Multigrid Methods in Reservoir Simulation

Abstract The equations we solve are those for incompressible two phase flow in two dimensions. We use two general techniques, both using multigrid, one called IMPES and the other called the simultaneous solution (SS) method. The problem is discretized by the standard finite difference method with reflection boundary conditions. The standard multigrid method will fail because of the large jump discontinuities in the coefficients. The difference operator itself is used for the interpolation operator as originally proposed by Alcouffe, Brandt Dendy, and Painter. Although the finite difference operator on the finest grid has a five point star, the operators on the coarser grids all have nine point stars. For relaxation steps we use point, line and alternating line Gauss Seidel methods. With the W-cycles we use, the algorithm exhibits the usual multigrid efficiency. The average error reduction factor per work unit ranges from 0.25 to 0.6. The system we use in the simultaneous solution method is symmetric. The finite difference discretization of the system leads to a symmetric, bloc-pentadiagonal system where the blocks are 2×2 submatrices. The off-diagonal blocks are diagonal matrices. The Discretization is backward in time with explicit mobilities. The scheme is unconditionally stable. We use a generalization of the interpolation procedure we use for the IMPES method applied to systems, which was originally proposed by Dendy. The coarse grid operator is still the Galerkin approximation. The interpolation and restriction operators, however, consist of submatrices. The relaxation is done by collective point, line or alternating line Gauss Seidel methods. The convergence of this multigrid method seems to be about the same as for the IMPES method. Because of the large accumulation term which appears on the coarser grids (not found with the IMPES method), the coarsest grid must be taken find enough to assure that the operator equation is nonsingular. This seems to impose no practical restriction on the method. We are currently testing different more efficient methods of obtaining the coarse grid operators. These promise to save much computational effort over the Galerkin approximation, particularly in three dimension.