Monolithic and splitting based solution schemes for fully coupled quasi-static thermo-poroelasticity with nonlinear convective transport

TL;DR

This paper develops and compares splitting-based iterative schemes for solving a complex, fully coupled thermo-poroelasticity model with nonlinear convective transport, extending the model to five fields and proving convergence.

Contribution

It introduces a five-field formulation for thermo-poroelasticity with nonlinear convection and compares three splitting schemes, providing convergence proofs and numerical validation.

Findings

The proposed schemes effectively decouple the nonlinear problem into simpler subproblems.

Convergence of the iterative algorithms is rigorously proven.

Numerical examples validate the accuracy and efficiency of the methods.

Abstract

This paper concerns splitting-based iterative procedures for the coupled nonlinear thermo-poroelasticity model problem. The thermo-poroelastic model problem we consider is formulated as a three-field system of PDE's, consisting of an energy balance equation, a mass balance equation and a momentum balance equation, where the primary variables are temperature, fluid pressure, and elastic displacement. Due to the presence of a nonlinear convective transport term in the energy balance equation, it is convenient to have access to both the pressure and temperature gradients. Hence, we introduce these as two additional variables and extend the original three-field model to a five-field model. For the numerical solution of this five-field formulation, we compare three approaches that differ by how we treat the coupling/decoupling between the flow and/from heat and/from mechanics; these approaches have in common a simultaneous application of the fixed-stress splitting scheme on both the non-linearity and the coupling structure of the problem. More precisely, the derived procedures transform a nonlinear and fully coupled problem into a set of simpler subproblems to be solved sequentially in an iterative fashion. We provide a convergence proof for the derived algorithms, and validate our results through several numerical examples.