Efficient CPR-type preconditioner and its adaptive strategies for large-scale parallel reservoir simulations

Abstract

With the rapid development of high performance parallel computers and the increasing demands of large-scale reservoir simulation, the development of effective and fast solvers and preconditioners for parallel reservoir simulation has become increasingly important. The incomplete LU factorization (ILU) preconditioners are the most commonly used preconditioners in reservoir simulations due to their low computational cost. They have been implemented in commercial simulators as the default preconditioners. However, for reservoirs with highly heterogeneous permeability, the CPR (constrained pressure residual)-type preconditioners are more effective than the ILU preconditioners, especially when parallel computing is employed. The CPR-type preconditioners consist of multi-stages of preconditioning processes, which lead to huge computational costs. Employing the algebraic multigrid (AMG) method to solve a pressure system and the ILU method to solve the whole system has two indispensable stages. For parallel computation, due to the serial nature of the ILU method, the ILU method is usually combined with the restricted additive Schwarz (RAS) method and used as a subdomain solver. Different settings of the AMG and RAS methods, such as a grid coarsening strategy, interpolation and smoothing in the AMG method, and a subdomain solver and an overlap level in the RAS method, as well as different CPR-type preconditioners, yield significantly different performance. This paper first gives an overview of the CPR-type preconditioners and the AMG and RAS methods. Then a detailed comparison between different CPR-type preconditioners with different settings of the AMG and RAS methods is presented. Based on the comparison, the efficient settings of the AMG and RAS methods and the most efficient CPR-type preconditioner are given. Furthermore, an adaptive preconditioning strategy is designed to save the computational time. The adaptive preconditioning strategy introduces an indicator to measure the difficulty level of solving a linear system. According to difficulties, a RAS–ILU preconditioner or a CPR-type preconditioner is automatically and dynamically selected to achieve both efficiency and effectiveness.