Nonlinear Preconditioning Strategies for Two-Phase Flows in Porous Media Discretized by a Fully Implicit Discontinuous Galerkin Method

Abstract

We consider numerical simulation of two-phase flows in porous media using implicit methods. Because of the complex features involving heterogeneous permeability and nonlinear capillary effects the nonlinear algebraic systems arising from the discretization are very difficult to solve. The traditional Newton method suffers from slow convergence in the form of a long stagnation or sometimes does not converge at all. In the paper, we develop some nonlinear preconditioning strategies for the system of two-phase flows discretized by a fully implicit discontinuous Galerkin method. The preconditioners identify and approximately eliminate the local high nonlinearities that cause the Newton method to take small updates. Specifically, we propose two elimination strategies, one is based on exploring the unbalanced nonlinearities of the pressure and the saturation fields, and the other is based on identifying certain elements of the finite element space that have much higher nonlinearities than the rest of the elements. We compare the performance and robustness of the proposed algorithms with an existing single-field elimination approach and the classical inexact Newton method with respect to some physical and numerical parameters. Experiments on three-dimensional porous media applications show that the proposed algorithms are superior to the other methods in terms of the robustness and parallel efficiency.

Comparison of nonlinear field-split preconditioners for two-phase flow in heterogeneous porous media

“…The multiplicative Schwarz method has been used to split boundary value problems (BVP) into subproblems solver on smaller physical domains [29][30][31][32][33]. It has also been used to split a coupled BVP into subproblems based on the physics [34][35][36], each subproblem being solved on the full domain to update one of the fields (here, pressure and saturation). The motivation for splitting a coupled problem according to the physics is to solve the physical subproblems one at a time (ideally, with a specialized solver) and use the individually updated fields to precondition the nonlinear iteration, yielding a faster nonlinear convergence.…”

“…A class of nonlinear preconditioners leverages domain decomposition methods [29][30][31][32][33] to precondition the nonlinear system in a computationally inexpensive way and speed up convergence. In this work, we focus on nonlinear preconditioners obtained by splitting the system by physical field [34][35][36]. Instead of decomposing the domain in space, field-split preconditioners exploit the mathematical structure of the nonlinear system to split the problem into multiple subproblems solved nonlinearly at a loose tolerance before the computation of a global update for all the degrees of freedom.…”