Application of Preconditioned Conjugate-Gradient-Like Methods in Reservoir Simulation

Abstract

·Conjugate-gradient-like (CGL) methods were used to solve the matrix equations for several large-field and fine-grid models. The methods used incomplete LU (lLU) or modified incomplete LU (MILU) preconditioning coupled with orthomin acceleration. The CGL methods were applied to the original matrix or to a reduced system from red/black ordering. Test results from a black-oil model with fault throws, a fractured reservoir model, and three steamflood models are given. It was found that the CGL methods are effective for solving the matrix equations. A more powerful preconditioner improves reliability but may require more work. Guidelines for selecting the most effective method are provided.

Ordering Methods for Preconditioned Conjugate Gradient Methods Applied to Unstructured Grid Problems

“…the convergence behavior of preconditioned conjugate gradient methods. There have been many studies of the use of various ordering techniques coupled with incomplete LU (ILU) factorization preconditioners [3,4,5,7,11,12,13,15,16,17,25,29,30,34,35,38,39].…”

“…Reduced systems are very eective (red/black ordering of a bipartite graph, and exact elimination of the red nodes). Incomplete factorizations can be classied by the allowed level of ll [17,29,38,39]. A rigorous denition of the level is given in x 2.2 .…”

“…for convergence [3,4,17,25,29,38]. Consequently, at least for ve or seven-point molecules, a reduced system level 1 or level 2 ILU is a popular choice.…”

Colloid‐facilitated contaminant transport in discretely fractured porous media: 1. Numerical formulation and sensitivity analysis

“…Many preconditioned conjugate gradient type methods use incomplete lower-upper factorization (ILU), which is similar to Gaussian elimination except that decomposition is terminated at a certain level of fill. Previous studies have shown that an increase in the level of fill (Q produces higher-quality preconditioning, which therefore reduces the number of iterations required for convergence [e.g., Simon, 1988;Northrup and Woo, 1988]; however, an increase in the level of fill requires more work per iteration, and consequently it may not be costeffective. Generally, the optimal level of fill is •' -1, 2, or 3 for two-dimensional problems and •' = 0 or 1 for threedimensional problems.…”