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

Abstract

An iterative method is proposed for the inversion of sparse band-structured matrices of the type that are common in numerical reservoir simulators. The method utilizes orthogonalizations and minimizations to achieve a fast convergence rate. Tests have shown that the method is highly competitive compared with other iterative techniques. The rate of convergence is insensitive to the use of iteration parameters, non-symmetry of the matrix, and ratios of off-diagonal bands in symmetrical matrices.

1.

Introduction

A time-consuming part of the calculation in numerical reservoir simulators is the inversion of a large set of simultaneous linear equations.

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For example when effecting a solution by the Implicit Potential - Explicit Saturation method x would represent the pressure or potential field, whilst A would be a matrix of transmissibility coefficients describing the interflow of fluids between grid blocks. In the common finite difference approaches, A is a sparse banded matrix, as depicted in figure 1 for a five-point finite difference representation of a two dimensional system. Often, the diagonal element of A is approximately equal to the negative sum of the off-diagonal elements in a given row.

Because of the sparseness of A, iterative methods appear at first sight attractive compared with direct methods. However, specialized direct routines with increased efficiency have become available in recent years. Of the iterative methods SIP (strongly implicit procedure) is possibly the most widely used. SIP has the possibly the most widely used. SIP has the disadvantage that its convergence rate depends strongly on a set of iteration parameters whose optimal values can be difficult to find.

A new highly competitive iterative method for symmetric matrices, a minimization process conceptually based on the conjugate-gradient technique, has lately been developed. The numerical examples and proofs quoted in reference 4 indicate that minimization processes can be expected to offer the following advantages: