Comparison of Alternating Direction Explicit and Implicit Procedures in Two-Dimensional Flow Calculations

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Abstract
Analysis and example applications have been performed to compare the accuracy and computing speed of alternating-direction explicit and implicit procedures (ADEP and ADIP) in numerical solution of reservoir fluid flow problems. ADIP yields significantly greater accuracy and requires about 60 per cent more computing time than ADEP, not 300 or 500 per cent more as reported elsewhere. Introduction Several recent papers discuss an alternating-direction explicit difference approximation (ADEP) to the diffusion equation. Example applications of ADEP and ADIP were reported to support conclusions that ADEP is comparable in accuracy to ADIP and requires one-fifth to one-third the computing time of ADIP. Applications of ADEP in calculation of two-phase flow in reservoirs was also proposed. This study was performed to compare further the relative merits of ADEP and ADIP in simulation of two-dimensional flow of one and two fluid phases in reservoirs. Since two-phase flow equations are often essentially elliptic rather than parabolic, the efficiency of ADEP in solving the elliptic equation was also examined. ADIP AND ADEP DIFFERENCE EQUATIONS The diffusion equation: ...................(1) governs heat conduction, molecular diffusion and slightly compressible fluid flow through porous media for the case of homogeneous, isotropic media. The ADEP procedure involves replacement of Eq. 1 at odd time steps by: ,.................(2) and at even time steps by: ,.................(3) where Sweeping a two-dimensional grid from southwest to northeast using Eq. 2 and from northeast to southwest using Eq. 3 allows direct (explicit) calculation of u at the new time step at each grid point. SPEJ P. 350ˆ

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