Fully implicit bound-preserving discontinuous Galerkin algorithm with unstructured block preconditioners for multiphase flows in porous media

Abstract

Massively parallel simulation of multiphase flows in porous media is challenging due to the highly nonlinear governing equations with the complexity of various modeling features and the significant heterogeneity of material coefficients. In this paper, we introduce a fully implicit discontinuous Galerkin (DG) reservoir simulator designed for parallel computers to handle large-scale multiphase flow simulations. Our approach formulates a fully implicit DG scheme on 3D unstructured grids to discretize the nonlinear governing equations, which take into account capillary effects, permeability heterogeneity, gravity, and injection/production wells. A vertex-based slope limiter within the fully implicit DG framework is adopted to suppress oscillations near discontinuities and ensure the boundedness of the solution. We address the resulting nonlinear systems from the fully implicit discretization using a family of inexact Newton–Krylov algorithms with an analytic Jacobian. Moreover, we employ an unstructured block-type preconditioner with an efficient overlapping additive Schwarz approximation to accelerate the convergence of iterations and enhance the scalability of the simulator. Large-scale simulations of both benchmark and realistic reservoir problems on 3D unstructured grids are conducted to demonstrate the efficiency and scalability of the proposed reservoir simulator.