         
                          
                         
                           μ 
                           w 
                         
                          
                         
                           B 
                           w 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     46 
                   
                   ) 
                 
               
             
           
         
       
     
     where p fi (a,t) is the fracture pressure at the wellbore during the buildup period obtained from either Eq. 10 or Eq. 21 by setting r=a. 
     The accuracy of the method is dependent upon the accuracy of the computation of (p DC )r D=1 . The new mathematical model is used to compute the pressure and then the pressure is normalized with respect to rate. It is implicit that for the above skin computation, the skin is not incorporated in the model, whereas the measurement is. 
     As the fracture grows, the skin will continue to decrease until it stabilizes to a point beyond which any further increase in the fracture length has no additional benefit. The proposed general purpose model takes into account this variable skin subject to two reasonable assumptions: a) the tip of the fracture is very near the flood front, and b) the permeability per unit length of the fracture is constant. The skin due to the fracture is computed using the well known Hawkins formula, which is 
     
       
         
           
             
               
                 
                   s 
                   = 
                   
                     
                       ( 
                       
                         
                           k 
                           
                             k 
                             s 
                           
                         
                         - 
                         1 
                       
                       ) 
                     
                      
                     ln 
                      
                     
                       
                         
                           r 
                           s 
                         
                         
                           r 
                           ω 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     47 
                   
                   ) 
                 
               
             
           
         
       
     
     If radius of the modified zone, r s , is fixed, then the skin, s, is constant. In this case r s  moves with the flood front, k=k m   a , and k s =k fi   a . Since k&lt;k s , the skin is negative. Also since r s  is inside the logarithmic function, its influence in changing the value of the skin progressively reduces. 
     In the model according to some embodiments, the variable skin is modeled by using a different absolute permeability in the invaded zone compared to the reservoir. Non-linear regression considering the injection and subsequent fall off data will yield, in addition to other reservoir parameters, invaded and uninvaded zone permeability values which can be substituted in the above equation to determine the skin. 
     Instructions of software described above (including the analysis software  316  of  FIG. 3 ) are loaded for execution on a processor (such as processor  318  in  FIG. 3 ). The processor includes microprocessors, microcontrollers, processor modules or subsystems (including one or more microprocessors or microcontrollers), or other control or computing devices. A “processor” can refer to a single component or to plural components (e.g., one or multiple CPUs in one or multiple computers). 
     Data and instructions (of the software) are stored in respective storage devices, which are implemented as one or more computer-readable or computer-usable storage media. The storage media include different forms of memory including semiconductor memory devices such as dynamic or static random access memories (DRAMs or SRAMs), erasable and programmable read-only memories (EPROMs), electrically erasable and programmable read-only memories (EEPROMs) and flash memories; magnetic disks such as fixed, floppy and removable disks; other magnetic media including tape; and optical media such as compact disks (CDs) or digital video disks (DVDs). 
     While the invention has been disclosed with respect to a limited number of embodiments, those skilled in the art, having the benefit of this disclosure, will appreciate numerous modifications and variations therefrom. It is intended that the appended claims cover such modifications and variations as fall within the true spirit and scope of the invention.