TY - JOUR

T1 - Mathematical theory and simulations of thermoporoelasticity

AU - van Duijn, Cornelis J.

AU - Mikelić, Andro

AU - Wick, Thomas

N1 - Funding information: CJvD acknowledges the support of the Darcy Center (Utrecht University - Eindhoven University of Technology), The Netherlands and of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for supporting this work by funding SFB 1313, Project Number327154368.This work of A.M. benefited from the support of the project UPGEO ?ANR-19-CU05-032? of the French National Research Agency (ANR) and from the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program ”Investissements d'Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

PY - 2020/7/1

Y1 - 2020/7/1

N2 - In this paper we study the equations of semi-linear thermoporoelasticity. Starting point is the dimensionless formulation in van Duijn et al. (2019), which was obtained by a formal two-scale expansion. Nonlinearities in the equations arise through the fluid viscosity and the thermal conductivity, both may depend on temperature, and through the coupling in the heat convection by the Darcy discharge in the energy equation. The coupled system of equations involves as unknowns the skeleton displacement, Darcy discharge, fluid pressure and temperature. We treat the system in its incremental (i.e. time-discrete) form. We prove existence by applying a fundamental theorem of Brézis on pseudo-monotone operators. Moreover we show that the free energy of the system acts as a Lyapunov functional. This yields global stability in the time-stepping process. Our theoretical results are substantiated with two-dimensional numerical tests using a monolithic formulation. Temporal discretization is based on the backward Euler scheme and finite elements are employed for the spatial discretization. The semi-linear discrete system is solved with Newton's method. In the proposed numerical examples, different source terms are employed and spatial mesh refinement studies show computational convergence.

AB - In this paper we study the equations of semi-linear thermoporoelasticity. Starting point is the dimensionless formulation in van Duijn et al. (2019), which was obtained by a formal two-scale expansion. Nonlinearities in the equations arise through the fluid viscosity and the thermal conductivity, both may depend on temperature, and through the coupling in the heat convection by the Darcy discharge in the energy equation. The coupled system of equations involves as unknowns the skeleton displacement, Darcy discharge, fluid pressure and temperature. We treat the system in its incremental (i.e. time-discrete) form. We prove existence by applying a fundamental theorem of Brézis on pseudo-monotone operators. Moreover we show that the free energy of the system acts as a Lyapunov functional. This yields global stability in the time-stepping process. Our theoretical results are substantiated with two-dimensional numerical tests using a monolithic formulation. Temporal discretization is based on the backward Euler scheme and finite elements are employed for the spatial discretization. The semi-linear discrete system is solved with Newton's method. In the proposed numerical examples, different source terms are employed and spatial mesh refinement studies show computational convergence.

KW - Free energy and stability

KW - Heat convection by Darcy's velocity

KW - Monolithic numerical scheme

KW - Thermoporoelasticity equations

UR - http://www.scopus.com/inward/record.url?scp=85083668572&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2020.113048

DO - 10.1016/j.cma.2020.113048

M3 - Article

AN - SCOPUS:85083668572

VL - 366

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

M1 - 113048

ER -