Well-posedness of the fully coupled quasi-static thermo-poro-elastic equations with nonlinear convective transport

TL;DR

This paper proves the existence and uniqueness of solutions for a nonlinear, fully coupled thermo-poroelastic model with convective heat transfer, using mixed formulations, Galerkin methods, and iterative convergence analysis.

Contribution

It introduces a rigorous analysis of the well-posedness of a nonlinear thermo-poroelastic model with convective transport, including linearization and iterative solution convergence.

Findings

Existence and uniqueness of weak solutions established.

Convergence of the iterative solution procedure proven.

Well-posedness demonstrated for the fully coupled nonlinear system.

Abstract

This paper is concerned with the analysis of the quasi-static thermo-poroelastic model. This model is nonlinear and includes thermal effects compared to the classical quasi-static poroelastic model (also known as Biot's model). It consists of a momentum balance equation, a mass balance equation, and an energy balance equation, fully coupled and nonlinear due to a convective transport term in the energy balance equation. The aim of this article is to investigate, in the context of mixed formulations, the existence and uniqueness of a weak solution to this model problem. The primary variables in these formulations are the fluid pressure, temperature and elastic displacement as well as the Darcy flux, heat flux and total stress. The well-posedness of a linearized formulation is addressed first through the use of a Galerkin method and suitable a priori estimates. This is used next to study the well-posedness of an iterative solution procedure for the full nonlinear problem. A convergence proof for this algorithm is then inferred by a contraction of successive difference functions of the iterates using suitable norms.