Well-posedness of the fully coupled quasi-static thermo-poroelastic equations with nonlinear convective transport
Peer reviewed, Journal article
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Original versionBrun MK, Radu FA, Ahmed E, Nordbotten JM. Well-posedness of the fully coupled quasi-static thermo-poroelastic equations with nonlinear convective transport. Journal of Mathematical Analysis and Applications. 2019;471(1-2):239-266 https://doi.org/10.1016/j.jmaa.2018.10.074
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.