Abstract

The objective of this work was to develop improved variations of the BEPS, FAC, and Sequential preconditioning methods used to solve composite grid linear systems. These methods are attractive since they use standard subgrid approximate factorizations in constructing the composite grid preconditioning. This paper describes two algebraic, multilevel grid coarsening procedures which allow efficient "black box" implementation of FAC and BEPS. The coarsening procedures are compared with regard to their effect upon convergence and computer efficiency. In addition all three methods may be modified to use variable step subgrid solutions to enhance convergence.

Introduction

Composite grids often require significantly fewer cells to model localized phenomena in reservoir simulation than conventional grids; however, the linear systems associated with composite grids can be more difficult to solve iteratively. One of the reasons is that refined grids can have large transmissibility variations in more than one spatial direction. This adversely affects the convergence of methods sensitive to the ordering of the cells. Another reason for slow convergence is that connections between grids are difficult to treat rigorously without adversely affecting computational efficiency. For example the irregular structure caused by these connections makes vectorization more difficult.

This paper examines the algebraic formulation of several multilevel iterative methods and compares their performance on IMPES-type matrix problems associated with locally refined grids. These methods share the idea of constructing the approximate factorization so that solution steps are performed on a grid by grid basis. This idea offers advantages in ease of implementation and efficiency with regards to vectorization and parallelization. Although beyond the scope of this paper, these methods can be adapted for use in solving implicit and mixed implicit composite grid linear systems.

We initially impose a coarse grid of level 1 cells on the region of interest. Any level n cell may be refined by a subgrid of level n+1 cells. We require that all subgrids of the same level are independent. In this paper we assume the 5-point (7-point in 3D) finite difference operator is used within each subgrid.

P. 59^

This content is only available via PDF.
You can access this article if you purchase or spend a download.