American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc.
In the past efforts in reservoir simulation have been directed to efficient techniques for solving the set of simultaneous equations arising in multi-phase multi-dimensional fluid flow. Techniques which have been investigated have included both direct solution and iterative methods such as SOR, ADIP and SIP. Iterative techniques have received greater attention since direct methods usually require significantly more computer storage and computing time. Direct methods have been the only alternative for problems which are insolvable with iterative methods, and the direct solution approach becomes competitive for those problems which are slow to converge. problems which are slow to converge. A semi-direct iterative approach is described which extends the capability of iterative methods to solve the more complex problems which result from severe heterogeneities due to geometries or reservoir properties. The semi-direct methods investigated in this paper determines the advantages of including additional diagonals in the triangular decomposition matrices. If all diagonals are included in the triangular matrices the semi-direct method becomes equivalent to the direct method. The investigation showed the asymptotic convergence rate on a computational work required basis was approximately equal to SIP for the model problem proposed by Stone. It achieved faster convergence for the heterogeneous model problem. For some complex problems the method will obtain convergence where SIP fails to converge or converges slowly.
Finite difference approximations to the parabolic and elliptic partial differential equations usually lead to a system of simultaneous equations. These equations may be solved directly by elimination methods or by iterative methods such as relaxation, successive over-relaxation, alternating direction iteration or the strongly implicit procedure. procedure. The direct method is efficient for small problems but computer time requirements become excessive for multidimensional problems involving several thousand mesh points. The direct method requires computational effort of the order of IJ for IJ equations in two dimensions and of the order IJ K computations in three dimensions.