Locally Conservative Continuous Galerkin FEM for Pressure Equation in Two-Phase Flow Model in Subsurfaces

TL;DR

This paper introduces a combined finite element and finite volume method for two-phase subsurface flow, ensuring locally conservative fluxes and improved accuracy in pressure and saturation simulations.

Contribution

It develops a novel approach combining CGFEM with post-processing and FVM with upwind schemes for nonlinear two-phase flow models in subsurfaces.

Findings

The method achieves locally conservative fluxes.

It effectively eliminates non-physical oscillations.

Numerical examples demonstrate robustness and accuracy.

Abstract

A typical two-phase model for subsurface flow couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. In this paper, we study a combined method consisting of continuous Galerkin finite element methods (CGFEMs) followed by a post-processing technique for Darcy equation and finite volume method (FVM) with upwind schemes for the saturation transport equation, in which the coupled nonlinear problem is solved in the framework of operator decomposition. The postprocessing technique is applied to CGFEM solutions to obtain locally conservative fluxes which ensures accuracy and robustness of the FVM solver for the saturation transport equation. We applied both upwind scheme and upwind scheme with slope limiter for FVM on triangular meshes in order to eliminate the non-physical oscillations. Various numerical examples are presented to demonstrate the performance of the overall methodology.