A Hybrid Method Combining Mimetic Finite Difference and Discontinuous Galerkin for Two-Phase Reservoir Flow Problems

Abstract

We introduce a new hybrid numerical approach that integrates the Mimetic Finite Difference (MFD) and Discontinuous Galerkin (DG) methods, termed the MFD-DG method. This technique leverages the MFD method to adeptly manage arbitrary quadrilateral meshes and full permeability tensors, addressing the flow equation for both edge-center and cell-center pressures. It also provides an approximation for phase fluxes across interfaces and within cells. Subsequently, the DG scheme, equipped with a slope limiter, is applied to the convection-dominated transport equation to compute nodal and cell-average water saturations. We present two numerical examples that demonstrate the MFD's capability to deliver high-precision approximations of pressure and flux distributions across a broad spectrum of grid types. Furthermore, our proposed hybrid MFD-DG method demonstrates a significantly enhanced ability to capture sharp water flooding fronts with greater accuracy compared to the traditional Finite Difference (FD) Method. To further demonstrate the efficacy of our approach, four numerical examples are provided to illustrate the MFD-DG method's superiority over the classical Finite Volume (FV) method and MFDM, particularly in scenarios characterized by anisotropic permeability tensors and intricate geometries.