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.
Highlights
The field of poroelasticity is concerned with describing the interaction between viscous fluid flow and elastic solid deformation within a porous material, and goes back to the works of K
Porous materials are by definition solid materials comprising a great number of interconnected pores, typically at the order of micrometers, where the interconnectivity of the pores is sufficient to allow for fluid flow through the material
The aim of the present work is to establish the well-posedness of the nonlinear thermo-poroelastic model as described in [10], where we provide a priori energy estimates and regularity properties of the solutions
Summary
The field of poroelasticity is concerned with describing the interaction between viscous fluid flow and elastic solid deformation within a porous material, and goes back to the works of K. [12,18] for a detailed review of this method), a thermo-poroelastic model was derived in [10], which accounts for fluid pressure, elastic displacement, and temperature distribution within a fine-grained, fully saturated poroelastic material within the framework of quasi-static deformation. This model is similar to other thermo-poroelastic models which exists in the literature; see e.g. We summarize the main contribution of the article as follows: under a natural hypothesis on the regularity of the convective term, we give a proof of existence and uniqueness of a weak solution to the fully coupled six-field thermo-poroelastic problem within the quasi-static framework. For easy reference of the notation used in this article we provide some tables in Appendix B
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