Task 2. Mathematical model for the quasi-static Thermo-Hydro-Mechanics (THM) via upscaling

There will be two main objectives of this task: (1) derive, using homogenization, the equations of quasi-static thermoporoelasticity and (2) develop, based on the geological data for the limestones and sandstones, a reliable high-fidelity numerical solver for the computation of the effective coefficients (permeability, Biot’s coefficient, Gassmann tensor, thermal dispersivity and effective heat capacity). We will develop a model based on the fluid-structure thermodynamically compatible pore scale equations, corresponding to realistic rock mechanics parameters [1]. Then, we will homogenize the dimensionless equations with respect to the random microscopic geometry [2]. We expect to derive an analogous upscaled system for a random pore structure and we will develop its analysis, study its discretization in time and space and write a robust numerical solver [2]. Our objective will be to develop a flexible solver that will provide, for given statistically homogeneous geometrical structure, computations of the permeability, Biot’s coefficient, Gassman’s tensor and the thermal diffusivity. Finally, we will implement it as a plug-in for Eclipse, PumaFlow, Tough2 and DuMuX /Dune software packages, which will be used in task 1 and 3.

[1]           C. J. van Duijn, A. Mikelić, M. F. Wheeler, and T. Wick, “Thermoporoelasticity via homogenization: Modeling and formal two-scale expansions,” Int. J. Eng. Sci., vol. 138, pp. 1–25, May 2019, doi: 10.1016/j.ijengsci.2019.02.005.

[2]           C. J. van Duijn, A. Mikelić, and T. Wick, “Mathematical theory and simulations of thermoporoelasticity,” Comput. Methods Appl. Mech. Eng., vol. 366, p. 113048, Jul. 2020, doi: 10.1016/j.cma.2020.113048.