Thermodynamically Consistent Modeling and Energy Stable Numerical Simulation of Multicomponent Compressible Flow in Poroelastic Media

Abstract

Modeling and numerical simulation of coupled poromechanical problems with multicomponent compressible flow are of particular importance in many fields including shale and natural gas engineering, carbon dioxide sequestration and geotechnical engineering. In this paper, using the second law of thermodynamics, we rigorously derive a thermodynamically consistent model for multicomponent compressible flow in deformable porous media coupled with poroelasticity. The model herein takes molar densities as the primary unknowns rather than pressure and molar fractions as well as introduces fluid and solid free energies, so that it naturally follows an energy dissipation law. Additionally, the Maxwell-Stefan model of multicomponent diffusion is generalized as a multicomponent fluid-solid coupling model accounting for the solid deformation, which not only satisfies Onsager’s reciprocal principle but also yields a thermodynamically consistent poro-visco-elastic equation. For numerical simulation, we propose a novel energy stable and mass conservative numerical scheme for the model. We first design the semi-discrete time scheme using the stabilized energy factorization approach to deal with the multicomponent Helmholtz free energy as well as subtle semi-implicit treatments for the coupling between multicomponent fluids and solids. A nontrivial treatment is the use of the discrete Gibbs-Duhem equation substituting for the pressure gradient contributed by the multicomponent fluids in the solid mechanical balance equation, which establishes the thermodynamically consistent relation between fluids and solids at the discrete level. Based on the cell-centered finite volume method on staggered grids, the fully discrete scheme is constructed using the upwind strategy for both molar densities and porosity. The scheme is proved to preserve the discrete energy dissipation law and Onsager’s reciprocal principle as well as to conserve the mass of fluid components and solids. Numerical experiments are performed to confirm our theories, especially to demonstrate the good performance of the proposed scheme in energy stability and mass conservation as expected from our theoretical analysis.