Upwinding and artificial viscosity for robust discontinuous Galerkin schemes of two-phase flow in mass conservation form

Abstract

High-order discretizations have become increasingly popular across a wide range of applications, including reservoir simulation. However, the lack of stability and robustness of these discretizations for advection-dominant problems prevent them from being widely adopted. This paper presents work towards improving the stability and robustness of the discontinuous Galerkin (DG) finite element scheme, for advection-dominant two-phase flow problems in particular. A linearized analysis of the two-phase flow equations is used to show that a standard DG discretization of the two-phase flow equations in mass conservation form results in a neutrally stable semi-discrete system in the advection-dominant limit. Furthermore, the analysis is also used to propose additional terms to the DG method which linearly stabilize the discretization. These additional terms are derived by comparing the linearized equations in mass conservation form against an upwinded pressure-saturation form of the equations. Next, a partial differential equation-based artificial viscosity method is proposed for the Buckley-Leverett and two-phase flow equations, as a means of mitigating Gibbs oscillations in high-order discretizations and ensuring convergence to physical solutions. The modified DG method with artificial viscosity is demonstrated on a two-phase flow problem with heterogeneous rock permeabilities, where the high-order discretizations significantly outperform a conventional first-order approach in terms of computational cost required to achieve a given level of error in an output of interest.