Multigrid Reduction for Coupled Flow Problems with Application to Reservoir Simulation

Abstract

Abstract Applications in geosciences, such as reservoir modeling, continue to grow in both size and complexity. Simulations are increasingly complex as they couple more physical phenomena over larger physical domains. As a result, the linear system that arises from the numerical solution of these problems can be challenging to solve by iterative methods. Simulators require advanced algebraic solvers that are robust enough to handle the anisotropies, heterogeneities and coupling between the physical variables, and scalable enough to handle large-scale solution on high performance parallel systems. Multigrid solvers are a class of iterative solvers that are scalable and efficient for solving linear systems that arise from large-scale applications. However, applications with multiple physical unknowns pose a challenge for standard multigrid techniques, particularly when the coupling between the unknowns is strong. In this paper, we present our efforts to develop a multigrid-preconditioned Krylov solver, where the preconditioner is based on the multigrid reduction framework. This preconditioner is designed to represent the coupling between the physical variables of the reservoir modeling equations and account for the underlying physics of the system. Two-stage preconditioners, such as the well-known constrained pressure residual (CPR) approach and its variants like CPR-AMG, have been commonly used in reservoir simulation applications. We discuss how these current solver strategies may be interpreted within the multigrid reduction framework to better understand the different variations. Finally, we present techniques for improving the MGR approach and present results on solver performance on examples from reservoir modeling.