On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow

Abstract

This paper presents studies on applying the novel weak Galerkin finite element method (WGFEM) to a two-phase model for subsurface flow, which couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. The coupled problem is solved in the framework of operator decomposition. Specifically, the Darcy equation is solved by the WGFEM, whereas the saturation is solved by a finite volume method. The numerical velocity obtained from solving the Darcy equation by the WGFEM is locally conservative and has continuous normal components across element interfaces. This ensures accuracy and robustness of the finite volume solver for the saturation equation. Numerical experiments on benchmarks demonstrate that the combined methods can handle very well two-phase flow problems in high-contrast heterogeneous porous media.

Less is often more: Applied inverse problems using hp-forward models

“…To compute p given in Eqs. 15-18, we use the mixed variational formulation described in [92,93]. For this, we conduct numerical integration over the hp discretization using Gaussian Quadrature and solve the symmetric indefinite linear system by implementing an adaptation of the open source finite element code…”

Nonlinearly preconditioned semismooth Newton methods for variational inequality solution of two-phase flow in porous media

“…It is worth mentioning that most used algorithms in practice do not guarantee the computed saturation to stay within this range [10,37,38]. Very few efforts have been made in the design of numerical methods to insure positivity for two-phase flow problems; a recent example is [26].…”

Low Regularity Error Analysis for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

“…The weak Galerkin finite element method is an effective and flexible numerical technique for solving partial differential equations. The WG method was first introduced in [16] and then has been applied to solve various partial differential equations such as second order elliptic equations, biharmonic equations, Stokes equations, convection dominant problems, two-phase flow problems and Maxwell's equations [1,2,[4][5][6][7][8][10][11][12][13][14][17][18][19]. However, the standard a priori error analysis of weak Galerkin finite element methods requires additional regularity on solutions.…”