Potential benefits in increased productivity and recovery efficiency coupled with advances in drilling and completion technology have roused considerable interest in horizontal well applications. Since horizontal well systems are characteristically less symmetric (compared to vertical ones), and horizontal wells typically penetrate the reservoir more and in paths that are more tortuous, the flow field and pressure distribution created by such wells can be expected to be three-dimensional in nature and thus be more susceptible to irregularities in geometry and other system properties in three dimensions. There is thus a need for developing predictive tools that adequately describe three-dimensional fluid flow through porous media in realistically complex situations. This need represents the main motivation of this work.

Through the years, both analytical and numerical methods have been used to solve partial differential equations governing engineering problems. Analytical solutions are the most desirable since they are exact and continuous. Unfortunately, they can be obtained only for relatively simple and idealized cases. This has led to the development of various numerical methods which are able to solve more complex and realistic problems. The finite-difference and finite-element methods (FDM and FEM) have been the most widely used numerical techniques in general; in petroleum engineering, however, the use of the former is almost exclusive. The boundary-element method (BEM) is an emerging technique that has gained considerable research attention and popularity in recent years. It offers several potential advantages over the FDM and/or FEM, including

  • Lesser modeling effort, due to the reduction of the dimensionality of multi-dimensional problems by one;

  • Greater numerical accuracy, due to its "semi-analytic" nature and the elimination of grid orientation effects and numerical dispersion; and

  • Better problem representation, due to its ability to conform better to complex geometries and boundary conditions.

The technique has been successfully applied to many diverse fields. It has only been used to a limited extent in petroleum engineering, however.

Potential benefits in increased productivity and recovery efficiency coupled with advances in drilling and completion technology have roused considerable interest in horizontal well applications. Since horizontal well systems are characteristically less symmetric (compared to vertical ones), and horizontal wells typically penetrate the reservoir more and in paths that are more tortuous, the flow field and pressure distribution created by such wells can be expected to be three-dimensional in nature and thus be more susceptible to irregularities in geometry and other system properties in three dimensions. There is thus a need for developing predictive tools that adequately describe three-dimensional fluid flow through porous media in realistically complex situations. This need represents the main motivation of this work.

Through the years, both analytical and numerical methods have been used to solve partial differential equations governing engineering problems. Analytical solutions are the most desirable since they are exact and continuous. Unfortunately, they can be obtained only for relatively simple and idealized cases. This has led to the development of various numerical methods which are able to solve more complex and realistic problems. The finite-difference and finite-element methods (FDM and FEM) have been the most widely used numerical techniques in general; in petroleum engineering, however, the use of the former is almost exclusive. The boundary-element method (BEM) is an emerging technique that has gained considerable research attention and popularity in recent years. It offers several potential advantages over the FDM and/or FEM, including

  • Lesser modeling effort, due to the reduction of the dimensionality of multi-dimensional problems by one;

  • Greater numerical accuracy, due to its "semi-analytic" nature and the elimination of grid orientation effects and numerical dispersion; and

  • Better problem representation, due to its ability to conform better to complex geometries and boundary conditions.

The technique has been successfully applied to many diverse fields. It has only been used to a limited extent in petroleum engineering, however.

In this work, we developed a boundary-element algorithm for the solution of three-dimensional pressure transient problems in homogeneous and isotropic or anisotropic reservoirs. It represents one of our efforts to contribute to petroleum engineering technology by capitalizing on the recent advances in boundary-element technology. In formulating the problem, we have made the following assumptions: single-phase fluid, homogeneous porous medium, applicability of Darcy’s law, negligible gravity effects, constant viscosity, and small and constant fluid compressibility. The implementation is capable of handling (1) reservoir of arbitrary geometry, (2) finite-radius well(s) of arbitrary geometry and orientation, and (3) arbitrary combination of the two common types of boundary conditions (Dirichlet and Neumann). Although the algorithm is not specifically formulated to solve steady-state problems, it can also be used to solve such problems since the solution of a steady-state system is numerically the same as the long-time solution of a similar transient system. The algorithm can thus be used to solve a wide variety of three-dimensional fluid flow through porous media problems in relatively complex situations, such as those associated with horizontal wells.

A summary of the methodology used is as follows. We began by adopting the usual governing equations to describe the fluid flow through porous media problem. Next, we used Laplace transformation to remove the time dependency of the problem. The transformed problem was then formulated in a form amenable to solution (by BEM) using the weighted residual approach. In numerical implementation, isoparametric linear triangular elements were used, resulting in integrals that are either regular or weakly singular. The regular integrals were evaluated numerically using Gaussian quadratures, whereas the weakly singular integrals were evaluated semi-analytically. To solve the systems of algebraic equations assembled, direct Gaussian elimination was used. Finally, the solutions in Laplace space were inverted to the real space using Stehfest algorithm.

To verify its validity and demonstrate its application, the algorithm was used to model various cases with known analytical solutions. Good agreement between model and analytical solutions was obtained except at very small dimensionless times for cases where the well is within the reservoir. A possible explanation for the early-time departures is that whereas the analytical approach assumes a line-source well, the well in the boundary-element model has a finite radius.

To obtain accurate results using the implementation, it was found necessary that the quadratures used in the numerical evaluation of areal integrals be of significantly higher order than those provided in standard boundary-element references. Since the evaluation of such integrals represents the core operation of the boundary-element procedure, the above necessity makes the implementation computationally intensive. This should not be a major concern, however, in view of the prevailing rapid advances in computer technology.

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