Application of Sparse Matrix Techniques in Reservoir Simulation

Abstract

This paper was prepared for the 48th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Las Vegas, Nev., Sept. 30-Oct. 3, 1973. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgement of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract The matrices arising in reservoir simulation are sparse in that most elements are zeros. These matrix equations can be solved efficiently by sparse matrix techniques, which take advantage of the zeros. The approach consists of two steps:The matrix or grid is reordered to preserve the sparsity.The preserve the sparsity. (2) The system of equations with the reordered matrix is then solved by using an efficient Gaussian elimination or factorization algorithm that avoids operating on zeros. This paper compares several schemes that are effective for reordering the matrix. It also presents two general purpose sparse matrix algorithms that efficiently solve any type of reservoir simulation equations with any ordering. One of the algorithms is up to 8 times faster than the conventional band matrix algorithm; and the other, up to 18 times faster. Sparse matrix techniques are also competitive with iterative techniques. For large matrices, the algorithms' storage requirement can be readily met by a computer with virtual storage. Background The linear pressure and saturation equations in reservoir simulation usually have been solved with iterative rather than direct methods. This was thought to save time and storage. However, the increasing complexity of simulation problems has substantially increased the number of iterations required. We have found that in some situations, such as coning or compositional simulation, many iterative methods (including SIP 22) will not converge to an acceptable solution within a reasonable number of iterations. Even in situations where iterative methods converge, selecting an optimum iterative method and optimum acceleration parameters is tedious and sometimes parameters is tedious and sometimes expensive.