Accelerated nonlinear domain decomposition solver for multi-phase flow and transport in porous media

Abstract

Development of robust and efficient nonlinear solution strategies for multi-phase flow and transport within natural porous media is challenging. Fully Implicit Method (FIM) is widely used in reservoir simulation for solving nonlinear algebraic systems. Previous studies showed that nonlinearities of flow and transport problems could exhibit large degrees of locality across timesteps and nonlinear iterations. Nonlinear domain decomposition (NDD) is an attractive class of methods for achieving localization and resolving spatially unbalanced nonlinearities. However, NDD is not robust as iterative solvers, because the outer iterations are essentially fixed-point type.We design a solution framework to accelerate the outer-loop convergence of NDD. We first apply a nonlinear acceleration (NA) technique to the fixed-point iterations, for correcting the boundary conditions of subdomains. Because flow problems are parabolic and exhibit global behaviors, long-range information is essential to capture main features of the flow field. We incorporate a preconditioning step based on CPR-AMG before each timestep, to provide better initial solutions. Additionally, smoother iterations can be taken to improve the boundary conditions during the NDD process.We evaluate the NDD methods using heterogeneous multi-phase models. The nonlinearities associated with the flow, transport and the coupling are locally resolved through the NDD iterations. Results demonstrate that the basic NDD solver is afflicted with severe nonlinear difficulties. By comparison, the accelerated NDD solver greatly improves the outer convergence performance. Moreover, the new solver can effectively exploit locality on the Newton level, for both the flow and transport problems.