Adaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization

TL;DR

This paper introduces an adaptive enriched Galerkin finite element method with entropy residual stabilization for simulating miscible displacement, effectively capturing viscous fingering in heterogeneous media with improved efficiency and stability.

Contribution

The paper develops a novel adaptive enriched Galerkin approach with entropy stabilization, offering conservative fluxes, fewer degrees of freedom, and suitability for large-scale 3D flow and transport simulations.

Findings

Successfully simulates viscous fingering instabilities.

Achieves computational efficiency with adaptive mesh refinement.

Prevents spurious oscillations in high-order transport systems.

Abstract

We present a novel approach to the simulation of miscible displacement by employing adaptive enriched Galerkin finite element methods (EG) coupled with entropy residual stabilization for transport. In particular, numerical simulations of viscous fingering instabilities in heterogeneous porous media and Hele-Shaw cells are illustrated. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions. The method provides locally and globally conservative fluxes, which is crucial for coupled flow and transport problems. Moreover, EG has fewer degrees of freedom in comparison with discontinuous Galerkin (DG) and an efficient flow solver has been derived which allows for higher order schemes. Dynamic adaptive mesh refinement is applied in order to save computational cost for large-scale three dimensional applications. In addition, entropy residual based stabilization for high order EG transport systems prevents any spurious oscillations. Numerical tests are presented to show the capabilities of EG applied to flow and transport.