Bound-preserving discontinuous Galerkin methods for compressible two-phase flows in porous media

TL;DR

This paper develops a stable, bound-preserving discontinuous Galerkin method for simulating compressible two-phase flows in heterogeneous porous media, ensuring physical saturation bounds and local mass conservation.

Contribution

It introduces a flux limiter and slope limiter within a fully implicit DG framework to enforce maximum principles and bound-preserving properties for two-phase flow simulations.

Findings

The method respects saturation bounds and maximum principles.

The flux limiter does not increase local mass error or nonlinear iterations.

Numerical results confirm stability and physical accuracy.

Abstract

This paper presents a numerical study of immiscible, compressible two-phase flows in porous media, that takes into account heterogeneity, gravity, anisotropy, and injection/production wells. We formulate a fully implicit stable discontinuous Galerkin solver for this system that is accurate, that respects the maximum principle for the approximation of saturation, and that is locally mass conservative. To completely eliminate the overshoot and undershoot phenomena, we construct a flux limiter that produces bound-preserving elementwise average of the saturation. The addition of a slope limiter allows to recover a pointwise bound-preserving discrete saturation. Numerical results show that both maximum principle and monotonicity of the solution are satisfied. The proposed flux limiter does not impact the local mass error and the number of nonlinear solver iterations.