Upwinding and artificial viscosity for robust discontinuous Galerkin schemes of two-phase flow in mass conservation form
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ORIGINAL PAPER
Upwinding and artificial viscosity for robust discontinuous Galerkin schemes of two-phase flow in mass conservation form Savithru Jayasinghe1
· David L. Darmofal1 · Steven R. Allmaras1 · Eric Dow2 · Marshall C. Galbraith1
Received: 29 October 2019 / Accepted: 9 September 2020 © Springer Nature Switzerland AG 2020
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 highorder 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. Keywords Two-phase flow · Discontinuous Galerkin · Linear stability · Upwinding · Artificial viscosity · High-order
1 Introduction Over the last few decades, porous media flow simulations have primarily been computed using low-order discretizations such as the finite volume (FV) [1, 2] and finite difference methods [3]. Although the term “loworder” typically refers to numerical methods that have at most second-order accuracy in space and time [4], most reservoir simulation tools used in practice are first-order accurate in both space and time for transport quantities (e.g., single-point upstream-weighted FV, backward Euler time-marching). Conventional methods such as the widely used two-point flux approximation (TPFA) finite volume method are in fact only consistent on K-orthogonal grids [5], where the grid cells are aligned with the principal directions
Savithru Jayasinghe
[email protected]
Extended author information available on the last page of the article.
of the permeability tensor. Multi-point flux approximation (MPFA) methods and its derivatives overcome this limitation by producing consistent discretizati
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