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Discretisations of mixed-dimensional Thermo-Hydro-Mechanical models preserving energy estimates

Jerome Droniou, Mohamed Laaziri, Roland Masson

TL;DR

This work develops energy-preserving discretisations for mixed-dimensional Thermo-Hydro-Mechanical models in fractured porous media, incorporating Coulomb friction at matrix–fracture interfaces. It compares two energy formulations—one based on energy conservation and one on entropy balance—within a unified HFV/mixed-FEM framework, ensuring discrete energy estimates and dissipativity of fracture contacts. Through 2D tests, including a Discrete Fracture Matrix model with convection-dominated heat transport and both weakly compressible liquids and perfect gases, the study demonstrates similar spatial convergence and solver robustness for both formulations, while highlighting the importance of including V_m·∇p terms in the entropy-based approach under gas-like conditions. The results have practical implications for geothermal stimulation and CO2 storage by enabling stable, energy-consistent simulations of THM processes in complex fracture networks.

Abstract

In this study, we explore mixed-dimensional Thermo-Hydro-Mechanical (THM) models in fractured porous media accounting for Coulomb frictional contact at matrix fracture interfaces. The simulation of such models plays an important role in many applications such as hydraulic stimulation in deep geothermal systems and assessing induced seismic risks in CO2 storage. We first extend to the mixed-dimensional framework the thermodynamically consistent THM models derived in [16] based on first and second principles of thermodynamics. Two formulations of the energy equation will be considered based either on energy conservation or on the entropy balance, assuming a vanishing thermo-poro-elastic dissipation. Our focus is on space time discretisations preserving energy estimates for both types of formulations and for a general single phase fluid thermodynamical model. This is achieved by a Finite Volume discretisation of the non-isothermal flow based on coercive fluxes and a tailored discretisation of the non-conservative convective terms. It is combined with a mixed Finite Element formulation of the contact-mechanical model with face-wise constant Lagrange multipliers accounting for the surface tractions, which preserves the dissipative properties of the contact terms. The discretisations of both THM formulations are investigated and compared in terms of convergence, accuracy and robustness on 2D test cases. It includes a Discrete Fracture Matrix model with a convection dominated thermal regime, and either a weakly compressible liquid or a highly compressible gas thermodynamical model.

Discretisations of mixed-dimensional Thermo-Hydro-Mechanical models preserving energy estimates

TL;DR

This work develops energy-preserving discretisations for mixed-dimensional Thermo-Hydro-Mechanical models in fractured porous media, incorporating Coulomb friction at matrix–fracture interfaces. It compares two energy formulations—one based on energy conservation and one on entropy balance—within a unified HFV/mixed-FEM framework, ensuring discrete energy estimates and dissipativity of fracture contacts. Through 2D tests, including a Discrete Fracture Matrix model with convection-dominated heat transport and both weakly compressible liquids and perfect gases, the study demonstrates similar spatial convergence and solver robustness for both formulations, while highlighting the importance of including V_m·∇p terms in the entropy-based approach under gas-like conditions. The results have practical implications for geothermal stimulation and CO2 storage by enabling stable, energy-consistent simulations of THM processes in complex fracture networks.

Abstract

In this study, we explore mixed-dimensional Thermo-Hydro-Mechanical (THM) models in fractured porous media accounting for Coulomb frictional contact at matrix fracture interfaces. The simulation of such models plays an important role in many applications such as hydraulic stimulation in deep geothermal systems and assessing induced seismic risks in CO2 storage. We first extend to the mixed-dimensional framework the thermodynamically consistent THM models derived in [16] based on first and second principles of thermodynamics. Two formulations of the energy equation will be considered based either on energy conservation or on the entropy balance, assuming a vanishing thermo-poro-elastic dissipation. Our focus is on space time discretisations preserving energy estimates for both types of formulations and for a general single phase fluid thermodynamical model. This is achieved by a Finite Volume discretisation of the non-isothermal flow based on coercive fluxes and a tailored discretisation of the non-conservative convective terms. It is combined with a mixed Finite Element formulation of the contact-mechanical model with face-wise constant Lagrange multipliers accounting for the surface tractions, which preserves the dissipative properties of the contact terms. The discretisations of both THM formulations are investigated and compared in terms of convergence, accuracy and robustness on 2D test cases. It includes a Discrete Fracture Matrix model with a convection dominated thermal regime, and either a weakly compressible liquid or a highly compressible gas thermodynamical model.
Paper Structure (19 sections, 3 theorems, 82 equations, 17 figures, 2 tables)

This paper contains 19 sections, 3 theorems, 82 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mixed-dimensional Thermo-Hydro-Mechanical models
  3. Mixed-dimensional geometry and function spaces
  4. Mixed-dimensional models
  5. Matrix model
  6. Fracture model
  7. Energy estimates
  8. Discretisation
  9. Space and time discretisations
  10. Discretisation of the mass and energy equations
  11. Mixed variational discretisation of the contact-mechanical model
  12. Summary of the schemes and discrete energy estimates
  13. Numerical experiments
  14. Manufactured solution without fractures
  15. 2D Discrete Fracture Matrix (DFM) model
  16. ...and 4 more sections

Key Result

Lemma 3.1

For all cell $K\in \mathcal{M}$, $T^n_K \times eq:entropy.discrete:matrix + h_K^n \timeseq:mass.discrete:matrix$ is equivalent to eq:energy.discrete:matrix upon correcting the left-hand side by adding For all fracture face $\sigma \in \mathcal{F}_\Gamma$, $T^n_\sigma \times eq:entropy.discrete:fracture + h_\sigma^n \timeseq:mass.discrete:fracture$ is equivalent to eq:energy.discrete:fracture upon

Figures (17)

  • Figure 1: Illustration of the dimension reduction in the fracture aperture for a 2D domain $\Omega$ with three intersecting fractures $\Gamma_i$, $i\in\{1,2,3\}$, with the equi-dimensional geometry on the left and the mixed-dimensional geometry on the right.
  • Figure 2: Conceptual fracture model with contact at asperities, $d_0$ being the fracture aperture at contact state.
  • Figure 3: Square domain $\Omega$ with its triangular mesh $m=2$ using $56 \times 4$ cells.
  • Figure 4: Convergence of the relative $L^2$ errors for the temperature $T$, pressure $p$, and displacement $\mathbf{u}$ and their gradients for the discretisation of the \ref{['model:entropy']} model and both the centred (for $\mathbb{K} = \mathbb{I}$) and upwind (for $\mathbb{K} = \mathbb{I},100 ~\mathbb{I}$) schemes.
  • Figure 5: Convergence of the relative $L^2$ errors for the temperature $T$, pressure $p$, and displacement $\mathbf{u}$ and their gradients for the discretisation of the \ref{['model:enthalpy']} model and both the centred (for $\mathbb{K} = \mathbb{I}$) and upwind (for $\mathbb{K} = \mathbb{I},100 ~\mathbb{I}$) schemes.
  • ...and 12 more figures

Theorems & Definitions (6)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof