Abstract
This paper concerns monolithic and 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 six approaches that differ by how we treat the coupling/decoupling between the flow and/from heat and/from the mechanics, suitable for varying coupling strength between the three physical processes. The approaches have in common a simultaneous application of the so-called L-scheme, which works both to stabilize iterative splitting as well as to linearize nonlinear problems, and can be seen as a generalization of the Undrained and Fixed-Stress Split algorithms. 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 demonstrate their performance through several numerical examples investigating different strengths of the coupling between the different processes.
Highlights
We consider here a thermo-poroelastic system which can be seen as a generalization of the Biot system to the non-isothermal case; i.e., the coupled processes are heat, flow, and geomechanics
The reason we propose six algorithms is the following: The coupling strength of the heat, flow and mechanics may vary depending on the physics at hand
Based on previous developments of iterative splitting schemes from linear poroelasticity, we have proposed six novel iterative procedures for nonlinear thermo-poroelasticity
Summary
The field of poroelasticity aims to describe the interaction between viscous fluid flow and elastic solid deformation within a porous material, and it was pioneered through the works of K. Berre et al / Computers and Mathematics with Applications 80 (2020) 1964–1984 saturated, quasi-static regime, the mathematical modeling of such processes constitutes a coupled two-field linear model where the primary variables are the fluid pressure and the elastic displacement of the solid. We consider here a thermo-poroelastic system which can be seen as a generalization of the Biot system to the non-isothermal case; i.e., the coupled processes are heat, flow, and geomechanics Since it is the cornerstone of many complex models, we focus on the following nonlinear and coupled quasi-static thermo-poroelastic equations as described in [4,5,6]: Find the temperature T , the pressure p, and the displacement u such that. All results presented in the sequel are valid for Neumann boundary conditions
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