Multigrid Solution of the Pressure Equation in Reservoir Simulation

Abstract

Abstract The multigrid technique is compared with the incomplete Cholesky conjugate gradient/modified incomplete Cholesky gradient (ICCG/MICCG) methods for solving implicit pressure, explicit saturation (IMPES)-type pressure equations. The numerical results of several test pressure equations. The numerical results of several test problems with widely varying transmissibilities are problems with widely varying transmissibilities are presented. presented. The multigrid algorithm is enhanced by pattern relaxation and acceleration. Optimization of the ICCG/ MICCG algorithms is investigated by high-order (up to tenth) decompositions and different ordering schemes. For large problems the multigrid method is superior in terms of scalar work. The multigrid scheme can also be highly vectorized, and is an O(N) algorithm even for problems with large jump discontinuities in equation problems with large jump discontinuities in equation coefficients. Results of the multigrid method applied to nonsymmetric problems are also presented. Introduction Simulation of large reservoirs or entire fields containing several thousand grid blocks entails solution of very large sets of linear equations. Because the amount of work required to solve a linear system by using a direct method increases as the square of the number of unknowns, it is clear that fast iterative methods would be preferable. Multigrid or multilevel methods have been developed recently to provide rapid numerical solution of partial differential equations. For smooth problems, these techniques will theoretically provide convergence in O(N) operations where N is the number of unknowns. In other words, the solution work per unknown does not increase as N increases. Straightforward application of multigrid methods to problems arising in reservoir simulation will generally fail if there are large differences in transmissibilities. However, techniques have recently been developed for problems with jump discontinuities in equation coefficients. This paper describes an improved multigrid technique for solving IMPES-type pressure equations. The basic algorithm described in Ref. 5 has been enhanced by various acceleration schemes and pattern relaxation methods. The resulting technique has been tested on several standard problems and compared with other methods. problems and compared with other methods. The basic idea of multigrid methods is to discretize the partial differential equation on a number of grids of partial differential equation on a number of grids of varying fineness. As is well known, relaxation processes are efficient at eliminating local or high-frequency errors that have a wavelength of the order of a grid spacing. However, relaxation is very inefficient at eliminating long wavelength error. This is usually manifested by an acceptable initial decrease in the residual after which the convergence rate becomes painfully slow. The multigrid method uses different grids to eliminate the different frequencies of error. For example, the finest grid removes high-frequency error, while the coarser grids eliminate low-frequency error. For a complete description of the multigrid method, see Refs. 1 through 3. In the following section we describe only the more recent developments in the multigrid method. SPEJ P. 623