Use of an Integral-Transform Technique for Comprehensive Solutions to Transient-Flow Problems in Homogeneous Domains

Abstract Analytical solutions in the petroleum engineering literature have always been treated as benchmarks for many complicated situations in reservoirs. But efforts to develop analytical solutions for some transient-flow problems have been hindered seriously due to a lack of appropriate mathematical tools. Nevertheless, no systematic attempts to take advantage of using an integral-transform technique (ITT) have been reported in the literature either. This study demonstrates that the use of the ITT results in the development of a huge number of useful solutions to transient-flow problems in a Cartesian coordinate system. New analytical solutions are presented comprehensively in a compact, convenient form. For instance, the closedform solutions in terms of the time and space variables can be written down for all 729 possible combinations of the conditions at the boundaries with constant-potential, no-flow and mixed boundary conditions in a parallelepiped. The parameters pertinent to the solutions are presented in tables. The generality of the solutions is greatly enhanced by considering the production rates through the wells to be time dependent. This situation is treated in a general but computationally efficient way. Thus, the use of the solutions is permissible even if there has been a production history in a system. Example problems with solutions are presented to illustrate the methodology. The new solutions are useful for analyzing drawdown-, buildup- and interference-test data. Introduction Mikhailov and Ozisik(1) have suggested that analytical solutions, when available, are advantageous in that they provide good insights into the significance of various parameters in the system affecting the transport phenomena, as well as accurate benchmarks for a numerical approach. Moreover, analytical solutions in the petroleum engineering literature have always been treated as benchmarks for many complicated situations in reservoirs. Also, these solutions have been used to complement numerical approaches to solving some of these problems. But efforts to develop analytical solutions to most transient-flow problems have been hindered due to a lack of easy-to-deal-with and appropriate mathematical tools. For example, the closed-form solution for flow due to production through a partially penetrating well from a finite, cylindrical-radial reservoir has not been presented in the literature. Finally, in the well-testing literature, analytical solutions are used to identify different flow regimes for computing different rock and fluid properties. Sporadic attempts to use the Fourier transforms for finite domains to develop solutions for some transientflow problems are reported in the literature.(2–12) In the reported studies, one-dimensional (1D) and twodimensional (2D), readily-solvable problems in finite domains were dealt with. Rahman(13) used the Fourier sine and cosine transforms to develop transient solutions for a rectangular parallelepiped for two sets of boundary conditions. Another popular method, the Laplacetransform technique, can be used only in semi-infinite domains; thus, it is limited to the time and semi-infinite space domains.(14) Rahman and Ambastha(15–18) successfully showed that the analytical solutions for transient flow in compartmentalized reservoirs can be developed using the ITT for finite, composite domains. Despite this development, to the best of the authors' knowledge, no systematic attempts have been made to exploit the advantages of using the ITT for finite, homogeneous domains.

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