A two-step finite volume method to discretize heterogeneous and anisotropic pressure equation on general grids

Abstract

A novel Two-Step cell-centered Finite Volume Method (TSFVM) is developed in this work to discretize the heterogeneous and anisotropic pressure equation on triangular and quadrilateral grids in 2D and hexahedral and tetrahedral grids in 3D. Physical properties such as permeability and porosity are piece-wise constant on each grid cell. In the first step, the Galerkin Finite Element Method (FEM) is utilized to compute pressure solutions at all cell vertices. In the second step, pressure values at cell vertices are used to derive continuous two-point flux stencils for cell faces. Mass conservation equations are then written for each cell to obtain a system of linear equations that can be solved for pressure at cell centers. Extensive numerical experiments are carried out to test the performance of our TSFVM. In particular, we compare TSFVM with the classical Multipoint Flux Approximation (MPFA-O) method as well as a more recently developed MPFA method with full pressure support called enhanced MPFA (eMPFA). The results show that the TSFVM compares well with eMPFA for challenging test cases for which MPFA-O breaks down. Specifically, and as a significant step forward, our TSFVM is quite robust for challenging problems involving heterogeneous and highly anisotropic permeability tensors when both MPFA-O and eMPFA suffer from unphysical oscillations. Finally, the numerical convergence study demonstrates that TSFVM has comparable convergence behavior to MPFA-O method for both homogeneous and discontinuous permeability fields.