A Direct Mixed-Enriched Galerkin Method on Quadrilaterals for Two-phase Darcy Flow

TL;DR

This paper introduces a novel finite element method combining mixed and enriched Galerkin techniques on quadrilaterals for two-phase Darcy flow, emphasizing minimal degrees of freedom and high accuracy.

Contribution

It develops a new locally conservative finite element approach using AC and direct serendipity elements tailored for quadrilaterals, improving accuracy and efficiency in two-phase flow simulations.

Findings

Accurate simulation results on quadrilaterals.

Stable computation of capillary flux without breakdown.

Extension to three-dimensional problems.

Abstract

We develop a locally conservative, finite element method for simulation of two-phase flow on quadrilateral meshes that minimize the number of degrees of freedom (DoFs) subject to accuracy requirements and the DoF continuity constraints. We use a mixed finite element method (MFEM) for the flow problem and an enriched Galerkin method (EG) for the transport, stabilized with an entropy viscosity. Standard elements for MFEM lose accuracy on quadrilaterals, so we use the newly developed AC elements which have our desired properties. Standard tensor product spaces used in EG have many excess DoFs, so we would like to use the minimal DoF serendipity elements. However, the standard elements lose accuracy on quadrilaterals, so we use the newly developed direct serendipity elements. We use the Hoteit-Firoozabadi formulation, which requires a capillary flux. We compute this in a novel way that does not break down when one of the saturations degenerate to its residual value. Extension to three dimensions is described. Numerical tests show that accurate results are obtained.