Thermodynamically Consistent Discretisation of a Thermo-Hydro-Mechanical Model

We consider in this work a Thermo-Hydro-Mechanical (THM) model coupling the non-isothermal single phase flow in the porous rock and the linear thermo-poro-elasticity. This type of models plays an important role in several applications such as e.g. the hydraulic stimulation of deep geothermal systems, or the risk assessment of induced seismicity in CO2 storages. Compared with the isothermal case, the thermal coupling induces additional difficulties related in particular to the nonlinear convection term. Starting from the pioneer work of Coussy [2], we introduce a thermodynamically consistent discretisation of the THM coupled model which naturally leads to a discrete energy estimate. Our approach applies to a large class of Finite Volume schemes for the flow and energy equations but to fix ideas we consider the Hybrid Finite Volume (HFV) discretisation [3]. It is combined with a conforming Galerkin approximation of the mechanics. Our methodology accounts for a wide range of thermodynamical single phase fluid model and of thermo-poro-elastic parameters, as well as for diffusive or convective dominated energy transport. The efficiency of our approach is assessed on a 2D analytical test case using the HFV scheme for the non-isothermal flow and a P₂ Finite Element method for the mechanics.