Energy stable finite volume discretization for a compressible THM model in porous media

In this work, we investigate a Thermo-Hydro-Mechanical (THM) model describing compressible flow in deformable porous media. Such models play a significant role in various areas of geomechanics, with applications ranging from underground energy storage to oil and gas reservoir engineering. The simulations of THM models are essential for the analysis and design of safe and efficient energy storage cavities. We begin by presenting the mathematical formulation of the model, which consists of a system of parabolic partial differential equations governing the conservation of fluid mass, entropy, and skeleton momentum. First, we derive energy estimates for the compressible model. Next, we focus in particular on the development of numerical discretizations that preserve the energy estimates of the continuous system at the discrete level. To this end, we employ the backward Euler scheme for the time discretization together with the finite volume two-point flux approximation (TPFA) method for the spatial discretization. Numerical experiments are presented to validate the proposed approach and to illustrate its accuracy, efficiency, and robustness.