Energy estimates and convergence analysis of a two-phase flow in deformable porous media

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In this work, we explore two-phase non-isothermal flows in deformable porous media. We consider a Thermo-Hydro-Mechanical (THM) model for a two-phase flow where small deformations and linear thermo-poro-elastic constitutive laws are assumed. These models are widely used in various areas of geomechanics, with applications ranging from underground energy storage to oil and gas reservoir engineering. We present the mathematical formulation of this model which is formulated as a strongly nonlinear system of parabolic partial differential equations governing the conservation of mass, conservation of entropy and momentum balance. Moreover, we derive some energy estimates of the continuous model. The discretization of our system relies on the backward Euler scheme in time and the finite volume two-point flux approximation (TPFA) scheme in space. We show that the energy estimates are well-preserved at the discrete level. These stability results allow us to establish the convergence of the proposed scheme to a weak solution of the nonlinear system. The proof is completed for the degenerate THM model.

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