American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc.

Abstract

A relaxation method called Slice Successive Overrelaxation has been developed for the solution of matrix problems occurring in three-dimensional reservoir simulation. The SSOR technique represents a significant improvement over previously used techniques and is considered to be a general method.

The mathematics of SSOR is a straight-forward extension of block SOR techniques in which the submatrix, or block, corresponds to a vertical plane (slice) through the reservoir. This technique has become practical by the application of a reordering practical by the application of a reordering scheme to the direct solution of each submatrix. On iterations subsequent to the first, the forward elimination on submatrices is not required, making succesive iteratons comparable to LSOR in speed. The Watts correction method is applied in its generalized manner similar to LSOR and dramatically enhances the speed of convergence.

Test results are shown for various reservoir problems. The method is shown to be particularly advantageous in a typical three-dimensional problem where vertical permeability barriers exist in portions of the permeability barriers exist in portions of the reservoir and high vertical transmissibilities occur elsewhere. It is demonstrated that the convergence rate of applying SSOR to a three-dimensional problem is essentially the same as the convergence rate of applying LSOR to the two-dimensional projection of the same problem. A method is presented which can be used to estimate the computer time required for an SSOR solution.

Introduction

One of the persistent problems in reservoir simulation is that of solving the matrix problem which arises from the "pressure problem which arises from the "pressure equation." Over the years, a number of methods have been developed and utilized with success for various types of problems.

For most reservoir simulation problems, we find that the solution of the matrix problems, we find that the solution of the matrix problem is very efficient and represents problem is very efficient and represents only a minor part of the cost of a reservoir simulation computer run. On other problems, however, the solution method can problems, however, the solution method can become sufficiently slow so that the matrix problem dominates the computer cost. problem dominates the computer cost.

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