Mathematical theory and simulations of thermoporoelasticity

Abstract

In this paper we study the equations of semi-linear thermoporoelasticity. Starting point is the dimensionless formulation in van Duijn et al. (2019), which was obtained by a formal two-scale expansion. Nonlinearities in the equations arise through the fluid viscosity and the thermal conductivity, both may depend on temperature, and through the coupling in the heat convection by the Darcy discharge in the energy equation. The coupled system of equations involves as unknowns the skeleton displacement, Darcy discharge, fluid pressure and temperature. We treat the system in its incremental (i.e. time-discrete) form. We prove existence by applying a fundamental theorem of Brézis on pseudo-monotone operators. Moreover we show that the free energy of the system acts as a Lyapunov functional. This yields global stability in the time-stepping process. Our theoretical results are substantiated with two-dimensional numerical tests using a monolithic formulation. Temporal discretization is based on the backward Euler scheme and finite elements are employed for the spatial discretization. The semi-linear discrete system is solved with Newton’s method. In the proposed numerical examples, different source terms are employed and spatial mesh refinement studies show computational convergence.