Abstract:
The disclosure is directed to a method of representing fluid flow response to imposed conditions in a physical fluid reservoir through wells. The invention utilizes techniques and formulas of unprecedented accuracy and speed for computer computation of Green&#39;s and Neumann functions in finite three-dimensional space for arbitrarily-oriented line sources in anisotropic media. The method includes the modeling of fluid flow in physical fluid reservoirs with an assemblage of linear well segments, characterizing arbitrary well trajectory, operating in unison with flux density coupled to flow rate within the well through a constitutive expression linking pressure distribution and flow. The method further includes generalization to complex fracture sets or fractured wells in modeling fluid flow in a reservoir, coupled use of such computations within a mesh representation of the physical fluid reservoir with isolation of well cell contributions, and extension to modeling of heterogeneous reservoirs and pressure transients.

Description:
REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a continuation-in-part (CIP) to previous application Ser. No. 12/436,779, filed May 7, 2009 by inventors Randy Doyle Hazlett and Desarazu Krishna Babu. 
     
    
     ACKNOWLEDGMENT OF GOVERNMENT SUPPORT 
       [0002]    Not Applicable 
       COMPACT DISC APPENDIX 
       [0003]    Not Applicable 
       BACKGROUND OF THE INVENTION 
       [0004]    1. Field of the Invention 
         [0005]    The invention pertains to the field of oil and gas well productivity modeling, and more particularly, to modeling the relationship between pressure gradient and fluid flux for the well and the surrounding hydrocarbon reservoir. More specifically, the invention relates to well productivity modeling for advanced well designs and to well connections for complex wells in numerical reservoir simulation methods. 
         [0006]    2. Description of Related Art 
         [0007]    The field of reservoir engineering includes modeling the capacity of wells to inject or withdraw fluids and the sustainability of production rates. The finite size and shape of hydrocarbon reservoirs dictates the long term relationship between withdrawal rate and pressure decline. Well equations try to capture this relationship for the dominant time period of well operation, known as the pseudo-steady state regime, as a function of reservoir characteristics and well properties, such as well radius and well location. Well equations are traditionally constructed for single phase flow and suitably modified through introduction of empirical relative permeability concepts for multiphase flow use. With adequate well models, numerical reservoir simulation is used to place wells optimally (U.S. Pat. No. 7,181,380). 
         [0008]    Early use of well equations was based upon mathematical solutions to the governing equations, but these were amenable for only the simplest of reservoir descriptions and well configurations. The industry turned to numerical methods and computers to solve the governing equations with more complex inputs. Examples of such simulation methods are found in U.S. Pat. No. 7,369,973, U.S. Pat. No. 7,324,929, U.S. Pat. No. 6,810,370, and U.S. Pat. No. 7,289,942. However, the physical size of a wellbore, nominally 6 inches, is far below the resolution of most numerical reservoir simulations. As such, the industry relies upon so-called well equations to relate the properties of a reservoir grid cell to observables, i.e. pressure and flow rate, for a well within the cell. Well equations, pioneered by Peaceman (1978), were constructed from fine scale numerical simulations for use at the practical scale, and as such, entertained only a small subset of possible configurations. In order to capture smaller scale effects, some practitioners use local grid refinement around wells (U.S. Pat. No. 6,907,392, U.S. Pat. No. 7,047,165, U.S. Pat. No. 7,451,066, U.S. Pat. No. 6,078,869, U.S. Pat. No. 6,018,497), dramatically increasing the computational overhead with greater risk of problem numerical instability. 
         [0009]    The application of horizontal drilling technology made the prior set of well equation rules obsolete, as hydrocarbon reservoirs are typically laterally extensive but thin. The proximity of reservoir boundaries in horizontal wells required new relationships. The industry returned to mathematical solutions of the governing equations to redefine well equations. While breakthroughs in modeling were found, this next generation of mathematical solutions was not without limitations regarding complexity of reservoir description and allowed well configurations. Some developments restricted the solution to the case where the well was aligned with a coordinate axis (Babu and Odeh, 1989). Such solutions applied to purely horizontal or vertical wells, and not the general slanted well of arbitrary inclination. Slanted wells were approximated by stair-step assembly of horizontal and vertical segments or by restricting well orientation (Jasti, Penmatcha, and Babu, 1997). Alternatively, Aavatsmark and Klausen (2003) use a coordinate transform for slanted and slightly curved wells. 
         [0010]    These equations also applied to the reservoir as a whole and could be localized for use in simulations only for uniform numerical grids in a homogeneous reservoir (Babu et al., 1991). Furthermore, the computational time is a substantial burden in numerical schemes for certain sets of input parameters. As mathematical solutions to differential equations, the models are specific to boundary conditions imposed at the physical limits to the reservoir. The boundary condition most often imposed is that of a sealed system; however, many reservoirs have leaky sides through which the influx of water is possible, resulting in delayed pressure decline. Older models exist for so-called bottom and edge water drive mechanisms; there remains a need to account for these options in advanced models. The industry typically defers to numerical simulation methods to incorporate these effects locally rather than globally. The restriction of analytic solutions to homogeneous systems has been somewhat overcome through the use of boundary integral methods to solve for pressure in a piece-wise homogeneous fashion (Hazlett and Babu, 2005). Still, prior art using semi-analytical solutions did not entirely replace the older Peaceman-type methods using empirical well connections. 
         [0011]    The concept of skin was introduced in modeling permeability alteration around wells resulting from stimulation or formation damage. Similar to well equations, skin represents an attempt to describe behavior without an explicit representation of the modified permeability feature or zone. A fractured well has a different fluid capture geometry than an unstimulated well. With increased industry activity in unconventional reservoirs invoking multiple hydraulic fracturing stages, there is further need for new models with enhanced capabilities. In addition to complex geometry in hydraulically fractured shale, early transient behavior dominates economic forecasts, and pseudo-steady state may never be attained. Microseismic mapping technology is yielding tremendous insight into the complex fracture patterns created (Mayerhofer et al., 2010; Fisher et al., 2004). While simpler models have gained attention, state-of-the-art numerical reservoir modeling (Cipolla et al., 2010; Mayerhofer et al., 2006) involves extensive adaptive gridding around discrete fracture patterns with cells approaching dimensions of actual fracture apertures. Such models require considerable computational resources and extensive inputs not generally amenable to field measurement or laboratory verification. 
         [0012]    A three-dimensional solution that honors the closed system boundary condition has been long recognized (Carslaw and Jaeger, 1959). However, these solutions, when applied to box-shaped rectangular domains, become computationally challenging. Of note, Durlofsky (2000, 2004) correctly writes the solution for one-dimensional, time-dependent potential for a point source in a closed system as unity plus an infinite sum of two cosine product terms with a third exponentially-damped factor in time. These authors then invoke Neumann&#39;s product rule to cast the solution in three dimensions as the product of three one-dimensional solutions. While this is indeed a statement of the problem solution, it has not been cast in a readily computable form. In particular, the triple infinite series contains mathematical singularities, converges very slowly, and becomes increasingly difficult to evaluate as the observation point, (x, y, z) approaches the location of the point source, (x o , y o , z o ). Unfortunately, the observation point of most interest is one extremely nearby corresponding to the wellbore radius. What is needed is a fast and accurate method to evaluate this known raw solution. 
         [0013]    The point source solution is not the solution most closely aligned with wells. Rather, we are interested in line sources in three-dimensional space. The line source can be cast as the integration of the point source solution along a trajectory. A number of researchers have posed the problem in this manner for application to horizontal (Economides et al., 1996) and non-conventional wells (Maizeret, 1996; Ouyang et al., 1997; Wolfsteiner et al., 1999, 2003; Durlofsky, 2004), where the term non-conventional implies wells at arbitrary angle or varied trajectory. All these approaches leverage the work of Economides et al. (1996) in which the authors elaborate upon the published efforts of others seeking direct analytical formulas, “The above works are outstanding contributions, presenting closed-form expressions and must be considered seminal work. We believe, though, that analytical approaches have reached their limitations.” In Appendix A, Economides et al. (1996, 261) reveal their approach to abandon the search for closed-form expressions and “ . . . integrate the solution for the point source numerically along the well trajectory.” The authors included a couple of cautionary notes. First, this approach “relies strongly on the ability to compute this point source solution accurately and fast.” Secondly, a term containing a singularity is substracted from the numerically evaluated integrand in order to remove much of the “spiked” character of the function being integrated. Maizeret (1996) followed a similar route in constructing solutions using numerical quadrature rather than seeking direct solutions of closed form. In Figure A.1 (p. 43), Maizeret plots the original function to be numerically integrated along with the modified function recommended to handle numerical problems due to the proximity to a mathematical singularity. Remarkably, both functions remain markedly spiked in nature, disclosing that challenges still remain in numerical evaluation, especially with quadrature methods built upon smooth polynomial approximation to functions. In Section B-2 (p. 50) on modifications needed, Maizeret cautions, “The objective is to increase the speed of the program which in its present state can be fairly slow for some cases.” Ouyang et al. (1997), Wolfsteiner et al. (1999, 2003), and Durlofsky (2004) all follow the Economides/Maizeret approximate numerical solution method. While Durlofsky (2004) does not give procedural detail before proceeding to cite further developments and applications for unconventional wells, methodology relying on numerical integration approximation is clear in companion documents (Ouyang &amp; Aziz, 2001; Wolfsteiner et al., 1999; Maizeret, 1996). What is needed is a method that avoids approximation in integration, accounts fully for the singular nature of functions encountered, and produces direct, closed-form analytical expressions for the fast and accurate evaluation of pressure, especially where the observation point corresponds to the wellbore radius when numerical integration methods are most challenged. 
         [0014]    A number of researchers have demonstrated approaches to couple the representative pressure of well segments in a multi-segment well representation to the flow within the wellbore (Ouyang &amp; Aziz, 2001; Durlofsky, 2004; Wolfsteiner et al., 1999). Thus, the pressure gradient along the well trajectory that pulls in fluid from the reservoir also satisfies approximations to physical laws for incoming fluid to flow within the well to a common collection point. What is needed is a computer method to couple the pressure drop due to fluid flow within a wellbore to a faster and more accurate computational method for well inflow distribution in complex wells. 
         [0015]    While the pseudo-steady state flow regime, in which average pressure continues to change but spatial gradients in pressure are constant, is the dominant period associated with the production life of a conventional well, and thus, is used in well equations, there is a need to compute production as a function of time, including early transients. Use of transient information provides the basis for so-called well testing to evaluate reservoir characteristics. Since singularities in the governing equations occur in terms of space, incorporation of temporal effects represents no new mathematical challenges to those in the field skilled in the art. Though not explicitly stated, but as is evident in related works, Durlofsky (2004) performed integrations in time with numerical approximation. In their analytic solution formulations, Odeh and Babu (1990) incorporated time and described the various time regimes possible leading up to pseudo-steady state production for strictly horizontal wells. What is needed is a temporal representation of production from wells of arbitrary trajectory in which inflow is computed from highly accurate direct formulas and rapidly convergent series produced by exact integrations in time and space using antiderivatives consistent with the Fundamental Theorem of Calculus and further simplification using mathematical identities for enhanced computational efficiency. 
         [0016]    There is a need for simple analytic productivity equations which can model the entire reservoir for different combinations of boundary conditions. It would be highly desirable to have a well productivity model which could quickly and accurately model pressure and flow rate for an arbitrary number of wells of arbitrary trajectory or of complex, multiple wellbore configuration for various imposed reservoir boundary conditions. 
         [0017]    There is further need to embed a fast-computing, accurate well equation that is capable of modeling wells of arbitrary trajectory or complex, multiple wellbore configurations below the resolution of the simulation in a numerical reservoir simulator. It would be highly desirable to have such a well productivity model that could restrict attention locally to the cell containing the well without loss of generality or accuracy and that could also be used within a feedback loop to restrict well pressure or production. 
       SUMMARY OF THE INVENTION 
       [0018]    The invention comprises the reduction of a generalized equation describing the productivity of an arbitrarily-oriented well segment within a subterranean fluid reservoir to a readily useable form of evaluation with unprecedented accuracy, consisting of a sum of direct, analytic formulas and fast-converging, exponentially-damped series summations, and subsequent use of this new method on a computer to accomplish new tasks, such as, but not limited to, productivity modeling of wells of arbitrary trajectory, including those with multiple intersecting wellbores. The invention also encompasses the interlacing of a new well equation in numerical reservoir simulators with wells below the resolution of the grid system used to define the reservoir size, shape, and heterogeneous property distribution and the use of such equations for feedback control on flux or well pressure within the simulator. The invention is capable of accurately modeling multiple wells simultaneously with sealed boundaries or with permeable boundaries through use of boundary integral equations. The invention is robust and adaptable to well constraints on either flux or pressure, allowing coupling to constitutive relations for fluid flow within the well. The invention applies also to two-dimensional simulation in thin reservoirs where horizontal and slanted wellbores behave as vertical, fully penetrating fractures with the orientation of their projections onto the XY plane. The invention further applies to production modeling from two-dimensional, complex discrete fracture sets induced in unconventional reservoirs for hydrocarbon production. The invention represents an unprecedented level of accuracy and flexibility in modeling wells with significant time savings over prior art due to a reduction of the solution to polynomial expressions and rapidly convergent infinite series summations. This is accomplished with exact integrations in time and space using antiderivatives according to the Fundamental Theorem of Calculus and subsequent simplification using mathematical identities. The invention replaces computationally burdensome and inaccurate prior art numerical approximations for temporal and spatial integration of the point source solution containing singularities to represent nonconventional wells. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0019]      FIG. 1  shows a schematic of a line source inside a three-dimensional box of dimensions a, b, and h with endpoints (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ), length, L, and direction cosines (α,β, γ). 
           [0020]      FIG. 2  shows the dimensionless pressure distribution, 
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           [0000]    within a unit cube of isotropic permeability for an XZ slice containing a well represented by a uniform flux line source with well endpoints (x 1 , y 1 , z 1 )=(0.25, 0.50, 0.25) and (x 2 , y 2 , z 2 )=(0.75, 0.50, 0.75): (a) Partially penetrating well configuration, and (b) Pressure distribution for the XZ plane containing the well shown with a banded look-up table in order to easily visualize isopotential contours. 
           [0021]      FIG. 3  shows validation of the time-dependent dimensionless well pressure as a function of dimensionless time for two partially-penetrating, horizontal wells centered in a square drainage area. Solid lines represent the results of Economides et al. (1996), while data points come from the described invention with an observation point 70% of the distance from midpoint to tip. The computations are for wells parallel to the x-axis with normalized length ratios, L/a, of 0.2 and 0.8, respectively, a square drainage area, a thickness ratio, h/a, of 0.05, an isotropic medium, and a well radius, r w /a, of 0.0004. 
           [0022]      FIG. 4  shows validation of the time-dependent dimensionless well drawdown pressure as a function of dimensionless time for variable length, partially-penetrating, uniform flux fractures centered in a square drainage area. Solid lines represent the results of Gringarten et al. (1974), while data points come from the two-dimensional subset of the described invention with midpoint observation selection. 
           [0023]      FIG. 5  shows dimensionless pressure maps of P(x,y)−P avg  for time-dependent discrete well modeling of fracture patterns interpreted from microseismic (Mayerhofer et al., 2006) in the stimulation of a vertical Barnett well. Production is modeled on a 3500 ft square with a uniform fracture pressure assumption. The grayscale for each image is self-normalized with maximum and minimum values for optimal viewing. a) t D =0.0001, b) t D =0.001, c) t D =0.01, and d) t D =0.1. 
           [0024]      FIG. 6  shows dimensionless stimulated well productivity as a function of dimensionless time for the fracture set interpreted from microseismic mapping by Mayerhofer et al. (2006) during single stage stimulation of a vertical Barnett well. Inset is the digitized fracture mapping used in the infinite conductivity fracture superposition model. 
           [0025]      FIG. 7  shows dimensionless pressure maps of P(x,y)−P avg  for time-dependent discrete well modeling of fracture patterns interpreted from microseismic (Fisher et al., 2004) in the multistage stimulation of a horizontal Barnett well. Production is modeled on a 6000 ft square with a uniform fracture pressure assumption. The grayscale for each image is self-normalized with maximum and minimum values for optimal viewing. a) t D =0.0001, b) t D =0.001, c) t D =0.01, and d) t D =0.1. 
           [0026]      FIG. 8  shows dimensionless stimulated well productivity for infinite conductivity and uniform flux fractures as a function of dimensionless time for the fracture set interpreted from microseismic mapping by Fisher et al. (2004) during multistage stimulation of a horizontal Barnett well. Inset is the digitized fracture mapping used in the infinite conductivity fracture superposition model. 
           [0027]      FIG. 9  shows a line source from (x 1 , y 1 , z 1 ) to (x 2 , y 2 , z 2 ) indicating the midpoint where the well pressure is typically evaluated along the locus of points on the plane normal to the well axis at a physical distance r w , the well radius, from the mathematical source. 
           [0028]      FIG. 10  shows an example of pressure computation for a well configuration within a unit cube leading to nonsymmetric isopotential surfaces normal to the well: (a) Partially penetrating well configuration, and (b) Pressure distribution for the XZ plane normal to the well at the midpoint, (c) Pressure distribution for the YZ plane containing the well, and (d) A close-up view of the pressure distribution in the immediate vicinity of the mathematical line source indicating a loss of symmetry as a function of radius and variation in pressure at the well radius, r w , as a function of position using r w /a=0.015. 
           [0029]      FIG. 11  shows XY planar slices of dimensionless pressure at different elevations through a cubic cell of homogeneous permeability in which a multilateral well with a motherbore and four laterals in the configuration shown in the schematic, demonstrating the capability of modeling advanced wells. In this example, the flux per unit of well length is held constant. 
           [0030]      FIG. 12  shows a multi-segmented well illustrating the construction of differential pressure constraints along the length of a well and relationships imposed. Each segment is 
           [0031]    characterized by the pressure at the midpoint at the well radius and an average flux in this piecewise uniform flux model. The internal flow rate increases progressively, starting from the tip in a production well, while the pressure drop between segments is a function of the Reynolds Number, N Re , and a wall roughness parameter. 
           [0032]      FIG. 13  shows the pressure distribution resulting from production through an advanced well forming a curved pathway within a homogeneous, 3D, cubic medium in which the well is composed of 29 adjoining, piecewise linear segments with either constant flux or constant pressure imposed at a chosen well radius (r w /a=0.00015) at the segment midpoints and graphs depicting the flux and pressure at those locations as a function of progressive length down the well: (a) Well configuration within the cell, (b) Central XZ slice through the computed pressure field for the plane containing the well with imposed uniform flux, (c) Central XZ slice through the computed pressure field for the plane containing the well with imposed uniform pressure, (d) Plot of pressure and flux as a function of cumulative well length for the uniform flux case, and (e) Plot of pressure and flux as a function of cumulative well length for the uniform pressure case. 
           [0033]      FIG. 14  shows a rectangular cell containing a line source representing a well from (x 1 , y 1 , z 1 ) to (x 2 , y 2 , z 2 ), an observation point, which is typically a location at the reservoir sandface, and the six external faces upon which a single value of flux is ascribed. Note the z values increase downward in accordance with industry standards, and all outward normal fluxes are defined as positive, as conventional in mathematics. 
           [0034]      FIG. 15  shows three-dimensional plots of pressure for the same curvilinear well trajectory as in the constant pressure case of  FIG. 13  with various combinations of open and closed boundaries: (a) the sealed boundary case in which the medium behaves as a volume-distributed source, supplying fluid to the well through fluid expansion as the average cell pressure declines, (b) the case with uniform flux on two lateral faces, each supplying half the well production, and (c) the case with uniform flux from the bottom face supplying the full well production. The choice of banded lookup table allows easy visualization of isopotential surfaces. Note that isopotential surfaces intersect sealed boundaries at right angles. The flux density weighted average pressure at a prescribed well radius is given as an annotation, indicating sensitivity of well observations to external boundary conditions. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0035]    The invention pertains to solutions to the three-dimensional heat equation, given in a Cartesian coordinate system in terms of potential, Φ, as 
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         [0000]    Here, (k x , k y , k z ) denote the directional permeabilities of the medium through which fluid moves, Φ, μ, and C t  represent the porosity, fluid viscosity, and system compressibility, respectively, and the last term indicates a source or sink. Potential is interpreted as pressure, P, once gravitational forces are included. In terms of the Dirac delta function, δ, a RHS point source term is represented as 
         [0000]        f ( x,y,z;x   o   ,y   o   ,z   o )˜δ( x−x   o )·δ( y−y   o )·δ( z−z   o ).  (2)
 
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         [0000]    where we introduce the dimensionless pressure due to a continuous point source, P DC , and dimensionless time, {circumflex over (t)} D . 
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         [0000]    This dimensionless time is related to a more common lumping with a second dimensionless scaling group, k/x e   2 , such that t D ≡{circumflex over (t)} D ·(k/x e   2 ), where x e  is a characteristic length scale we choose to represent with the longest box dimension, a. Other variants of the dimensionless time use area in place of a 2  for nonsquare domains. A second form of the same continuous point source solution is rooted in the method of images and catalogs the approach to pseudo-steady state. 
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                           ^ 
                         
                         D 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                          
                         t 
                       
                       · 
                       
                         [ 
                         
                           1 
                           + 
                           
                             2 
                              
                             
                               
                                 ∑ 
                                 
                                   l 
                                   = 
                                   1 
                                 
                                 ∞ 
                               
                                
                               
                                 
                                   
                                      
                                     
                                       - 
                                       
                                         
                                           
                                             π 
                                             2 
                                           
                                            
                                           
                                             l 
                                             2 
                                           
                                            
                                           
                                             k 
                                             x 
                                           
                                            
                                           t 
                                         
                                         
                                           a 
                                           2 
                                         
                                       
                                     
                                   
                                   · 
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           lx 
                                         
                                         a 
                                       
                                       ) 
                                     
                                   
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           lx 
                                           o 
                                         
                                       
                                       a 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       · 
                       
                           
                         
                           
                             [ 
                             
                               1 
                               + 
                               
                                 2 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       = 
                                       1 
                                     
                                     ∞ 
                                   
                                    
                                   
                                     
                                       
                                          
                                         
                                           - 
                                           
                                             
                                               
                                                 π 
                                                 2 
                                               
                                                
                                               
                                                 m 
                                                 2 
                                               
                                                
                                               
                                                 k 
                                                 y 
                                               
                                                
                                               t 
                                             
                                             
                                               b 
                                               2 
                                             
                                           
                                         
                                       
                                       · 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               π 
                                                
                                               
                                                   
                                               
                                                
                                               my 
                                             
                                             b 
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             π 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               my 
                                               o 
                                             
                                           
                                           b 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           · 
                           
                               
                             
                               [ 
                               
                                 1 
                                 + 
                                 
                                   2 
                                    
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         1 
                                       
                                       ∞ 
                                     
                                      
                                     
                                       
                                         
                                            
                                           
                                             - 
                                             
                                               
                                                 
                                                   π 
                                                   2 
                                                 
                                                  
                                                 
                                                   n 
                                                   2 
                                                 
                                                  
                                                 
                                                   k 
                                                   z 
                                                 
                                                  
                                                 t 
                                               
                                               
                                                 h 
                                                 2 
                                               
                                             
                                           
                                         
                                         · 
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 nz 
                                               
                                               h 
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               π 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 nz 
                                                 o 
                                               
                                             
                                             h 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    It is recognized that evaluation of Eq. 3 is more tractable for short times, while the exponential damping in time in Eq. 5 aids in evaluating transients at longer times. In either case, evaluation of the time independent contribution is problematic due to the presence of spatial singularities, leading to extremely slow convergence. 
         [0036]    While others have opted for integration using numerical schemes despite problems cited in evaluation of the integrand, the invention pertains to exact integration in both time and space. The Fundamental Theorem of Calculus links the processes of differentiation and integration. In particular, it allows exact integration with identification of an appropriate antiderivative. This invention identifies and incorporates such antiderivatives. As such, the singular nature of the integrand is handled without approximation or error. 
         [0037]    In particular, this invention pertains to a fast method to compute the solution for a line source term representing a well with arbitrary three-dimensional orientation within a sealed, rectangular, box-shaped cell. 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                     D 
                   
                   ≡ 
                   
                     
                       1 
                       L 
                     
                     · 
                     
                       
                         ∫ 
                         0 
                         L 
                       
                        
                       
                         
                           P 
                           
                             D 
                              
                             
                                 
                             
                              
                             C 
                           
                         
                         · 
                         
                             
                         
                          
                         
                            
                           s 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Such spatial integration can be carried out on either Equation (3) or Equation (5). The source is represented by a straight line of length L, with endpoints (x 1 , y 1 , z 1 ) and (x 2 , x 2 , y 2 ), located within the box as illustrated in  FIG. 1 . Let the direction cosines of this line be (α, β, γ), so that (Lα, Lβ, Lγ)≡[(x 2 −x 1 ),(y 2 −y 1 ),(z 2 −z 1 )]. Points on this line source are represented parametrically, 
         [0000]      ( x   o   ,y   o   ,z   o )≡[( x   1   +αs ),( y   1   +βs ),( z   1   +γs )],0 ≦s≦L,   (7)
 
         [0000]    where the parameter “s” measures the distance along its length from one end. 
         [0038]    Concentrating on the more problematic approach to pseudo-steady state given by Equation (5), integration with respect to time using antiderivatives yields 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             P 
                             D 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               
                                 z 
                                 ; 
                                 
                                   x 
                                   1 
                                 
                               
                               , 
                               
                                 y 
                                 1 
                               
                               , 
                               
                                 
                                   z 
                                   1 
                                 
                                 ; 
                                 α 
                               
                               , 
                               β 
                               , 
                               
                                 γ 
                                 ; 
                                 L 
                               
                               , 
                               t 
                             
                             ) 
                           
                         
                         ≡ 
                         
                           
                             ( 
                             
                               t 
                               
                                 φ 
                                  
                                 
                                     
                                 
                                  
                                 μ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   C 
                                   t 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             
                               1 
                               L 
                             
                             · 
                             
                               
                                 ∫ 
                                 0 
                                 L 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     l 
                                     , 
                                     m 
                                     , 
                                     
                                       n 
                                       ≠ 
                                       0 
                                     
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                     
                                       C 
                                       lmn 
                                     
                                     
                                       π 
                                       2 
                                     
                                   
                                    
                                   
                                     
                                       · 
                                       
                                         ( 
                                         trig 
                                         ) 
                                       
                                     
                                     
                                       D 
                                       lmn 
                                       2 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                      
                                     s 
                                   
                                 
                               
                             
                           
                           - 
                           
                             
                               1 
                               L 
                             
                             · 
                             
                               
                                 ∫ 
                                 0 
                                 L 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     l 
                                     , 
                                     m 
                                     , 
                                     
                                       n 
                                       ≠ 
                                       0 
                                     
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                     
                                       C 
                                       lmn 
                                     
                                     
                                       π 
                                       2 
                                     
                                   
                                    
                                   
                                     
                                       E 
                                       · 
                                       
                                         ( 
                                         trig 
                                         ) 
                                       
                                     
                                     
                                       D 
                                       lmn 
                                       2 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                      
                                     s 
                                   
                                 
                               
                             
                           
                         
                       
                       , 
                       
                         
 
                       
                        
                       
                           
                       
                        
                       where 
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         E 
                         ≡ 
                         
                           exp 
                            
                           
                             ( 
                             
                               
                                 
                                   - 
                                   
                                     π 
                                     2 
                                   
                                 
                                  
                                 
                                   D 
                                   lmn 
                                   2 
                                 
                                  
                                 t 
                               
                               
                                 φ 
                                  
                                 
                                     
                                 
                                  
                                 μ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   C 
                                   t 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ; 
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         D 
                         lmn 
                         2 
                       
                       ≡ 
                       
                         ( 
                         
                           
                             
                               
                                 k 
                                 x 
                               
                                
                               
                                 l 
                                 2 
                               
                             
                             
                               a 
                               2 
                             
                           
                           + 
                           
                             
                               
                                 k 
                                 y 
                               
                                
                               
                                 m 
                                 2 
                               
                             
                             
                               b 
                               2 
                             
                           
                           + 
                           
                             
                               
                                 k 
                                 z 
                               
                                
                               
                                 n 
                                 2 
                               
                             
                             
                               h 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     ; 
                   
                    
                   
                     
 
                   
                    
                   
                     trig 
                     ≡ 
                     
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               π 
                                
                               
                                   
                               
                                
                               lx 
                             
                             a 
                           
                           ) 
                         
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               π 
                                
                               
                                   
                               
                                
                               my 
                             
                             b 
                           
                           ) 
                         
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               π 
                                
                               
                                   
                               
                                
                               nz 
                             
                             h 
                           
                           ) 
                         
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               π 
                                
                               
                                   
                               
                                
                               
                                 lx 
                                 o 
                               
                             
                             a 
                           
                           ) 
                         
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               π 
                                
                               
                                   
                               
                                
                               
                                 my 
                                 o 
                               
                             
                             b 
                           
                           ) 
                         
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               π 
                                
                               
                                   
                               
                                
                               
                                 nz 
                                 o 
                               
                             
                             h 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The factor, C lmn , takes on values 2, 4, and 8, depending upon the dimensionality of the infinite series. On the RHS of Equation (8), the part containing the first two terms is identified as the PSS component of the dimensionless pressure drop, P D . The remaining term is the transient component of the pressure that vanishes in long time behavior. 
         [0039]    Focusing on the transient term, spatial integration using antiderivatives gives 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       P 
                       trans 
                     
                     = 
                     
                       
                         1 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             π 
                             3 
                           
                         
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             , 
                             m 
                             , 
                             
                               n 
                               ≠ 
                               0 
                             
                           
                           ∞ 
                         
                          
                         
                           
                             
                               
                                 C 
                                 lmn 
                               
                                
                               
                                 
                                   exp 
                                    
                                   
                                     ( 
                                     
                                       
                                         - 
                                         
                                           π 
                                           2 
                                         
                                       
                                        
                                       
                                         D 
                                         lmn 
                                         2 
                                       
                                        
                                       
                                         
                                           t 
                                           ^ 
                                         
                                         D 
                                       
                                     
                                     ) 
                                   
                                 
                                 · 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         lx 
                                       
                                       a 
                                     
                                     ) 
                                   
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       my 
                                     
                                     b 
                                   
                                   ) 
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       nz 
                                     
                                     h 
                                   
                                   ) 
                                 
                               
                             
                             
                               ( 
                               
                                 
                                   
                                     
                                       k 
                                       x 
                                     
                                      
                                     
                                       l 
                                       2 
                                     
                                   
                                   
                                     a 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       k 
                                       y 
                                     
                                      
                                     
                                       m 
                                       2 
                                     
                                   
                                   
                                     b 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       k 
                                       z 
                                     
                                      
                                     
                                       n 
                                       2 
                                     
                                   
                                   
                                     h 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             λ 
                              
                             
                               ( 
                               
                                 
                                   
                                     x 
                                     -&gt; 
                                   
                                   1 
                                 
                                 , 
                                 
                                   
                                     x 
                                     -&gt; 
                                   
                                   2 
                                 
                                 , 
                                 l 
                                 , 
                                 m 
                                 , 
                                 n 
                                 , 
                                 a 
                                 , 
                                 b 
                                 , 
                                 h 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where λ represents the sum of four different terms over (±) signs 
         [0000]    
       
         
           
             
               
                 
                   
                     λ 
                     ≡ 
                     
                       
                         ∑ 
                         ± 
                         4 
                       
                        
                       
                         
                           
                             sin 
                              
                             
                               [ 
                               
                                 
                                   π 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           l 
                                            
                                           
                                               
                                           
                                            
                                           Δ 
                                            
                                           
                                               
                                           
                                            
                                           x 
                                         
                                         a 
                                       
                                       ± 
                                       
                                         
                                           m 
                                            
                                           
                                               
                                           
                                            
                                           Δ 
                                            
                                           
                                               
                                           
                                            
                                           y 
                                         
                                         b 
                                       
                                     
                                     ± 
                                     
                                       
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         Δ 
                                          
                                         
                                             
                                         
                                          
                                         z 
                                       
                                       h 
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                            
                           
                             cos 
                              
                             
                               [ 
                               
                                 π 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           lx 
                                           m 
                                         
                                         a 
                                       
                                       ± 
                                       
                                         
                                           my 
                                           m 
                                         
                                         b 
                                       
                                     
                                     ± 
                                     
                                       
                                         nz 
                                         m 
                                       
                                       h 
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                         
                           ( 
                           
                             
                               
                                 
                                   l 
                                    
                                   
                                       
                                   
                                    
                                   Δ 
                                    
                                   
                                       
                                   
                                    
                                   x 
                                 
                                 a 
                               
                               ± 
                               
                                 
                                   m 
                                    
                                   
                                       
                                   
                                    
                                   Δ 
                                    
                                   
                                       
                                   
                                    
                                   y 
                                 
                                 b 
                               
                             
                             ± 
                             
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 z 
                               
                               h 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with Δx=x 2 −x 1 ; Δy=y 2 −y; Δz=z 2 −z 1 ; x m =(x 2 +x 1 )/2; y m =(y 2 +y 1 )/2; z m =(z 2 +z 1 )/2.
 
The exponential term in time induces rapid convergence of the triple series, except for very small time.
 
         [0040]    A similar development would follow for constant pressure external boundary formulations. Whereas a solution with a sealed boundary corresponds to a Neumann function, a constant pressure boundary yields a Green&#39;s function solution. The computation can be further generalized to represent a cell of spatially invariant properties within a larger heterogeneous reservoir system decomposed into intercommunicating blocks. 
         [0041]    The more difficult term in Equation (8) to evaluate is the undamped term independent of time that contains spatial singularities. In dimensioned units, this is represented by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       P 
                        
                       
                         ( 
                         
                           x 
                           , 
                           y 
                           , 
                           
                             z 
                             ; 
                             
                               x 
                               1 
                             
                           
                           , 
                           
                             y 
                             1 
                           
                           , 
                           
                             
                               z 
                               1 
                             
                             ; 
                             α 
                           
                           , 
                           β 
                           , 
                           
                             γ 
                             ; 
                             L 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       P 
                       _ 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         C 
                         L 
                       
                       ) 
                     
                      
                     
                       
                         ∫ 
                         0 
                         L 
                       
                        
                       
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   S 
                                   XYZ 
                                 
                                 + 
                                 
                                   S 
                                   YZ 
                                 
                               
                               ) 
                             
                             + 
                             
                               ( 
                               
                                 
                                   S 
                                   XZ 
                                 
                                 + 
                                 
                                   S 
                                   Z 
                                 
                               
                               ) 
                             
                             + 
                             
                               ( 
                               
                                 
                                   S 
                                   XY 
                                 
                                 + 
                                 
                                   S 
                                   Y 
                                 
                               
                               ) 
                             
                             + 
                             
                               S 
                               X 
                             
                           
                           ] 
                         
                          
                         
                             
                         
                          
                         
                            
                           s 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             C 
             = 
             
               
                 ( 
                 
                   
                     4 
                      
                     
                         
                     
                      
                     π 
                      
                     
                         
                     
                      
                     B 
                      
                     
                         
                     
                      
                     μ 
                      
                     
                         
                     
                      
                     q 
                   
                   abh 
                 
                 ) 
               
               . 
             
           
         
       
     
         [0000]    Here,  P  is the spatial average pressure, B is a volume correction factor for the fluid, since standard practice is to cite the production rate at surface conditions, μ is the fluid viscosity, q is the volumetric production rate, and a, b, and h are the box dimensions. If standard engineering units are used, i.e. viscosity in centipoises, production rate in barrels per day, permeability in millidarcies, pressure in psi, and lengths in feet, a conversion factor of 
         [0000]    
       
         
           
             70.6 
              
             
               
                 day 
                 · 
                 
                   ft 
                   3 
                 
                 · 
                 md 
               
               
                 cp 
                 · 
                 bbl 
               
             
           
         
       
     
         [0000]    is required to reconcile units. On the RHS of Equation (11), the point source solution consists of one triple infinite series (S XYZ ), three double infinite series, (S XY , S XZ , S YZ ), and three single infinite series (S X , S Y , S Z ). The triple infinite series, S XYZ , is given as 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                     XYZ 
                   
                   = 
                   
                     
                       ( 
                       
                         8 
                         
                           π 
                           2 
                         
                       
                       ) 
                     
                      
                     
                       
                         ∑ 
                         
                           l 
                           , 
                           m 
                           , 
                           
                             n 
                             = 
                             1 
                           
                         
                         ∞ 
                       
                        
                       
                         
                           
                             
                               
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         lx 
                                       
                                       a 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         my 
                                       
                                       b 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         nz 
                                       
                                       h 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           lx 
                                           o 
                                         
                                       
                                       a 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           my 
                                           o 
                                         
                                       
                                       b 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           nz 
                                           o 
                                         
                                       
                                       h 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 k 
                                 x 
                               
                                
                               
                                 ( 
                                 
                                   l 
                                   a 
                                 
                                 ) 
                               
                             
                             2 
                           
                           + 
                           
                             
                               
                                 k 
                                 y 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   b 
                                 
                                 ) 
                               
                             
                             2 
                           
                           + 
                           
                             
                               
                                 k 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   n 
                                   h 
                                 
                                 ) 
                               
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The double infinite series, S YZ , is given as 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                     YZ 
                   
                   = 
                   
                     
                       ( 
                       
                         4 
                         
                           π 
                           2 
                         
                       
                       ) 
                     
                      
                     
                       
                         ∑ 
                         
                           m 
                           , 
                           
                             n 
                             = 
                             1 
                           
                         
                         ∞ 
                       
                        
                       
                         
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   my 
                                 
                                 b 
                               
                               ) 
                             
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   nz 
                                 
                                 h 
                               
                               ) 
                             
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     my 
                                     o 
                                   
                                 
                                 b 
                               
                               ) 
                             
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     nz 
                                     o 
                                   
                                 
                                 h 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 k 
                                 y 
                               
                                
                               
                                 ( 
                                 
                                   m 
                                   b 
                                 
                                 ) 
                               
                             
                             2 
                           
                           + 
                           
                             
                               
                                 k 
                                 z 
                               
                                
                               
                                 ( 
                                 
                                   n 
                                   h 
                                 
                                 ) 
                               
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Analogous series S XY  and S XZ  are obtained by change of variables. The single infinite series, S X , is given as 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                     X 
                   
                   = 
                   
                     
                       ( 
                       
                         2 
                         
                           π 
                           2 
                         
                       
                       ) 
                     
                      
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           1 
                         
                         ∞ 
                       
                        
                       
                         
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   lx 
                                 
                                 a 
                               
                               ) 
                             
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     lx 
                                     o 
                                   
                                 
                                 a 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               k 
                               x 
                             
                              
                             
                               ( 
                               
                                 l 
                                 a 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Analogous series S Y  and S Z  are obtained by change of variables. 
         [0042]    The invention is not the delineation of Equation (8) but rather the reduction of these very slowly converging triple and double infinite series to readily computable parts for practical application on a computer. Any source term in Equation (11) without a dependency upon the integration parameter, s, is recognized as a trivial case, e.g. a well aligned with a principal axis, and is covered by developments in the public domain. The preferred embodiment of this invention would screen out the special cases where alternate computation schemes are advantageous, though these situations are tractable with the general formulas given, provided routine precautions are taken to avoid fatal computational errors, such as division by zero. For completeness, guidance regarding alternate schemes for these special cases is provided in a later section. The complementary solution with Dirichlet (constant pressure) boundary conditions is recovered by replacing all cosine terms with sine counterparts. Constant pressure conditions are especially relevant to cases of strong aquifer (large connected water reservoir) pressure support of hydrocarbon reservoirs. The Neumann case is further generalized herein to include material transport into or out of the cell through use of Green&#39;s Theorem and boundary integral equations. A comparable scheme can be generated starting with Equation (5). 
       DETAILED DEVELOPMENT 
       [0043]    For ease of notation, we divide the integral in Equation (11) into subintegrals. The insistence on direct integration using antiderivatives, also known to those skilled in the art as analytical integration, distinguishes the invention from prior art which approximates integration with numerical schemes. Analytical integration using antiderivatives successfully includes contributions from mathematical singularities without loss of accuracy, whereas strictly numerical approaches must evaluate integrands of spiked character, introducing unwarranted error and adding significantly to the computational burden. 
         [0000]        P ( x,y,Z;x   1   ,y   1   ,z   1   ;α,β,γ;L )−   P =C [( I   XYZ   +I   YZ )+( I   XZ   +I   Z )+( I   XY   +I   Y )+ I   X ]  (15)
 
         [0000]    The grouping of terms recognizes that certain expressions developed will contain lower dimensional series, eliminating the need for separate computations. The invention consists of a fast and accurate evaluation method for Equation (15) that avoids numerical quadrature schemes in favor of integration using antiderivatives with subsequent simplification using mathematical identities resulting in direct formulas and rapidly-convergent series and the successful implementation of such formulas and series summations on a computer for various industry needs requiring well productivity evaluation. 
       Integrated Triple Infinite Series (I XYZ ) 
       [0044]    Beginning with the most complicated term of Equation (15), I XYZ , we have 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       I 
                       XYZ 
                     
                     ≡ 
                     
                       
                         1 
                         L 
                       
                        
                       
                         
                           ∫ 
                           
                             s 
                             = 
                             0 
                           
                           L 
                         
                          
                         
                           
                             S 
                             XYZ 
                           
                            
                           
                               
                           
                            
                           
                              
                             s 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         8 
                         
                           
                             π 
                             2 
                           
                            
                           L 
                         
                       
                       ) 
                     
                      
                     
                       
                         ∫ 
                         0 
                         L 
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             , 
                             m 
                             , 
                             
                               n 
                               = 
                               1 
                             
                           
                           ∞ 
                         
                          
                         
                           
                             
                               
                                 
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           lx 
                                         
                                         a 
                                       
                                       ) 
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             π 
                                              
                                             
                                                 
                                             
                                              
                                             my 
                                           
                                           b 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             π 
                                              
                                             
                                                 
                                             
                                              
                                             nz 
                                           
                                           h 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             lx 
                                             o 
                                           
                                         
                                         a 
                                       
                                       ) 
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             π 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               my 
                                               o 
                                             
                                           
                                           b 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             π 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               nz 
                                               o 
                                             
                                           
                                           h 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     k 
                                     x 
                                   
                                    
                                   
                                     ( 
                                     
                                       l 
                                       a 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     k 
                                     y 
                                   
                                    
                                   
                                     ( 
                                     
                                       m 
                                       b 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     k 
                                     z 
                                   
                                    
                                   
                                     ( 
                                     
                                       n 
                                       h 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                             
                           
                            
                           
                               
                           
                            
                           
                              
                             s 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Next we reduce the triple series to a double series and then integrate with respect to s. Rearranging and using the identity, 2 cos A cos B=cos(A+B)+cos(A−B), 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     XYZ 
                   
                   = 
                   
                     
                       ( 
                       
                         4 
                         
                           
                             π 
                             2 
                           
                            
                           L 
                         
                       
                       ) 
                     
                      
                     
                       
                         ∫ 
                         0 
                         L 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             , 
                             
                               n 
                               = 
                               1 
                             
                           
                           ∞ 
                         
                          
                         
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 my 
                                               
                                               b 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 nz 
                                               
                                               h 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   my 
                                                   o 
                                                 
                                               
                                               b 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   nz 
                                                   o 
                                                 
                                               
                                               h 
                                             
                                             ) 
                                           
                                         
                                       
                                       
                                         
                                           k 
                                           x 
                                         
                                         
                                           a 
                                           2 
                                         
                                       
                                     
                                     · 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         l 
                                         = 
                                         1 
                                       
                                       ∞ 
                                     
                                      
                                     
                                       
                                         [ 
                                         
                                           
                                             cos 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   π 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     l 
                                                      
                                                     
                                                       ( 
                                                       
                                                         x 
                                                         + 
                                                         
                                                           x 
                                                           o 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 a 
                                               
                                               ) 
                                             
                                           
                                           + 
                                           
                                             cos 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   π 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     l 
                                                      
                                                     
                                                       ( 
                                                       
                                                         x 
                                                         - 
                                                         
                                                           x 
                                                           o 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 a 
                                               
                                               ) 
                                             
                                           
                                         
                                         ] 
                                       
                                       
                                         
                                           l 
                                           2 
                                         
                                         + 
                                         
                                           
                                             a 
                                             2 
                                           
                                            
                                           
                                             [ 
                                             
                                               
                                                 
                                                   
                                                     k 
                                                     y 
                                                   
                                                   
                                                     k 
                                                     x 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                     ( 
                                                     
                                                       m 
                                                       b 
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                               
                                               + 
                                               
                                                 
                                                   
                                                     k 
                                                     z 
                                                   
                                                   
                                                     k 
                                                     x 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                     ( 
                                                     
                                                       n 
                                                       h 
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             } 
                           
                            
                           
                               
                           
                            
                           
                              
                             s 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0045]    Applying the identity, 
         [0000]    
       
         
           
             
               
                 
                   ∑ 
                   
                     n 
                     = 
                     1 
                   
                   ∞ 
                 
                  
                 
                   
                     
                       cos 
                        
                       
                         ( 
                         
                           π 
                            
                           
                               
                           
                            
                           nz 
                         
                         ) 
                       
                     
                      
                     
                         
                     
                   
                   
                     
                       n 
                       2 
                     
                     + 
                     
                       β 
                       2 
                     
                   
                 
               
               = 
               
                 
                   
                     ( 
                     
                       π 
                       
                         2 
                          
                         
                             
                         
                          
                         β 
                       
                     
                     ) 
                   
                    
                   
                     
                       cosh 
                        
                       
                         [ 
                         
                           π 
                            
                           
                               
                           
                            
                           
                             β 
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 z 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     
                       sinh 
                        
                       
                         ( 
                         
                           π 
                            
                           
                               
                           
                            
                           β 
                         
                         ) 
                       
                     
                   
                 
                 - 
                 
                   1 
                   
                     2 
                      
                     
                         
                     
                      
                     
                       β 
                       2 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    0≦z≦2, from Gradshteyn and Rhyzik (2007, p. 47) and rewriting the hyperbolic functions in their exponential forms, we get 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     XYZ 
                   
                   = 
                   
                     
                       ( 
                       
                         4 
                         
                           
                             π 
                             2 
                           
                            
                           L 
                         
                       
                       ) 
                     
                      
                     
                       
                         ∫ 
                         0 
                         L 
                       
                        
                       
                         
                           { 
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         a 
                                       
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           x 
                                         
                                       
                                     
                                     ) 
                                   
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         m 
                                         , 
                                         
                                           n 
                                           = 
                                           1 
                                         
                                       
                                       ∞ 
                                     
                                      
                                     
                                       
                                         
                                           
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     π 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     my 
                                                   
                                                   b 
                                                 
                                                 ) 
                                               
                                                
                                               
                                                 cos 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       π 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       nz 
                                                     
                                                     h 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     π 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       my 
                                                       o 
                                                     
                                                   
                                                   b 
                                                 
                                                 ) 
                                               
                                                
                                               
                                                 cos 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       π 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         nz 
                                                         o 
                                                       
                                                     
                                                     h 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               ( 
                                               
                                                 1 
                                                 - 
                                                 
                                                    
                                                   
                                                     
                                                       
                                                         - 
                                                         2 
                                                       
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       π 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       a 
                                                        
                                                       
                                                         
                                                           
                                                             
                                                             
                                                             k 
                                                             y 
                                                             
                                                             
                                                             k 
                                                             x 
                                                             
                                                             
                                                              
                                                             
                                                             
                                                             m 
                                                             2 
                                                             
                                                             
                                                             b 
                                                             2 
                                                             
                                                             
                                                           
                                                           + 
                                                           
                                                             
                                                             
                                                             k 
                                                             z 
                                                             
                                                             
                                                             k 
                                                             x 
                                                             
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             2 
                                                             
                                                             
                                                             h 
                                                             2 
                                                             
                                                             
                                                           
                                                         
                                                       
                                                     
                                                      
                                                     
                                                         
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     
                                                       k 
                                                       y 
                                                     
                                                     
                                                       k 
                                                       x 
                                                     
                                                   
                                                    
                                                   
                                                     
                                                       m 
                                                       2 
                                                     
                                                     
                                                       b 
                                                       2 
                                                     
                                                   
                                                 
                                                 + 
                                                 
                                                   
                                                     
                                                       k 
                                                       z 
                                                     
                                                     
                                                       k 
                                                       x 
                                                     
                                                   
                                                    
                                                   
                                                     
                                                       n 
                                                       2 
                                                     
                                                     
                                                       h 
                                                       2 
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                                
                                               
                                                 
                                                   - 
                                                   π 
                                                 
                                                  
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           k 
                                                           y 
                                                         
                                                         
                                                           k 
                                                           x 
                                                         
                                                       
                                                        
                                                       
                                                         
                                                           m 
                                                           2 
                                                         
                                                         
                                                           b 
                                                           2 
                                                         
                                                       
                                                     
                                                     + 
                                                     
                                                       
                                                         
                                                           k 
                                                           z 
                                                         
                                                         
                                                           k 
                                                           x 
                                                         
                                                       
                                                        
                                                       
                                                         
                                                           n 
                                                           2 
                                                         
                                                         
                                                           h 
                                                           2 
                                                         
                                                       
                                                     
                                                   
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                    
                                                   
                                                     x 
                                                     ± 
                                                     
                                                       x 
                                                       o 
                                                     
                                                   
                                                    
                                                 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                       
                                         
                                           
                                              
                                             
                                               
                                                 - 
                                                 π 
                                               
                                                
                                               
                                                 
                                                   
                                                     
                                                       
                                                         k 
                                                         y 
                                                       
                                                       
                                                         k 
                                                         x 
                                                       
                                                     
                                                      
                                                     
                                                       
                                                         m 
                                                         2 
                                                       
                                                       
                                                         b 
                                                         2 
                                                       
                                                     
                                                   
                                                   + 
                                                   
                                                     
                                                       
                                                         k 
                                                         z 
                                                       
                                                       
                                                         k 
                                                         x 
                                                       
                                                     
                                                      
                                                     
                                                       
                                                         n 
                                                         2 
                                                       
                                                       
                                                         h 
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     2 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     a 
                                                   
                                                   - 
                                                   
                                                      
                                                     
                                                       x 
                                                       ± 
                                                       
                                                         x 
                                                         o 
                                                       
                                                     
                                                      
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       m 
                                       , 
                                       
                                         n 
                                         = 
                                         1 
                                       
                                     
                                     ∞ 
                                   
                                    
                                   
                                     
                                       
                                         
                                           
                                             cos 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   π 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   my 
                                                 
                                                 b 
                                               
                                               ) 
                                             
                                              
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     π 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     nz 
                                                   
                                                   h 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             cos 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   π 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     my 
                                                     o 
                                                   
                                                 
                                                 b 
                                               
                                               ) 
                                             
                                              
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     π 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       nz 
                                                       o 
                                                     
                                                   
                                                   h 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                     
                                     
                                       ( 
                                       
                                         
                                           
                                             k 
                                             y 
                                           
                                            
                                           
                                             
                                               m 
                                               2 
                                             
                                             
                                               b 
                                               2 
                                             
                                           
                                         
                                         + 
                                         
                                           
                                             k 
                                             z 
                                           
                                            
                                           
                                             
                                               n 
                                               2 
                                             
                                             
                                               h 
                                               2 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           } 
                         
                          
                         
                             
                         
                          
                         
                            
                           s 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the symbol “±” is used to denote the sum of two terms: one plus sign and the other with the minus sign. Here, the two “±” terms arise from the two cosine terms in the inner sum of Equation (17). We immediately identify the integral of second sum in Equation (18) as I YZ , eliminating the need for a separate expression. 
         [0046]    If we restrict our attention to geometries with orientations such that 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   a 
                   
                     
                       k 
                       x 
                     
                   
                 
                 ) 
               
               ≥ 
               
                 max 
                 ( 
                 
                   
                     b 
                     
                       
                         k 
                         y 
                       
                     
                   
                   , 
                   
                     h 
                     
                       
                         k 
                         z 
                       
                     
                   
                 
                 ) 
               
             
             , 
           
         
       
     
         [0000]    we note that, for m, n=1, 2, 3, . . . 
         [0000]    
       
         
           
             
               max 
               ( 
               
                 
                    
                   
                     
                       - 
                       
                         
                           2 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           am 
                         
                         b 
                       
                     
                      
                     
                       
                         
                           k 
                           y 
                         
                         
                           k 
                           x 
                         
                       
                     
                   
                 
                 , 
                 
                    
                   
                     
                       - 
                       
                         
                           2 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           an 
                         
                         h 
                       
                     
                      
                     
                       
                         k 
                         z 
                       
                       
                         k 
                         x 
                       
                     
                   
                 
                 , 
                 
                   
                      
                     
                       
                         - 
                         2 
                       
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       a 
                     
                   
                    
                   
                     
                       
                         
                           m 
                           
                             2 
                             
                               k 
                               y 
                             
                           
                         
                         
                           b 
                           
                             2 
                             
                               k 
                               x 
                             
                           
                         
                       
                       + 
                       
                         
                           n 
                           
                             2 
                             
                               k 
                               z 
                             
                           
                         
                         
                           h 
                           
                             2 
                             
                               k 
                               x 
                             
                           
                         
                       
                     
                   
                 
               
               ) 
             
             &lt; 
             
                
               
                 
                   - 
                   2 
                 
                  
                 
                     
                 
                  
                 π 
               
             
             &lt; 
             
               0.00187 
               . 
             
           
         
       
     
         [0000]    Thus, we can drop the exponential term in the denominator of Equation (18) with no practical loss of accuracy. This allows analytical reduction of the integral through identification of an appropriate antiderivative. We rewrite Equation (18) as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           I 
                           XYZ 
                         
                         + 
                         
                           I 
                           YZ 
                         
                       
                       ) 
                     
                     ≈ 
                     
                       
                         ( 
                         
                           a 
                           
                             
                               π 
                               2 
                             
                              
                             
                               Lk 
                               x 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           ∑ 
                           
                             m 
                             , 
                             
                               n 
                               = 
                               1 
                             
                           
                           ∞ 
                         
                          
                         
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   my 
                                 
                                 b 
                               
                               ) 
                             
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   nz 
                                 
                                 h 
                               
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 I 
                                 1 
                               
                               + 
                               
                                 I 
                                 2 
                               
                               + 
                               
                                 I 
                                 3 
                               
                               + 
                               
                                 I 
                                 4 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         I 
                         
                           
                             j 
                             = 
                             1 
                           
                           , 
                           4 
                         
                       
                       ≡ 
                       
                         2 
                          
                         
                             
                         
                          
                         π 
                          
                         
                           
                             ∫ 
                             
                               s 
                               = 
                               0 
                             
                             
                               s 
                               = 
                               L 
                             
                           
                            
                           
                             
                               
                                 E 
                                 j 
                               
                               · 
                               
                                 cos 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         m 
                                       
                                       b 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           y 
                                           1 
                                         
                                         + 
                                         
                                           β 
                                            
                                           
                                               
                                           
                                            
                                           s 
                                         
                                       
                                       ) 
                                     
                                   
                                   ] 
                                 
                               
                             
                              
                             
                               cos 
                                
                               
                                 [ 
                                 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       n 
                                     
                                     h 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         z 
                                         1 
                                       
                                       + 
                                       
                                         γ 
                                          
                                         
                                             
                                         
                                          
                                         s 
                                       
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                              
                             
                                 
                             
                              
                             
                                
                               s 
                             
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     or 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       I 
                       
                         
                           j 
                           = 
                           1 
                         
                         , 
                         4 
                       
                     
                     ≡ 
                     
                       π 
                        
                       
                         
                           ∫ 
                           
                             s 
                             = 
                             0 
                           
                           
                             s 
                             = 
                             L 
                           
                         
                          
                         
                           
                             
                               E 
                               j 
                             
                             · 
                             cos 
                           
                            
                           
                               
                           
                            
                           
                             π 
                              
                             
                               [ 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         m 
                                         
                                           y 
                                           1 
                                         
                                       
                                       b 
                                     
                                     ± 
                                     
                                       
                                         n 
                                         
                                           z 
                                           1 
                                         
                                       
                                       h 
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   s 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           m 
                                            
                                           
                                               
                                           
                                            
                                           β 
                                         
                                         b 
                                       
                                       ± 
                                       
                                         
                                           n 
                                            
                                           
                                               
                                           
                                            
                                           γ 
                                         
                                         h 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                               
                           
                            
                           
                              
                             s 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0047]    If we introduce the shorthand notation, 
         [0000]    
       
         
           
             
               
                 ξ 
                 k 
               
               ≡ 
               
                 
                   
                     
                       
                         k 
                         y 
                       
                        
                       
                         m 
                         2 
                       
                     
                     
                       
                         k 
                         x 
                       
                        
                       
                         b 
                         2 
                       
                     
                   
                   + 
                   
                     
                       
                         k 
                         z 
                       
                        
                       
                         n 
                         2 
                       
                     
                     
                       
                         k 
                         x 
                       
                        
                       
                         h 
                         2 
                       
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                     1 
                   
                   ≡ 
                   
                     
                       1 
                       
                         ξ 
                         k 
                       
                     
                      
                     
                        
                       
                         
                           - 
                           
                             π 
                              
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                 
                                 + 
                                 
                                   α 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                                 + 
                                 x 
                               
                               ) 
                             
                           
                         
                          
                         
                           ξ 
                           k 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     E 
                     2 
                   
                   ≡ 
                   
                     
                       1 
                       
                         ξ 
                         k 
                       
                     
                      
                     
                        
                       
                         
                           - 
                           
                             π 
                              
                             
                               [ 
                               
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   a 
                                 
                                 - 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       1 
                                     
                                     + 
                                     
                                       α 
                                        
                                       
                                           
                                       
                                        
                                       s 
                                     
                                     + 
                                     x 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                          
                         
                           ξ 
                           k 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     E 
                     3 
                   
                   ≡ 
                   
                     
                       1 
                       
                         ξ 
                         k 
                       
                     
                      
                     
                        
                       
                         
                           - 
                           π 
                         
                          
                         
                            
                           
                             
                               x 
                               1 
                             
                             + 
                             
                               α 
                                
                               
                                   
                               
                                
                               s 
                             
                             - 
                             x 
                           
                            
                         
                          
                         
                           ξ 
                           k 
                         
                       
                     
                      
                     
                         
                     
                      
                     and 
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   
                     E 
                     4 
                   
                   ≡ 
                   
                     
                       1 
                       
                         ξ 
                         k 
                       
                     
                      
                     
                        
                       
                         
                           - 
                           
                             π 
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   a 
                                 
                                 - 
                                 
                                    
                                   
                                     
                                       x 
                                       1 
                                     
                                     + 
                                     
                                       α 
                                        
                                       
                                           
                                       
                                        
                                       s 
                                     
                                     - 
                                     x 
                                   
                                    
                                 
                               
                               ) 
                             
                           
                         
                          
                         
                           ξ 
                           k 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using the identity 
         [0000]    
       
         
           
             
               
                 ∫ 
                 
                   
                     
                        
                       at 
                     
                     · 
                     
                       cos 
                        
                       
                         ( 
                         bt 
                         ) 
                       
                     
                   
                    
                   
                       
                   
                    
                   
                      
                     t 
                   
                 
               
               ≡ 
               
                 
                   
                      
                     at 
                   
                    
                   
                     ( 
                     
                       
                         b 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             bt 
                             ) 
                           
                         
                       
                       + 
                       
                         a 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             bt 
                             ) 
                           
                         
                       
                     
                     ) 
                   
                 
                 
                   
                     a 
                     2 
                   
                   + 
                   
                     b 
                     2 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    we evaluate the integrals I 1 , I ,  I 3 , and I 4 . Integration of the first two terms, I 1  and I 2 , is straightforward. 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     1 
                   
                   ≡ 
                   
                     
                       1 
                       
                         
                           
                             ( 
                             
                               
                                 
                                   m 
                                    
                                   
                                       
                                   
                                    
                                   β 
                                 
                                 b 
                               
                               ± 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   γ 
                                 
                                 h 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             α 
                             2 
                           
                            
                           
                             ξ 
                             k 
                             2 
                           
                         
                       
                     
                      
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                    
                                   
                                     
                                       - 
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             
                                               x 
                                               2 
                                             
                                             + 
                                             x 
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                     
                                       ξ 
                                       k 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           
                                             
                                               1 
                                               
                                                 ξ 
                                                 k 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     m 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     β 
                                                   
                                                   b 
                                                 
                                                 ± 
                                                 
                                                   
                                                     n 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     γ 
                                                   
                                                   h 
                                                 
                                               
                                               ) 
                                             
                                              
                                             sin 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               π 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       my 
                                                       2 
                                                     
                                                     b 
                                                   
                                                   ± 
                                                   
                                                     
                                                       nz 
                                                       2 
                                                     
                                                     h 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           - 
                                         
                                       
                                     
                                     
                                       
                                         
                                           α 
                                            
                                           
                                               
                                           
                                            
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     my 
                                                     2 
                                                   
                                                   b 
                                                 
                                                 ± 
                                                 
                                                   
                                                     nz 
                                                     2 
                                                   
                                                   h 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                  
                                 
                                   
                                     - 
                                     
                                       π 
                                        
                                       
                                         ( 
                                         
                                           
                                             x 
                                             1 
                                           
                                           + 
                                           x 
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     ξ 
                                     k 
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           
                                             1 
                                             
                                               ξ 
                                               k 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   m 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   β 
                                                 
                                                 b 
                                               
                                               ± 
                                               
                                                 
                                                   n 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   γ 
                                                 
                                                 h 
                                               
                                             
                                             ) 
                                           
                                            
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     my 
                                                     1 
                                                   
                                                   b 
                                                 
                                                 ± 
                                                 
                                                   
                                                     nz 
                                                     1 
                                                   
                                                   h 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         - 
                                       
                                     
                                   
                                   
                                     
                                       
                                         α 
                                          
                                         
                                             
                                         
                                          
                                         cos 
                                          
                                         
                                             
                                         
                                          
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   my 
                                                   1 
                                                 
                                                 b 
                                               
                                               ± 
                                               
                                                 
                                                   nz 
                                                   1 
                                                 
                                                 h 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     I 
                     2 
                   
                   ≡ 
                   
                     
                       1 
                       
                         
                           
                             ( 
                             
                               
                                 
                                   m 
                                    
                                   
                                       
                                   
                                    
                                   β 
                                 
                                 b 
                               
                               ± 
                               
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   γ 
                                 
                                 h 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             α 
                             2 
                           
                            
                           
                             ξ 
                             k 
                             2 
                           
                         
                       
                     
                      
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                    
                                   
                                     
                                       - 
                                       
                                         π 
                                          
                                         
                                           [ 
                                           
                                             
                                               2 
                                                
                                               
                                                   
                                               
                                                
                                               a 
                                             
                                             - 
                                             
                                               ( 
                                               
                                                 
                                                   x 
                                                   2 
                                                 
                                                 + 
                                                 x 
                                               
                                               ) 
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                      
                                     
                                       ξ 
                                       k 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           
                                             
                                               1 
                                               
                                                 ξ 
                                                 k 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     m 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     β 
                                                   
                                                   b 
                                                 
                                                 ± 
                                                 
                                                   
                                                     n 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     γ 
                                                   
                                                   h 
                                                 
                                               
                                               ) 
                                             
                                              
                                             sin 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               π 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       my 
                                                       2 
                                                     
                                                     b 
                                                   
                                                   ± 
                                                   
                                                     
                                                       nz 
                                                       2 
                                                     
                                                     h 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           + 
                                         
                                       
                                     
                                     
                                       
                                         
                                           α 
                                            
                                           
                                               
                                           
                                            
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     my 
                                                     2 
                                                   
                                                   b 
                                                 
                                                 ± 
                                                 
                                                   
                                                     nz 
                                                     2 
                                                   
                                                   h 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                  
                                 
                                   
                                     - 
                                     
                                       π 
                                        
                                       
                                         [ 
                                         
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             a 
                                           
                                           - 
                                           
                                             ( 
                                             
                                               
                                                 x 
                                                 1 
                                               
                                               + 
                                               x 
                                             
                                             ) 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                    
                                   
                                     ξ 
                                     k 
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           
                                             1 
                                             
                                               ξ 
                                               k 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   m 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   β 
                                                 
                                                 b 
                                               
                                               ± 
                                               
                                                 
                                                   n 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   γ 
                                                 
                                                 h 
                                               
                                             
                                             ) 
                                           
                                            
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     my 
                                                     1 
                                                   
                                                   b 
                                                 
                                                 ± 
                                                 
                                                   
                                                     nz 
                                                     1 
                                                   
                                                   h 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         α 
                                          
                                         
                                             
                                         
                                          
                                         cos 
                                          
                                         
                                             
                                         
                                          
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   my 
                                                   1 
                                                 
                                                 b 
                                               
                                               ± 
                                               
                                                 
                                                   nz 
                                                   1 
                                                 
                                                 h 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Integration of I 3  and I 4  is also straightforward if x&lt;x 1  or x&gt;x 2 . However, within the well interval x 1 ≦x≦x 2 , the integral is split to correctly represent absolute values. 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     3 
                   
                   ≡ 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 π 
                                 
                                   ξ 
                                   k 
                                 
                               
                               ) 
                             
                             · 
                             
                               
                                 ∫ 
                                 
                                   s 
                                   = 
                                   0 
                                 
                                 
                                   
                                     α 
                                      
                                     
                                         
                                     
                                      
                                     s 
                                   
                                   = 
                                   
                                     x 
                                     - 
                                     
                                       x 
                                       1 
                                     
                                   
                                 
                               
                                
                               
                                 
                                   
                                      
                                     
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             
                                               x 
                                               1 
                                             
                                             + 
                                             
                                               α 
                                                
                                               
                                                   
                                               
                                                
                                               s 
                                             
                                             - 
                                             x 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         ξ 
                                         k 
                                       
                                     
                                   
                                   · 
                                   cos 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   π 
                                   [ 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             my 
                                             1 
                                           
                                           b 
                                         
                                         ± 
                                         
                                           
                                             nz 
                                             1 
                                           
                                           h 
                                         
                                       
                                       ) 
                                     
                                     + 
                                     
                                       s 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               m 
                                                
                                               
                                                   
                                               
                                                
                                               β 
                                             
                                             b 
                                           
                                           ± 
                                           
                                             
                                               n 
                                                
                                               
                                                   
                                               
                                                
                                               γ 
                                             
                                             h 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                       
                                   
                                   ] 
                                 
                                  
                                 
                                    
                                   s 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               π 
                               
                                 ξ 
                                 k 
                               
                             
                             ) 
                           
                           · 
                           
                             
                               ∫ 
                               
                                 
                                   α 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                                 = 
                                 
                                   x 
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                               
                               
                                 
                                   α 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                 
                                 = 
                                 
                                   
                                     x 
                                     2 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                     
                                       π 
                                        
                                       
                                         ( 
                                         
                                           x 
                                           - 
                                           
                                             x 
                                             1 
                                           
                                           - 
                                           
                                             α 
                                              
                                             
                                                 
                                             
                                              
                                             s 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       ξ 
                                       k 
                                     
                                   
                                 
                                 · 
                                 cos 
                               
                                
                               
                                   
                               
                                
                               
                                 π 
                                 [ 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           my 
                                           1 
                                         
                                         b 
                                       
                                       ± 
                                       
                                         
                                           nz 
                                           1 
                                         
                                         h 
                                       
                                     
                                     ) 
                                   
                                   + 
                                   
                                     s 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             m 
                                              
                                             
                                                 
                                             
                                              
                                             β 
                                           
                                           b 
                                         
                                         ± 
                                         
                                           
                                             n 
                                              
                                             
                                                 
                                             
                                              
                                             γ 
                                           
                                           h 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                  
                                 
                                     
                                 
                                 ] 
                               
                                
                               
                                  
                                 s 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Integrating and simplifying, we obtain the general expression 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     3 
                   
                   ≡ 
                   
                     
                       1 
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     m 
                                      
                                     
                                         
                                     
                                      
                                     β 
                                   
                                   b 
                                 
                                 ± 
                                 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     γ 
                                   
                                   h 
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               α 
                               2 
                             
                              
                             
                               ξ 
                               k 
                               2 
                             
                           
                         
                         ] 
                       
                     
                     · 
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   
                                     H 
                                      
                                     
                                       ( 
                                       
                                         x 
                                         - 
                                         
                                           x 
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                   · 
                                   
                                     H 
                                      
                                     
                                       ( 
                                       
                                         
                                           x 
                                           2 
                                         
                                         - 
                                         x 
                                       
                                       ) 
                                     
                                   
                                   · 
                                   2 
                                 
                                  
                                 
                                     
                                 
                                  
                                 α 
                                  
                                 
                                     
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   π 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               
                                                 m 
                                                 b 
                                               
                                                
                                               
                                                 [ 
                                                 
                                                   
                                                     y 
                                                     1 
                                                   
                                                   + 
                                                   
                                                     
                                                       β 
                                                       α 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         x 
                                                         - 
                                                         
                                                           x 
                                                           1 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 ] 
                                               
                                             
                                             ± 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               n 
                                               h 
                                             
                                              
                                             
                                               [ 
                                               
                                                 
                                                   z 
                                                   1 
                                                 
                                                 + 
                                                 
                                                   
                                                     γ 
                                                     α 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       x 
                                                       - 
                                                       
                                                         x 
                                                         1 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ] 
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                    
                                   
                                     
                                       - 
                                       π 
                                     
                                      
                                     
                                        
                                       
                                         x 
                                         - 
                                         
                                           x 
                                           2 
                                         
                                       
                                        
                                     
                                      
                                     
                                       ξ 
                                       k 
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           
                                             
                                               1 
                                               
                                                 ξ 
                                                 k 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     m 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     β 
                                                   
                                                   b 
                                                 
                                                 ± 
                                                 
                                                   
                                                     n 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     γ 
                                                   
                                                   h 
                                                 
                                               
                                               ) 
                                             
                                              
                                             sin 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               π 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       m 
                                                       b 
                                                     
                                                      
                                                     
                                                       y 
                                                       2 
                                                     
                                                   
                                                   ± 
                                                   
                                                     
                                                       n 
                                                       h 
                                                     
                                                      
                                                     
                                                       z 
                                                       2 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           + 
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             α 
                                             · 
                                             
                                               sign 
                                                
                                               
                                                 ( 
                                                 
                                                   x 
                                                   - 
                                                   
                                                     x 
                                                     2 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                            
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     m 
                                                     b 
                                                   
                                                    
                                                   
                                                     y 
                                                     2 
                                                   
                                                 
                                                 ± 
                                                 
                                                   
                                                     n 
                                                     h 
                                                   
                                                    
                                                   
                                                     z 
                                                     2 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                  
                                 
                                   
                                     - 
                                     π 
                                   
                                    
                                   
                                      
                                     
                                       x 
                                       - 
                                       
                                         x 
                                         1 
                                       
                                     
                                      
                                   
                                    
                                   
                                     ξ 
                                     k 
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           
                                             1 
                                             
                                               ξ 
                                               k 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   m 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   β 
                                                 
                                                 b 
                                               
                                               ± 
                                               
                                                 
                                                   n 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   γ 
                                                 
                                                 h 
                                               
                                             
                                             ) 
                                           
                                            
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     m 
                                                     b 
                                                   
                                                    
                                                   
                                                     y 
                                                     1 
                                                   
                                                 
                                                 ± 
                                                 
                                                   
                                                     n 
                                                     h 
                                                   
                                                    
                                                   
                                                     z 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           α 
                                           · 
                                           
                                             sign 
                                              
                                             
                                               ( 
                                               
                                                 x 
                                                 - 
                                                 
                                                   x 
                                                   1 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                          
                                         cos 
                                          
                                         
                                             
                                         
                                          
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   m 
                                                   b 
                                                 
                                                  
                                                 
                                                   y 
                                                   1 
                                                 
                                               
                                               ± 
                                               
                                                 
                                                   n 
                                                   h 
                                                 
                                                  
                                                 
                                                   z 
                                                   1 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the Heavyside function, H(x−x 1 ), was introduced to capture the first term only when x is within the interval, [x 1 , x 2 ]. Following a similar approach and dropping the exponential term, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           H 
                            
                           
                             ( 
                             
                               x 
                               - 
                               
                                 x 
                                 1 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           H 
                            
                           
                             ( 
                             
                               
                                 x 
                                 2 
                               
                               - 
                               x 
                             
                             ) 
                           
                         
                         · 
                         2 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         · 
                         
                            
                           
                             
                               - 
                               2 
                             
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             a 
                              
                             
                                 
                             
                              
                             
                               ξ 
                               k 
                             
                           
                         
                         · 
                       
                     
                   
                 
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         π 
                          
                         
                           ( 
                           
                             
                               
                                 m 
                                 b 
                               
                                
                               
                                 [ 
                                 
                                   
                                     y 
                                     1 
                                   
                                   + 
                                   
                                     
                                       β 
                                       α 
                                     
                                      
                                     
                                       ( 
                                       
                                         x 
                                         - 
                                         
                                           x 
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                             ± 
                             
                               
                                 n 
                                 h 
                               
                                
                               
                                 [ 
                                 
                                   
                                     z 
                                     1 
                                   
                                   + 
                                   
                                     
                                       γ 
                                       α 
                                     
                                      
                                     
                                       ( 
                                       
                                         x 
                                         - 
                                         
                                           x 
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 [ 
                 
                   
                     
                       ( 
                       
                         
                           
                             m 
                              
                             
                                 
                             
                              
                             β 
                           
                           b 
                         
                         ± 
                         
                           
                             n 
                              
                             
                                 
                             
                              
                             γ 
                           
                           h 
                         
                       
                       ) 
                     
                     2 
                   
                   + 
                   
                     
                       α 
                       2 
                     
                      
                     
                       ξ 
                       k 
                       2 
                     
                   
                 
                 ] 
               
             
             , 
           
         
       
     
       we get I 4 . 
       [0048]    
       
         
           
             
               
                 
                   
                     I 
                     4 
                   
                   ≡ 
                   
                     
                       1 
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     m 
                                      
                                     
                                         
                                     
                                      
                                     β 
                                   
                                   b 
                                 
                                 ± 
                                 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     γ 
                                   
                                   h 
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               α 
                               2 
                             
                              
                             
                               ξ 
                               k 
                               2 
                             
                           
                         
                         ] 
                       
                     
                     · 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                    
                                   
                                     - 
                                     
                                       π 
                                       ( 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           a 
                                         
                                         - 
                                         
                                             
                                         
                                          
                                         
                                           
                                              
                                             
                                               x 
                                               - 
                                               
                                                 x 
                                                 2 
                                               
                                             
                                              
                                           
                                            
                                           
                                             ξ 
                                             k 
                                           
                                         
                                       
                                     
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           
                                             
                                               1 
                                               
                                                 ξ 
                                                 k 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     m 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     β 
                                                   
                                                   b 
                                                 
                                                 ± 
                                                 
                                                   
                                                     n 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     γ 
                                                   
                                                   h 
                                                 
                                               
                                               ) 
                                             
                                              
                                             sin 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               π 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       m 
                                                       b 
                                                     
                                                      
                                                     
                                                       y 
                                                       2 
                                                     
                                                   
                                                   ± 
                                                   
                                                     
                                                       n 
                                                       h 
                                                     
                                                      
                                                     
                                                       z 
                                                       2 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           + 
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             α 
                                             · 
                                             
                                               sign 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     x 
                                                     2 
                                                   
                                                   - 
                                                   x 
                                                 
                                                 ) 
                                               
                                             
                                           
                                            
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     m 
                                                     b 
                                                   
                                                    
                                                   
                                                     y 
                                                     2 
                                                   
                                                 
                                                 ± 
                                                 
                                                   
                                                     n 
                                                     h 
                                                   
                                                    
                                                   
                                                     z 
                                                     2 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                  
                                 
                                   - 
                                   
                                     π 
                                     ( 
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         a 
                                       
                                       - 
                                       
                                         
                                            
                                           
                                             x 
                                             - 
                                             
                                               x 
                                               1 
                                             
                                           
                                            
                                         
                                          
                                         
                                           ξ 
                                           k 
                                         
                                       
                                     
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           
                                             1 
                                             
                                               ξ 
                                               k 
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   m 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   β 
                                                 
                                                 b 
                                               
                                               ± 
                                               
                                                 
                                                   n 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   γ 
                                                 
                                                 h 
                                               
                                             
                                             ) 
                                           
                                            
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     m 
                                                     b 
                                                   
                                                    
                                                   
                                                     y 
                                                     1 
                                                   
                                                 
                                                 ± 
                                                 
                                                   
                                                     n 
                                                     h 
                                                   
                                                    
                                                   
                                                     z 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           α 
                                           · 
                                           
                                             sign 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   x 
                                                   1 
                                                 
                                                 - 
                                                 x 
                                               
                                               ) 
                                             
                                           
                                         
                                          
                                         cos 
                                          
                                         
                                             
                                         
                                          
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   m 
                                                   b 
                                                 
                                                  
                                                 
                                                   y 
                                                   1 
                                                 
                                               
                                               ± 
                                               
                                                 
                                                   n 
                                                   h 
                                                 
                                                  
                                                 
                                                   z 
                                                   1 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    We have thus derived fast-computing formulas for the integrals, (I XYZ +I YZ ), involving the triple infinite series (S XYZ +S YZ ) of Equations (11) and (15), using Equations (26), (27), (29), and (30). 
         [0049]    In order to further distance the described invention from prior art using numerical methods, we note that once recast in the form of Eq (29), it should be obvious to one skilled in the art that the leading term in the I 3  component contains no exponential damping to expedite convergence, but rather it can be expressed by direct analytical formulas containing logarithmic terms upon application of the standard reduction formula given after Eq (17) and subsequent use of generalized analytical reduction procedures as demonstrated in prior art application for wells of restricted orientation and lower dimensionality (McCann et al., 2001). Proper and expedient evaluation of the term containing I 3  is necessary, as this terms typically contributes significantly in the vicinity of the wellbore. 
       Integrated Double Infinite Series (I XY  and I XZ ) 
       [0050]    Recalling I YZ  is exactly cancelled by a term in I XYZ , we focus on the terms (I XY +I Y ) and (I XZ +I Z ). Fast-computing terms for the double infinite series could be accomplished by following a similar procedure; however, they can more easily be obtained as special cases of the prior development. For example, taking n=0, eliminates the series in z. We note that the coefficient automatically takes on ½ values by omitting the repetition of “±” when n=0. Thus, with n=0 (and z absent) in Equation (19), we get 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         I 
                         XY 
                       
                       + 
                       
                         I 
                         Y 
                       
                     
                     ) 
                   
                   ≈ 
                   
                     
                       ( 
                       
                         a 
                         
                           
                             π 
                             2 
                           
                            
                           
                             Lk 
                             x 
                           
                         
                       
                       ) 
                     
                      
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           1 
                         
                         ∞ 
                       
                        
                       
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 my 
                               
                               b 
                             
                             ) 
                           
                         
                          
                         
                           ( 
                           
                             
                               I 
                               1 
                             
                             + 
                             
                               I 
                               2 
                             
                             + 
                             
                               I 
                               3 
                             
                             + 
                             
                               I 
                               4 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
         [0000]    or, in terms of the exponential functions used in Equations (26-30), with 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                             F 
                             1 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               
                                 
                                   x 
                                   i 
                                 
                                 ; 
                                 m 
                               
                               , 
                               n 
                             
                             ) 
                           
                         
                         ≡ 
                         
                           
                             
                                
                               
                                 - 
                                 
                                   π 
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         i 
                                       
                                       + 
                                       x 
                                     
                                     ) 
                                   
                                 
                               
                             
                              
                             
                               
                                 
                                   
                                     
                                       k 
                                       y 
                                     
                                      
                                     
                                       m 
                                       2 
                                     
                                   
                                   
                                     
                                       k 
                                       x 
                                     
                                      
                                     
                                       b 
                                       2 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       k 
                                       z 
                                     
                                      
                                     
                                       n 
                                       2 
                                     
                                   
                                   
                                     
                                       k 
                                       x 
                                     
                                      
                                     
                                       h 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     k 
                                     y 
                                   
                                    
                                   
                                     m 
                                     2 
                                   
                                 
                                 
                                   
                                     k 
                                     x 
                                   
                                    
                                   
                                     b 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     k 
                                     z 
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                 
                                 
                                   
                                     k 
                                     x 
                                   
                                    
                                   
                                     h 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                       , 
                       
                         
 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             F 
                             2 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               
                                 
                                   x 
                                   i 
                                 
                                 ; 
                                 m 
                               
                               , 
                               n 
                             
                             ) 
                           
                         
                         ≡ 
                         
                           
                             
                                
                               
                                 - 
                                 
                                   π 
                                    
                                   
                                     [ 
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         a 
                                       
                                       - 
                                       
                                         ( 
                                         
                                           
                                             x 
                                             i 
                                           
                                           + 
                                           x 
                                         
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                              
                             
                               
                                 
                                   
                                     
                                       k 
                                       y 
                                     
                                      
                                     
                                       m 
                                       2 
                                     
                                   
                                   
                                     
                                       k 
                                       x 
                                     
                                      
                                     
                                       b 
                                       2 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       k 
                                       z 
                                     
                                      
                                     
                                       n 
                                       2 
                                     
                                   
                                   
                                     
                                       k 
                                       x 
                                     
                                      
                                     
                                       h 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     k 
                                     y 
                                   
                                    
                                   
                                     m 
                                     2 
                                   
                                 
                                 
                                   
                                     k 
                                     x 
                                   
                                    
                                   
                                     b 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     k 
                                     z 
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                 
                                 
                                   
                                     k 
                                     x 
                                   
                                    
                                   
                                     h 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                       , 
                       
                         
 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             F 
                             3 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               
                                 
                                   x 
                                   i 
                                 
                                 ; 
                                 m 
                               
                               , 
                               n 
                             
                             ) 
                           
                         
                         ≡ 
                         
                           
                             
                                
                               
                                 
                                   - 
                                   π 
                                 
                                  
                                 
                                    
                                   
                                     
                                       x 
                                       i 
                                     
                                     + 
                                     x 
                                   
                                    
                                 
                               
                             
                              
                             
                               
                                 
                                   
                                     
                                       k 
                                       y 
                                     
                                      
                                     
                                       m 
                                       2 
                                     
                                   
                                   
                                     
                                       k 
                                       x 
                                     
                                      
                                     
                                       b 
                                       2 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       k 
                                       z 
                                     
                                      
                                     
                                       n 
                                       2 
                                     
                                   
                                   
                                     
                                       k 
                                       x 
                                     
                                      
                                     
                                       h 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     k 
                                     y 
                                   
                                    
                                   
                                     m 
                                     2 
                                   
                                 
                                 
                                   
                                     k 
                                     x 
                                   
                                    
                                   
                                     b 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     k 
                                     z 
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                 
                                 
                                   
                                     k 
                                     x 
                                   
                                    
                                   
                                     h 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                       , 
                       and 
                     
                      
                     
                       
 
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           
                             F 
                             4 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               
                                 
                                   x 
                                   i 
                                 
                                 ; 
                                 m 
                               
                               , 
                               n 
                             
                             ) 
                           
                         
                         ≡ 
                         
                           
                             
                                
                               
                                 - 
                                 
                                   π 
                                    
                                   
                                     ( 
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         a 
                                       
                                       - 
                                       
                                          
                                         
                                           
                                             x 
                                             i 
                                           
                                           - 
                                           x 
                                         
                                          
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                              
                             
                               
                                 
                                   
                                     
                                       k 
                                       y 
                                     
                                      
                                     
                                       m 
                                       2 
                                     
                                   
                                   
                                     
                                       k 
                                       x 
                                     
                                      
                                     
                                       b 
                                       2 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       k 
                                       z 
                                     
                                      
                                     
                                       n 
                                       2 
                                     
                                   
                                   
                                     
                                       k 
                                       x 
                                     
                                      
                                     
                                       h 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     k 
                                     y 
                                   
                                    
                                   
                                     m 
                                     2 
                                   
                                 
                                 
                                   
                                     k 
                                     x 
                                   
                                    
                                   
                                     b 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     k 
                                     z 
                                   
                                    
                                   
                                     n 
                                     2 
                                   
                                 
                                 
                                   
                                     k 
                                     x 
                                   
                                    
                                   
                                     h 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                       , 
                       
                         
                           for 
                            
                           
                               
                           
                            
                           i 
                         
                         = 
                         1 
                       
                       , 
                       2 
                       , 
                       
                         
 
                       
                        
                       
                         
                           ( 
                           
                             
                               I 
                               XY 
                             
                             + 
                             
                               I 
                               Y 
                             
                           
                           ) 
                         
                         ≈ 
                         
                           
                             
                               ab 
                               2 
                             
                             
                               
                                 π 
                                 2 
                               
                                
                               
                                 L 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         α 
                                         2 
                                       
                                        
                                       
                                         k 
                                         y 
                                       
                                     
                                     + 
                                     
                                       
                                         β 
                                         2 
                                       
                                        
                                       
                                         k 
                                         x 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                            
                           
                             { 
                             
                               
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       · 
                                       
                                         H 
                                          
                                         
                                           ( 
                                           
                                             x 
                                             - 
                                             
                                               x 
                                               1 
                                             
                                           
                                           ) 
                                         
                                       
                                       · 
                                       
                                         H 
                                          
                                         
                                           ( 
                                           
                                             
                                               x 
                                               2 
                                             
                                             - 
                                             x 
                                           
                                           ) 
                                         
                                       
                                       · 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         ∑ 
                                         
                                           m 
                                           = 
                                           1 
                                         
                                         ∞ 
                                       
                                        
                                       
                                         
                                           1 
                                           
                                             m 
                                             2 
                                           
                                         
                                          
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 my 
                                               
                                               b 
                                             
                                             ) 
                                           
                                         
                                          
                                         cos 
                                          
                                         
                                             
                                         
                                          
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               
                                                 m 
                                                 b 
                                               
                                                
                                               
                                                 [ 
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           y 
                                                           1 
                                                         
                                                         + 
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         
                                                           β 
                                                           α 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             x 
                                                             - 
                                                             
                                                             x 
                                                             1 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                 
                                                 ] 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         j 
                                         = 
                                         1 
                                       
                                       4 
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           m 
                                           = 
                                           1 
                                         
                                         ∞ 
                                       
                                        
                                       
                                         
                                           ( 
                                           
                                             1 
                                             
                                               m 
                                               2 
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                           
                                             F 
                                             j 
                                           
                                            
                                           
                                             ( 
                                             
                                               x 
                                               , 
                                               
                                                 
                                                   x 
                                                   2 
                                                 
                                                 ; 
                                                 m 
                                               
                                               , 
                                               0 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 my 
                                               
                                               b 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                 β 
                                                  
                                                 
                                                   
                                                     
                                                       
                                                         k 
                                                         x 
                                                       
                                                       
                                                         k 
                                                         y 
                                                       
                                                     
                                                   
                                                   · 
                                                   sin 
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   π 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         my 
                                                         2 
                                                       
                                                       b 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 α 
                                                 · 
                                                 
                                                   
                                                     λ 
                                                     j 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       x 
                                                       , 
                                                       
                                                         x 
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                 · 
                                                 cos 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 π 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       my 
                                                       2 
                                                     
                                                     b 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         j 
                                         = 
                                         1 
                                       
                                       4 
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           m 
                                           = 
                                           1 
                                         
                                         ∞ 
                                       
                                        
                                       
                                         
                                           ( 
                                           
                                             1 
                                             
                                               m 
                                               2 
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                           
                                             F 
                                             j 
                                           
                                            
                                           
                                             ( 
                                             
                                               x 
                                               , 
                                               
                                                 
                                                   x 
                                                   1 
                                                 
                                                 ; 
                                                 m 
                                               
                                               , 
                                               0 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 my 
                                               
                                               b 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             
                                               β 
                                                
                                               
                                                 
                                                   
                                                     
                                                       k 
                                                       x 
                                                     
                                                     
                                                       k 
                                                       y 
                                                     
                                                   
                                                 
                                                 · 
                                                 sin 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 π 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       my 
                                                       1 
                                                     
                                                     b 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               α 
                                               · 
                                               
                                                 
                                                   λ 
                                                   j 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     x 
                                                     , 
                                                     
                                                       x 
                                                       1 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               · 
                                               cos 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               π 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     my 
                                                     1 
                                                   
                                                   b 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where ξ k  was simplified by setting n=0, and 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 λ 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   
                                     x 
                                     i 
                                   
                                 
                                 ) 
                               
                             
                             = 
                             
                               - 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 λ 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   
                                     x 
                                     i 
                                   
                                 
                                 ) 
                               
                             
                             = 
                             1 
                           
                         
                       
                       
                         
                           
                             
                               
                                 λ 
                                 3 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   
                                     x 
                                     i 
                                   
                                 
                                 ) 
                               
                             
                             = 
                             
                               sign 
                                
                               
                                 ( 
                                 
                                   x 
                                   - 
                                   
                                     x 
                                     i 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 λ 
                                 4 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   
                                     x 
                                     i 
                                   
                                 
                                 ) 
                               
                             
                             = 
                             
                               sign 
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     i 
                                   
                                   - 
                                   x 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     } 
                   
                   , 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   2 
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
         [0000]    We get an analogous expression for (I XZ +I Z ). Thus, with m=0 (and y absent), in Equation (19) 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         I 
                         XZ 
                       
                       + 
                       
                         I 
                         Z 
                       
                     
                     ) 
                   
                   ≈ 
                   
                     
                       
                         ah 
                         2 
                       
                       
                         
                           π 
                           2 
                         
                          
                         
                           L 
                            
                           
                             ( 
                             
                               
                                 
                                   α 
                                   2 
                                 
                                  
                                 
                                   k 
                                   z 
                                 
                               
                               + 
                               
                                 
                                   γ 
                                   2 
                                 
                                  
                                 
                                   k 
                                   x 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                      
                     
                       { 
                       
                         
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 · 
                                 
                                   H 
                                    
                                   
                                     ( 
                                     
                                       x 
                                       - 
                                       
                                         x 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                                 · 
                                 
                                   H 
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         2 
                                       
                                       - 
                                       x 
                                     
                                     ) 
                                   
                                 
                                 · 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                     1 
                                     
                                       n 
                                       2 
                                     
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           nz 
                                         
                                         h 
                                       
                                       ) 
                                     
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     π 
                                      
                                     
                                       ( 
                                       
                                         
                                           n 
                                           h 
                                         
                                          
                                         
                                           [ 
                                           
                                             
                                               z 
                                               1 
                                             
                                             + 
                                             
                                               
                                                 γ 
                                                 α 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   x 
                                                   - 
                                                   
                                                     x 
                                                     1 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 ∑ 
                                 
                                   j 
                                   = 
                                   1 
                                 
                                 4 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       1 
                                       
                                         n 
                                         2 
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     
                                       F 
                                       j 
                                     
                                      
                                     
                                       ( 
                                       
                                         x 
                                         , 
                                         
                                           
                                             x 
                                             2 
                                           
                                           ; 
                                           0 
                                         
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           nz 
                                         
                                         h 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           γ 
                                            
                                           
                                             
                                               
                                                 
                                                   k 
                                                   x 
                                                 
                                                 
                                                   k 
                                                   z 
                                                 
                                               
                                             
                                             · 
                                             sin 
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   nz 
                                                   2 
                                                 
                                                 h 
                                               
                                               ) 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           α 
                                           · 
                                           
                                             
                                               λ 
                                               j 
                                             
                                              
                                             
                                               ( 
                                               
                                                 x 
                                                 , 
                                                 
                                                   x 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                           
                                           · 
                                           cos 
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               
                                                 nz 
                                                 2 
                                               
                                               h 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 ∑ 
                                 
                                   j 
                                   = 
                                   1 
                                 
                                 4 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       1 
                                       
                                         n 
                                         2 
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     
                                       F 
                                       j 
                                     
                                      
                                     
                                       ( 
                                       
                                         x 
                                         , 
                                         
                                           
                                             x 
                                             1 
                                           
                                           ; 
                                           0 
                                         
                                         , 
                                         n 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           nz 
                                         
                                         h 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         γ 
                                          
                                         
                                           
                                             
                                               
                                                 k 
                                                 x 
                                               
                                               
                                                 k 
                                                 z 
                                               
                                             
                                           
                                           · 
                                           sin 
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               
                                                 nz 
                                                 1 
                                               
                                               h 
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         α 
                                         · 
                                         
                                           
                                             λ 
                                             j 
                                           
                                            
                                           
                                             ( 
                                             
                                               x 
                                               , 
                                               
                                                 x 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                         · 
                                         cos 
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             
                                               nz 
                                               1 
                                             
                                             h 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where F j s are as previously defined with m=0.
 
Note that the first sums of the RHS of Equations (32) and (34) admit polynomial formulas, recognizing
 
         [0000]    
       
         
           
             
               
                 
                   
                     2 
                     · 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         ∞ 
                       
                        
                       
                         
                           
                             cos 
                              
                             
                               ( 
                               
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   
                                     x 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   
                                     x 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           n 
                           2 
                         
                       
                     
                   
                   ≡ 
                   
                     
                       
                         π 
                         2 
                       
                       [ 
                       
                         
                           1 
                           3 
                         
                         - 
                         
                           max 
                            
                           
                             ( 
                             
                               
                                 x 
                                 1 
                               
                               , 
                               
                                 x 
                                 2 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             
                               x 
                               1 
                               2 
                             
                             + 
                             
                               x 
                               2 
                               2 
                             
                           
                           2 
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
       Integrated Single Infinite Series (I X ) 
       [0051]    The integration of single series solution, S X , is straightforward. 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     X 
                   
                   ≡ 
                   
                     
                       ( 
                       
                         
                           2 
                           a 
                           2 
                         
                         
                           
                             π 
                             2 
                           
                            
                           
                             L 
                             
                               k 
                               x 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       
                         ∫ 
                         0 
                         L 
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             1 
                           
                           ∞ 
                         
                          
                         
                           
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       lx 
                                     
                                     a 
                                   
                                   ) 
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       
                                         l 
                                          
                                         
                                           ( 
                                           
                                             
                                               x 
                                               1 
                                             
                                             + 
                                             
                                               α 
                                                
                                               
                                                   
                                               
                                                
                                               s 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     a 
                                   
                                   ) 
                                 
                               
                             
                             
                               l 
                               2 
                             
                           
                            
                           
                               
                           
                            
                           
                              
                             s 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
             
               
                 
                   
                     I 
                     X 
                   
                   ≡ 
                   
                     
                       ( 
                       
                         
                           2 
                           a 
                           2 
                         
                         
                           
                             π 
                             2 
                           
                            
                           
                             L 
                             
                               k 
                               x 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           1 
                         
                         ∞ 
                       
                        
                       
                         
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   lx 
                                 
                                 a 
                               
                               ) 
                             
                           
                            
                           
                             [ 
                             
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       
                                         l 
                                         
                                           x 
                                           2 
                                         
                                       
                                     
                                     a 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       
                                         l 
                                         
                                           x 
                                           1 
                                         
                                       
                                     
                                     a 
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                         
                           
                             π 
                              
                             
                                 
                             
                              
                             α 
                              
                             
                                 
                             
                              
                             
                               l 
                               3 
                             
                           
                           a 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       I 
                       X 
                     
                     ≡ 
                     
                       
                         ( 
                         
                           
                             a 
                             3 
                           
                           
                             
                               π 
                               3 
                             
                              
                             α 
                              
                             
                                 
                             
                              
                             
                               L 
                               
                                 k 
                                 x 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             1 
                           
                           ∞ 
                         
                          
                         
                           
                             
                               sin 
                                
                               
                                 [ 
                                 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       l 
                                     
                                     a 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         2 
                                       
                                       ± 
                                       x 
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               sin 
                                
                               
                                 [ 
                                 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       l 
                                     
                                     a 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         1 
                                       
                                       ± 
                                       x 
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                           
                           
                             
                                 
                             
                              
                             
                               l 
                               3 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     Thus 
                     , 
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       I 
                       X 
                     
                     ≡ 
                     
                       
                         ( 
                         
                           
                             a 
                             3 
                           
                           
                             
                               π 
                               3 
                             
                              
                             α 
                              
                             
                                 
                             
                              
                             
                               L 
                               
                                 k 
                                 x 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     F 
                                     3 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           x 
                                           2 
                                         
                                         + 
                                         x 
                                       
                                       a 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     F 
                                     3 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           x 
                                           2 
                                         
                                         - 
                                         x 
                                       
                                       a 
                                     
                                     ) 
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     F 
                                     3 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           x 
                                           1 
                                         
                                         + 
                                         x 
                                       
                                       a 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     F 
                                     3 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           x 
                                           1 
                                         
                                         - 
                                         x 
                                       
                                       a 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           F 
                           3 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                       ≡ 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           ∞ 
                         
                          
                         
                           
                             sin 
                              
                             
                               ( 
                               
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 nz 
                               
                               ) 
                             
                           
                           
                             n 
                             3 
                           
                         
                       
                     
                     = 
                     
                       
                         π 
                         3 
                       
                       · 
                       z 
                       · 
                       
                         [ 
                         
                           
                             1 
                             6 
                           
                           - 
                           
                             
                               
                                  
                                 z 
                                  
                               
                               4 
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                      
                                     z 
                                      
                                   
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     0 
                     ≤ 
                     
                        
                       z 
                        
                     
                     ≤ 
                     2 
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
         [0000]    according to Gradshteyn and Rhyzik (2007, p. 47). In summary, the pressure at an arbitrary observation point (x, y, z) in a sealed 3-D box with spatially invariant, but possibly anisotropic, permeability (k x , k y , k z ) due to a line source segment of unit strength from (x 1 , y 1 , z 1 ) to (x 2 , y 2 , z 2 ) is given by Equation (8) and the combination of elements reduced to easily computable expressions represented by Equations (9, 15, 26, 27, 29, 30, 32, 34, and 39) for evaluation by a computer. To illustrate, the dimensionless pressure distribution, 
         [0000]    
       
         
           
             
               
                 
                   P 
                   ~ 
                 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
               ≡ 
               
                 
                   
                     ( 
                     
                       
                         P 
                          
                         
                           ( 
                           
                             x 
                             , 
                             y 
                             , 
                             z 
                           
                           ) 
                         
                       
                       - 
                       
                         P 
                         _ 
                       
                     
                     ) 
                   
                    
                   abh 
                 
                 
                   4 
                    
                   
                       
                   
                    
                   π 
                    
                   
                       
                   
                    
                   B 
                    
                   
                       
                   
                    
                   μ 
                    
                   
                       
                   
                    
                   q 
                 
               
             
             , 
           
         
       
     
         [0000]    for the central XZ slice in the plane containing a well with endpoints (x 1 , y 1 , z 1 )=(0.25, 0.5, 0.25) and (x 2 , y 2 , z 2 )=(0.75, 0.5, 0.75) in a cube with isotropic permeability is shown in  FIG. 2 . 
       Validation 
       [0052]    We validate the computation scheme against literature solutions for degenerate cases, i.e. simpler special cases of the new invention not utilizing its generality. In particular, we compare the invention with the three-dimensional solution given by Economides et al. (1996) for the transient behavior of centralized partially-penetrating horizontal wells.  FIG. 3  shows the dimensionless well pressure versus time, as defined by these authors, for the case of uniform wellbore pressure, for wells of length L/a=0.2 and 0.8, centered in a square drainage area of thickness h/a=0.05, with isotropic permeability and well radius r w /a=0.0004. The solid lines represent the digitized data of Economides et al. (1996), while the data points represent computations from the described invention. Of particular note is the generation of this data in under 0.25 seconds on a 2 GHz PC. 
         [0053]    We further validate the invention through comparison with the two-dimensional solution for transient behavior of centralized, partially-penetrating fractures given by Gringarten et al. (1974).  FIG. 4  gives the results of Gringarten et al. recast as the dimensionless pressure difference between the system average pressure and the pressure observed at the midpoint versus dimensionless time for uniform flux fractures of varying length. The solid lines represent the tabular data of Gringarten et al. (1974), while the data points indicate computations from a two-dimensional subset of the described invention. 
       Two Dimensional Application 
       [0054]    The two-dimensional case is also of practical interest and is recognized as a contained embodiment of the above development. When the Neumann function for a point source is integrated along a linear segment in the XY plane (which constitutes a planar source in 3-D), we merely pick up a subset of the developed terms for three dimensions. 
         [0000]        P ( x,y;x   1   ,y   1   ,α,β,L )−   P =C ·[( I   XY   +I   Y )+ I   X ]  (41)
 
         [0000]    The 2-D result is particularly useful in modeling of fractured wells or inclined wells in thin reservoirs for dimensionless values of the ratio 
         [0000]    
       
         
           
             ( 
             
               
                 a 
                 h 
               
                
               
                 
                   
                     k 
                     z 
                   
                   
                     k 
                     x 
                   
                 
               
             
             ) 
           
         
       
     
         [0000]    exceeding 5.  FIGS. 5-8  show application to two-dimensional modeling the dimensionless pressure with respect to the average pressure in complex fracture networks using literature interpretations of fracture patterns in Barnett shale.  FIGS. 5 and 6  pertain to a stimulated vertical Barnett well (Mayerhofer et al., 2006), while  FIGS. 7 and 8  are for a fractured horizontal well in the Barnett shale (Fisher et al., 2004). In unconventional resources, the reservoir permeability is low such that the practical lifetime of well operation is captured in the early transients. These results are a demonstration of a reduction to practice of the described invention. While the obtained dimensionless well productivity indices were obtained on the basis a unit volume of fluid withdrawal per unit time, they may be converted for variable rate interpretation using standard convolution methods to capture the high initial production rate (IPR) and rapid decline common in such unconventional reservoirs. 
       Well Pressure and Well Productivity 
       [0055]    Of particular interest is the observation point a distance r w , the well radius, from the mathematical line sink. Of historical interest has been the value at the midpoint of the well (see  FIG. 9 ) and another at roughly 70% of the distance from the midpoint to the well tip, where the uniform flux well boundary condition yields approximately the same characteristic value for pressure as does a uniform pressure boundary condition for symmetrically-placed wells with respect to cell boundaries, though the method allows for pressure determination at any specified point or sets of points. The determination of the pressure observable at a well location in this manner has been reduced to practice. In  FIG. 10 , we see the pressure distribution for a partially penetrating horizontal well near one of the boundaries and an exploded view of the pressure distribution in the area immediately surrounding the mathematical line source for the plane normal to the well passing through the well midpoint. For the line source with endpoints (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) and direction cosines (α, β, γ), the plane normal to the line passing through the midpoints, 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     x 
                     m 
                   
                   , 
                   
                     y 
                     m 
                   
                   , 
                   
                     z 
                     m 
                   
                 
                 ) 
               
               ≡ 
               
                 ( 
                 
                   
                     
                       
                         x 
                         1 
                       
                       + 
                       
                         x 
                         2 
                       
                     
                     2 
                   
                   , 
                   
                     
                       
                         y 
                         1 
                       
                       + 
                       
                         y 
                         2 
                       
                     
                     2 
                   
                   , 
                   
                     
                       
                         z 
                         1 
                       
                       + 
                       
                         z 
                         2 
                       
                     
                     2 
                   
                 
                 ) 
               
             
             , 
           
         
       
     
         [0000]    is represented by the equation 
         [0000]      α( x−x   m )+β( y−y   m )+γ( z−z   m )=0  (42)
 
         [0000]    Points on this plane a radial distance equal to the well radius, r w , from the midpoint are given by 
         [0000]      ( x−x   m ) 2 +( y−y   m ) 2 +( z−z   m ) 2   =r   2   w   (43)
 
         [0000]    In one embodiment of this invention, we take a limited number of such points, setting successively x=x m , y=y m , and  z =z m , yielding the observation points 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         x 
                         m 
                       
                       , 
                       
                         
                           y 
                           m 
                         
                         ∓ 
                         
                           
                             γ 
                             
                               r 
                               w 
                             
                           
                           
                             
                               
                                 β 
                                 2 
                               
                               + 
                               
                                 γ 
                                 2 
                               
                             
                           
                         
                       
                       , 
                       
                         
                           z 
                           m 
                         
                         ± 
                         
                           
                             
                               β 
                               
                                 r 
                                 w 
                               
                             
                              
                             
                                 
                             
                           
                           
                             
                               
                                 β 
                                 2 
                               
                               + 
                               
                                 γ 
                                 2 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                    
                   
                     
 
                   
                    
                   
                     ( 
                     
                       
                         
                           x 
                           m 
                         
                         ∓ 
                         
                           
                             γ 
                             
                               r 
                               w 
                             
                           
                           
                             
                               
                                 α 
                                 2 
                               
                               + 
                               
                                 γ 
                                 2 
                               
                             
                           
                         
                       
                       , 
                       
                         y 
                         m 
                       
                       , 
                       
                         
                           z 
                           m 
                         
                         ± 
                         
                           
                             α 
                             
                               r 
                               w 
                             
                           
                           
                             
                               
                                 α 
                                 2 
                               
                               + 
                               
                                 γ 
                                 2 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                    
                   
                     
 
                   
                    
                   
                     ( 
                     
                       
                         
                           x 
                           m 
                         
                         ∓ 
                         
                           
                             β 
                             
                               r 
                               w 
                             
                           
                           
                             
                               
                                 α 
                                 2 
                               
                               + 
                               
                                 β 
                                 2 
                               
                             
                           
                         
                       
                       , 
                       
                         
                           y 
                           m 
                         
                         ± 
                         
                           
                             α 
                             
                               r 
                               w 
                             
                           
                           
                             
                               
                                 α 
                                 2 
                               
                               + 
                               
                                 β 
                                 2 
                               
                             
                           
                         
                       
                       , 
                       
                         z 
                         m 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
         [0000]    We recommend taking an average of unique observation points along the wellpath and averaging those, if desired, to get an overall representative average for a well composed of multiple segments. 
       Advanced Well Designs 
       [0056]    A well trajectory can be approximated by any number of linear segments. By the superposition principle, the effects of all such line source segments on the pressure at an observation point (x, y, z) are additive. Thus, for any number of line source segments, N seg , within the box, we can write the expression for pressure as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       P 
                        
                       
                         ( 
                         
                           x 
                           , 
                           y 
                           , 
                           
                             z 
                             ; 
                             
                               x 
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                             
                           
                           , 
                           
                             y 
                             
                               1 
                                
                               
                                   
                               
                                
                               i 
                             
                           
                           , 
                           
                             
                               z 
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                             
                             ; 
                             
                               α 
                               i 
                             
                           
                           , 
                           
                             β 
                             i 
                           
                           , 
                           
                             
                               γ 
                               i 
                             
                             ; 
                             
                               L 
                               i 
                             
                           
                           , 
                           
                             i 
                             = 
                             1 
                           
                           , 
                           
                             N 
                             seg 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       P 
                       _ 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       Nseg 
                     
                      
                     
                       
                         C 
                         i 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       I 
                                       
                                         XYZ 
                                         i 
                                       
                                     
                                     + 
                                     
                                       I 
                                       
                                         YZ 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       I 
                                       
                                         XZ 
                                         i 
                                       
                                     
                                     + 
                                     
                                       I 
                                       
                                         Z 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   ( 
                                   
                                     
                                       I 
                                       
                                         XY 
                                         i 
                                       
                                     
                                     + 
                                     
                                       I 
                                       
                                         Y 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   I 
                                   
                                     X 
                                     i 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The method where multiple line segments approximate an arbitrary wellbore trajectory and act in unison to determine the pressure distribution has been reduced to practice. In  FIG. 11 , we see the dimensionless pressure for different XY slices through a three-dimensional cube of isotropic permeability with a partially penetrating motherbore and four lateral wellbores as illustrated. The fractional length of the line segment in comparison with the total well length is an appropriate weighting factor for this uniform flux approach. 
       Infinite and Finite Conductivity Wells 
       [0057]    The segmented well can be further modified to describe the case where pressure rather than flux is specified. Using a constitutive relationship between pressure drop and flow rate within a well, one can pose and solve a set of equations to describe the pressure distribution along a well with unknown segment flux but known overall production rate, as illustrated in  FIG. 12 . The problem could likewise be posed with a pressure constraint, such as a pump mechanical limit, and unknown overall flux. The so-called infinite conductivity well (uniform pressure) is a special case of this algorithm where pressure is unknown but uniform along the well. An expression for pressure difference between adjacent segments, along with the overall material balance for the well, closes the system of equations. This has been demonstrated in prior art (Ouyang &amp; Aziz, 2001) using less accurate, strictly numerically integrated point source formulas to compute inflow for nonconventional wells. 
         [0058]    For a uniform pressure well boundary condition, a set of pressure matching expressions using the described invention could be solved directly by any number of standard linear matrix solvers. Alternatively, uniform pressure cases can be readily evaluated within the context of the present invention using an iterative procedure. A uniform flux initial guess returns the productivity index for each segment. The reciprocal of the computed productivity index is normalized and used as the updated flux. The process is repeated until convergence. Multiple passes are required only to account for well interference effects. The described iterative procedure has been reduced to practice in software to give the flux in each well segment. 
         [0059]    Realistic pressure drop within wellbores can be computed directly by passing a set of pressure and flux matching expressions along with a material balance and constitutive relationship between pressure gradient and flow rate to any number of standard linear matrix solvers. The solver can likewise be bypassed in finite conductivity well cases by computing a productivity index for each segment with a uniform flux initial guess, computing the flux required to give the specified pressure drop along the well segment using a constitutive equation, such as that for pipe flow using friction factor concepts, and repeating the process until convergence. 
         [0060]      FIG. 13  illustrates the difference between uniform flux and uniform pressure cases with the same curvilinear well trajectory decomposed into 29 piecewise linear segments.  FIG. 13   a  shows the dimensionless pressure distribution for the central plane containing the well with identical flux per unit length.  FIG. 13   b  shows the dimensionless pressure for the same slice and same well trajectory after the iterative procedure to yield equal pressure at each segment midpoint with adjustment of flux. The pressure and flux distributions for these two cases are presented in  FIGS. 13   c  &amp;  13   d.    
       Generalization to Open Systems 
       [0061]    The Neumann function solution can be further generalized to include material transport at the faces of the rectangular cell using Green&#39;s Theorem. In this method, the equation becomes 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                      
                     
                       ( 
                       
                         x 
                         _ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       P 
                       _ 
                     
                     - 
                     
                       
                         ( 
                         
                           1 
                           
                             q 
                             o 
                           
                         
                         ) 
                       
                       · 
                       
                         
                           ∑ 
                           i 
                         
                          
                         
                           
                             q 
                             wi 
                           
                           · 
                           
                             
                               N 
                               L 
                             
                              
                             
                               ( 
                               
                                 
                                   x 
                                   _ 
                                 
                                 , 
                                 
                                   
                                     x 
                                     _ 
                                   
                                   
                                     1 
                                      
                                     
                                         
                                     
                                      
                                     i 
                                   
                                 
                                 , 
                                 
                                   
                                     x 
                                     _ 
                                   
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     i 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     - 
                     
                       
                         ( 
                         
                           1 
                           
                             q 
                             o 
                           
                         
                         ) 
                       
                        
                       
                         
                           ∫ 
                           S 
                           
                               
                           
                         
                          
                         
                           
                             
                               N 
                               o 
                             
                              
                             
                               ( 
                               
                                 
                                   x 
                                   _ 
                                 
                                 , 
                                 σ 
                               
                               ) 
                             
                           
                            
                           
                             g 
                              
                             
                               ( 
                               σ 
                               ) 
                             
                           
                            
                           
                              
                             σ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where  x  is the observation point location, q o  is a reference production rate, q wi  is the production rate for segment i, N L  are the Neumann functions for the linear segments (  x   1i ,  x   2i ), and the product of the normal flux, g(σ), and point source Neumann function, N o , is integrated over open boundary surfaces, S. Equation (46) is especially relevant for the choice of observation point for well pressure characterization,  r   w . If all six faces were open for material transport, the integration would be broken into six separate integrations, one for each face. Partial integrations are allowed with the flux defined to be zero (and hence noncontributing) on closed portions of the boundary. The integrations in Equation (46) can be estimated numerically by any one of a variety of standard techniques, such as Gaussian quadrature or the trapezoidal rule. In this fashion, pressure support to the cell by water influx is accommodated. Flux can be easily included from any of the faces to model physical cases. In this invention, the fast computing Equation (8) is used for N L . 
         [0062]    In the limit of strong aquifer support, the pressure at the boundary remains constant, as water is supplied to fully account for fluid withdrawn from the cell via the well. In the strong aquifer limit, the Green&#39;s function is recommended over the Neumann function. The derivation is repeated for the analogous case with Sine functions in place of Cosines in Equations (12), (13), and (14). 
         [0063]    An important special case using Equation (46) for open systems is when the flux is presumed to be uniformly distributed across a fully open face. In traditional numerical reservoir simulators, only a single value of flux is associated with each planar interface of adjoining cells, as illustrated in  FIG. 14 . In the absence of a detailed flux distribution pattern, a uniform flux distribution allows analytic reduction of the integral in Equation (46). For example, replacing the normal flux with a representative average value,          g         , 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       〈 
                       g 
                       〉 
                     
                      
                     
                       
                         ∫ 
                         0 
                         b 
                       
                        
                       
                         
                           ∫ 
                           0 
                           h 
                         
                          
                         
                           
                             
                               N 
                               o 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 
                                   z 
                                   ; 
                                   
                                     
                                       x 
                                       o 
                                     
                                     = 
                                     0 
                                   
                                 
                                 , 
                                 
                                   y 
                                   o 
                                 
                                 , 
                                 
                                   z 
                                   o 
                                 
                               
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                              
                             
                               y 
                               o 
                             
                           
                            
                           
                               
                           
                            
                           
                              
                             
                               z 
                               o 
                             
                           
                            
                           
                             
                               〈 
                               g 
                               〉 
                             
                             bh 
                           
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   a 
                                   2 
                                 
                               
                               
                                 
                                   π 
                                   2 
                                 
                                  
                                 
                                   k 
                                   x 
                                 
                               
                             
                             ) 
                           
                            
                           
                             
                               ∫ 
                               0 
                               b 
                             
                              
                             
                               
                                 ∫ 
                                 0 
                                 h 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     l 
                                     = 
                                     1 
                                   
                                   ∞ 
                                 
                                  
                                 
                                   
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             π 
                                              
                                             
                                                 
                                             
                                              
                                             lx 
                                           
                                           a 
                                         
                                         ) 
                                       
                                     
                                     
                                       l 
                                       2 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                      
                                     
                                       y 
                                       o 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                      
                                     
                                       z 
                                       o 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       〈 
                       g 
                       〉 
                     
                      
                     
                       
                         ( 
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               a 
                               2 
                             
                           
                           
                             k 
                             x 
                           
                         
                         ) 
                       
                        
                       
                         [ 
                         
                           
                             1 
                             6 
                           
                           - 
                           
                             x 
                             
                               2 
                                
                               
                                   
                               
                                
                               a 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 x 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   a 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Similar expressions apply for the other five faces with switch of variables and the dummy variable, x, representing the normal distance to the face over which the Neumann function is integrated.  FIG. 15  shows the pressure distribution across the central slice containing the well for the uniform pressure case developed in  FIG. 13  for three external boundary situations: (a) sealed boundaries, (b) open lateral boundaries, and (c) influx from the bottom face. The well pressure, which is actually the driving force for production since we have used a degree of freedom to set the average reservoir pressure to zero in these calculations, is sensitive to the boundary conditions imposed, as boundaries strongly influence the direction and convergent nature of fluid flow towards wellbores. The observed flow weighted average well pressures are given as insets for the three boundary condition cases of  FIG. 15 . Sensitivity to the integrity of external boundaries is anticipated to heighten between multiple wellbores, e.g. well interference tests. 
         [0064]    Of particular commercial interest is when the rectangular cell is a well block within a numerical reservoir simulation. The invention can then be used to describe well productivity (the relationship between pressure and production rate) for any complex well trajectory or wellbore cluster below the resolution of the simulation. The invention can be used for explicit computation of well properties or as feedback for an ongoing numerical simulation as a constraint on the pressure or total flux for the well block. 
         [0065]    Accordingly, it is to be understood that the embodiments of the invention herein described are merely illustrative of the application of the principles of the invention. Reference herein to details of the illustrated embodiments is not intended to limit the scope of the claims, which themselves recite those features regarded as essential to the invention. For example, potential and pressure have been used interchangeably in the development herein. The solutions are actually in terms of potential, with pressure being the commonly measured property. Potentials are routinely converted to pressures, as any skilled in the art would know, with the inclusion of gravitational forces. This is particularly relevant to pressure relationships for flow within a non-horizontal wellbore where gravitational head is an important contribution to the driving force for flow. 
       Note on Special Limiting Cases (Wells Parallel to Coordinate Axes) 
       [0066]    When the wells are oriented strictly parallel to one of the coordinate axes (horizontal and vertical wells), two of the direction cosines (α, β, γ) become zero. In such limiting cases, the generalized equations will present computing and coding challenges. By a proper grouping of the terms, and some algebraic simplifications, the relevant computations can be handled. Slow convergence rates often are the challenge in these limiting cases. Alternately, direct methods based upon components from published literature are available to address such simpler version of the general problem (Muskat, 1949; Babu and Odeh, 1989). For example, consider the following infinite series Bessel function solution to the Laplace Equation given by Muskat (1949, p. 208). 
         [0000]    
       
         
           
             
               
                 
                   
                     Pressure 
                     ∼ 
                     
                       
                         2 
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                             
                               
                                 
                                   K 
                                   o 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       nr 
                                     
                                     h 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       nz 
                                     
                                     h 
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     
                                       nz 
                                       o 
                                     
                                   
                                   h 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       + 
                       
                         ln 
                          
                         
                           ( 
                           
                             
                               4 
                                
                               
                                   
                               
                                
                               h 
                             
                             r 
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     0 
                     &lt; 
                     r 
                     &lt; 
                     ∞ 
                   
                   , 
                   
                     0 
                     ≤ 
                     z 
                     ≤ 
                     h 
                   
                   , 
                   
                     0 
                     ≤ 
                     
                       z 
                       o 
                     
                     ≤ 
                     h 
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This equation describes fluid flow due to a point sink/source at (r=0, z=z o ) in a slab-like reservoir of thickness h, extending laterally to infinity, and with impermeable top and bottom boundaries (z=0, z=h). The convergence of the series in Equation (48) is extremely fast, and in most cases, only a few terms of the series are needed to achieve highly accurate results. 
         [0067]    Integration with respect to z o , from z 1  to z 2 , solves the problem of a partially penetrating vertical well. Problems of partially penetrating wells in rectangular box-shaped (bounded) reservoirs can be addressed by summing up all reflections of K o  across the sides of the rectangle. Change of variables, between (x, y, z), will solve problems of horizontal wells parallel to coordinate axes. 
         [0068]    Next, to deal with small magnitudes of the variables (r/h) or (z/h), in Eq (48), the following expansions are available (Gradshteyn and Rhyzik, 2007, p. 939). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         1 
                       
                       ∞ 
                     
                      
                     
                       
                         
                           K 
                           o 
                         
                          
                         
                           ( 
                           
                             π 
                              
                             
                                 
                             
                              
                             nX 
                           
                           ) 
                         
                       
                       · 
                       
                         cos 
                          
                         
                           ( 
                           
                             π 
                              
                             
                                 
                             
                              
                             nZ 
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       [ 
                       
                         0.5772 
                         + 
                         
                           ln 
                            
                           
                             ( 
                             
                               X 
                               4 
                             
                             ) 
                           
                         
                         + 
                         
                           1 
                           
                             
                               
                                 X 
                                 2 
                               
                               + 
                               
                                 Z 
                                 2 
                               
                             
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       1 
                                       
                                         
                                           
                                             X 
                                             2 
                                           
                                           + 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   l 
                                                 
                                                 - 
                                                 Z 
                                               
                                               ) 
                                             
                                             2 
                                           
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       1 
                                       
                                         
                                           
                                             X 
                                             2 
                                           
                                           + 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   l 
                                                 
                                                 + 
                                                 Z 
                                               
                                               ) 
                                             
                                             2 
                                           
                                         
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     1 
                                     l 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
           
         
       
     
         [0000]    For fast computations involving Equations (32) and (34) in the case of degenerate (limiting) versions of the general problem, the following formulas facilitate switching between the two types of series: Eq (48) in K o , and the trigonometric series of Eqs (32) and (34). To separate variables α and β, use the integral (Gradshteyn and Rhyzik, 2007, p. 719) 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       0 
                       ∞ 
                     
                      
                     
                       
                         
                           
                             K 
                             o 
                           
                            
                           
                             ( 
                             
                               π 
                                
                               
                                   
                               
                                
                               α 
                                
                               
                                   
                               
                                
                               t 
                             
                             ) 
                           
                         
                         · 
                         
                           cos 
                            
                           
                             ( 
                             
                               π 
                                
                               
                                   
                               
                                
                               β 
                                
                               
                                   
                               
                                
                               t 
                             
                             ) 
                           
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         t 
                       
                     
                   
                   = 
                   
                     
                       1 
                       
                         2 
                          
                         
                           
                             
                               α 
                               2 
                             
                             + 
                             
                               β 
                               2 
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In summing series that converge extremely slowly, we use the spectral representation of the Dirac Delta function, in conjunction with Eq (50), and subsequently achieve rapid convergence rates. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       1 
                       + 
                       
                         2 
                         · 
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     mt 
                                   
                                   a 
                                 
                                 ) 
                               
                             
                             · 
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     mx 
                                   
                                   a 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       a 
                       · 
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             0 
                           
                           ∞ 
                         
                          
                         
                           δ 
                            
                           
                             [ 
                             
                               t 
                               - 
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     al 
                                   
                                   ± 
                                   x 
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       for 
                        
                       
                           
                       
                        
                       0 
                     
                     ≤ 
                     x 
                     ≤ 
                     a 
                   
                   , 
                   
                     0 
                     &lt; 
                     t 
                     &lt; 
                     ∞ 
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
     
         [0000]    While elements leading to fast converging formulas for pressure computation in special case well orientations exist in the public domain, we incorporate the method of using these identities in tandem as a preferred embodiment of handling also the special cases as limiting simplified versions of this invention.