Summary We describe a collection of block preconditioners for use in solving large, sparse, linear systems of equations by iterative methods, and we compare their performance with several point preconditioners in solving some systems arising in numerical reservoir simulation. We consider block preconditioners that handle either lines or planes in an implicit manner and pointwise incomplete LU (ILU) factorizations combined with partial elimination preprocessors. We conclude that the best of the pointwise methods are both more robust and faster, but that the best of the block methods are competitive for certain orderings of unknowns and require less storage.
Introduction Numerical simulation of petroleum reservoirs using finite-different methods requires the use of large numbers of gridblocks for accurate models. The resultant computational process typically entails an iteration over time in which each timestep requires the solution of a system of nonlinear algebraic equations. When Newtonian methods are used for these nonlinear systems, the main computation-intensive task is the solution of sets of sparse linear systems of equations. For large problems, especially in three dimensions (3D), direct methods (Gaussian elimination) are prohibitively expensive. Thus, there is a great need for robust iterative methods for solving large, sparse, linear systems. An important set of tools making up these methods are preconditioners - most notably, incomplete factorizations of the coefficient matrices - which can be combined with acceleration schemes such as orthomin, and have the effect of improving the conditioning of the linear system. Pointwise incomplete factorization methods, in which the triangular factors of the preconditioners have nonzero patterns similar in form to the lower or upper triangle of the coefficient matrix, have proved to be successful in reservoir simulation.
In recent years, a collection of block-oriented preconditioners has been proposed for solving the sparse linear systems arising from the finite-difference discretization of elliptic partial differential equations. Each of these methods computes an incomplete factorization of the coefficient matrix in which either lines or planes are handled in an implicit manner. Examples are the nested factorization of Appleyard and Cheshire and Appleyard et al., the INV and MINV factorizations of Concus et al. [for two-dimensional (2D) problems], and some generalizations of the latter methods for 3D problems suggested by Meijerink (see also Ref. 10). Numerical comparisons with pointwise factorizations such as ILU and modified ILU (MILU) factorizations suggest that block preconditioners result in faster convergence of iterative methods such as the conjugate-gradient method or orthomin (see, e.g., Refs. 7 and 8).
In this paper, we examine the performance of a collection of block preconditioners used in conjunction with orthomin for solving several linear systems arising from numerical reservoir simulation. We consider four examples of such preconditioners:the nested factorization for both 2D and 3D problems;
INV(1) and MINV(1) for 2D problems;
generalizations of INV(1) and MINV(1) for 3D problems in which lines (i.e., tridiagonal blocks) are treated in an implicit manner; and
generalizations of INV(1) and MINV(1) for 3D problems in which planes (pentadiagonal blocks) are treated in an implicit manner.
Generalizations 3 and 4 have been suggested by Meijerink. We compare the performance of these preconditioners with the pointwise ILU and MILU factorizations and with the use of a partial elimination preprocessing step for two-cyclic matrices (forming "reduced system") combined with the MILU factorization. We test these methods on one 2D problem and four 3D problems, all arising from black-oil simulations with an implicit pressure, explicit saturation (IMPES) discretization. We describe the block factorizations considered, give a brief description of the point methods considered, present the results of numerical experiments, and outline our conclusions.
Block Methods In this section, we given an overview of the block methods considered. We briefly review the nested factorization, present in some detail the largely untested generalizations of INV and MINV for 3D problems, and outline the work and storage requirements for all the methods considered.