The Hybrid-Dimensional Darcy's Law: A Non-Conforming Reinterpreted Discrete Fracture Model (RDFM) for the Compressible Miscible Displacement and Multicomponent Gas Flow in Fractured Media

Abstract

In this work, we develop a non-conforming reinterpreted discrete fracture model for the compressible miscible displacement and multicomponent gas flow in porous media containing high-permeability fractures and/or low-permeability barriers based on the hybrid-dimensional Darcy's law established in our previous work. The key idea of the model is to describe the permeability of codimension-one fractures and barriers by the Dirac-delta functions. When there are only fractures, delta functions are added to the permeability tensor on the right-hand side of the Darcy's law. In contrast, when there are only barriers, delta functions are added to the inverse of the permeability tensor, which represents the resistance to fluids, on the left-hand side of the Darcy's law. When both appear, delta functions are contained on both sides by the principle of superposition. Thereby, we establish partial differential equations (PDEs) to model fluid flow in fractured porous media, which exempts any requirements on meshes. We adopt the discontinuous Galerkin (DG) method to discretize the model in space and the second order implicit pressure explicit concentration (SIMPEC) method to march in time. The resulting non-conforming discrete fracture model is local mass conservative, flexible for complex geometry and easy to implement. The good performance of the method is demonstrated by several numerical examples.