Energy stable finite element approximations of gas flow in poroelastic media.

In this paper, we consider numerical modeling of gas flow in porous media with compressible gas and rock. To develop an effective numerical method for simulating this problem, we propose an alternative equation by introducing the poroelasticity equation and the porosity variation equation to account for the influence of rock deformation on porosity. In addition, we introduce a free energy for the skeleton of rocks and take into account rock compressibility as well. Instead of the pressure gradient, we utilize the chemical potential gradient as the primary driving force. This formulation is proved to satisfy an energy dissipation law. By applying the improved energy factorization method to handle the Helmholtz free energy density, we propose a semi-implicit time discretization scheme. The discrete pressure is carefully calculated by the discrete chemical potential and Helmholtz free energy so as to inherit the energy dissipation law. A fully discrete scheme is constructed based on the discontinuous Galerkin and mixed finite element methods with the upwind strategy. We prove that the fully discrete scheme still satisfies the energy dissipation law as well as possesses the feature of mass conservation. Additionally, we can prove the boundedness of density under reasonable assumptions on porosity. Numerical results are provided to show the performance of the proposed scheme and validate our theoretical analysis. [ABSTRACT FROM AUTHOR]