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A phase-field fracture model in thermo-poro-elastic media with micromechanical strain energy degradation

Yuhao Liu, Keita Yoshioka, Tao You, Hanzhang Li, Fengshou Zhang

TL;DR

This work develops a thermo-poro-elastic phase-field fracture framework that integrates thermal strain and micromechanical degradation of Biot's coefficient through an energy-decomposition approach. A strain-based porosity update is introduced to avoid phase-field length-scale artifacts, enabling consistent computation of $M_p(\upsilon,\varepsilon)$ and $M_T(\upsilon,\varepsilon)$ that govern hydraulic and thermal responses as material transitions from intact to fractured. Numerical stability is achieved via an isotropic-diffusion stabilization for advection-dominated heat transfer and a staggered solution with fixed-stress stabilization for the hydro-mechanical part. The model is verified against Terzaghi’s and thermal consolidation, and against the KGD fracture problem, showing that the new porosity formulation more accurately captures sharp-fracture behavior. Numerical experiments reveal pronounced thermo-hydro-mechanical interactions, including interface effects under various temperature differences, with implications for geothermal and CO$_2$ sequestration applications.

Abstract

This work extends the hydro-mechanical phase-field fracture model to non-isothermal conditions with micromechanics based poroelasticity, which degrades Biot's coefficient not only with the phase-field variable (damage) but also with the energy decomposition scheme. Furthermore, we propose a new approach to update porosity solely determined by the strain change rather than damage evolution as in the existing models. As such, these poroelastic behaviors of Biot's coefficient and the porosity dictate Biot's modulus and the thermal expansion coefficient. For numerical implementation, we employ an isotropic diffusion method to stabilize the advection-dominated heat flux and adapt the fixed stress split method to account for the thermal stress. We verify our model against a series of analytical solutions such as Terzaghi's consolidation, thermal consolidation, and the plane strain hydraulic fracture propagation, known as the KGD fracture. Finally, numerical experiments demonstrate the effectiveness of the stabilization method and intricate thermo-hydro-mechanical interactions during hydraulic fracturing with and without a pre-existing weak interface.

A phase-field fracture model in thermo-poro-elastic media with micromechanical strain energy degradation

TL;DR

This work develops a thermo-poro-elastic phase-field fracture framework that integrates thermal strain and micromechanical degradation of Biot's coefficient through an energy-decomposition approach. A strain-based porosity update is introduced to avoid phase-field length-scale artifacts, enabling consistent computation of and that govern hydraulic and thermal responses as material transitions from intact to fractured. Numerical stability is achieved via an isotropic-diffusion stabilization for advection-dominated heat transfer and a staggered solution with fixed-stress stabilization for the hydro-mechanical part. The model is verified against Terzaghi’s and thermal consolidation, and against the KGD fracture problem, showing that the new porosity formulation more accurately captures sharp-fracture behavior. Numerical experiments reveal pronounced thermo-hydro-mechanical interactions, including interface effects under various temperature differences, with implications for geothermal and CO sequestration applications.

Abstract

This work extends the hydro-mechanical phase-field fracture model to non-isothermal conditions with micromechanics based poroelasticity, which degrades Biot's coefficient not only with the phase-field variable (damage) but also with the energy decomposition scheme. Furthermore, we propose a new approach to update porosity solely determined by the strain change rather than damage evolution as in the existing models. As such, these poroelastic behaviors of Biot's coefficient and the porosity dictate Biot's modulus and the thermal expansion coefficient. For numerical implementation, we employ an isotropic diffusion method to stabilize the advection-dominated heat flux and adapt the fixed stress split method to account for the thermal stress. We verify our model against a series of analytical solutions such as Terzaghi's consolidation, thermal consolidation, and the plane strain hydraulic fracture propagation, known as the KGD fracture. Finally, numerical experiments demonstrate the effectiveness of the stabilization method and intricate thermo-hydro-mechanical interactions during hydraulic fracturing with and without a pre-existing weak interface.
Paper Structure (21 sections, 91 equations, 17 figures, 4 tables)

This paper contains 21 sections, 91 equations, 17 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Mechanical equilibrium with discrete cracks
  4. Mechanical equilibrium with phase-field approximation
  5. Fluid flow model
  6. Heat transfer model
  7. Numerical implementation
  8. The staggered solution scheme
  9. Isotropic diffusion stabilization method
  10. Model verification
  11. Terzaghi's consolidation problem
  12. Thermal consolidation problem
  13. Fluid-driven fracture propagation in plane-strain
  14. Numerical experiments and discussion
  15. Stabilization for the advection-dominated problem
  16. ...and 6 more sections

Figures (17)

  • Figure 1: (a) The schematic of porous medium with an existing fracture and (b) the diffused representation of a fracture $\mit\mit{\Gamma}$ in the poroelastic medium.
  • Figure 2: Physical crack opening within quadrilateral elements.
  • Figure 3: Schematic of Terzaghi's consolidation problem.
  • Figure 4: Comparisons of numerical results and analytical solution of Terzaghi’s consolidation problem for (a) pressure and (b) displacement.
  • Figure 5: Schematic of the thermal consolidation problem.
  • ...and 12 more figures