A thermodynamically consistent phase-field model for anisotropic hydraulic fracturing

Hydraulic fracturing in anisotropic poroelastic media involves strongly coupled deformation, pore-pressure diffusion, and crack growth, all of which are direction dependent. Although phase-field methods have advanced hydraulic-fracture simulation, their application to anisotropic reservoirs remains limited. This study proposes a thermodynamically consistent hydro-mechanical phase-field formulation for hydraulic fracturing in anisotropic porous media. The model combines anisotropic Helmholtz free energy with an energy-orthogonal volumetric-deviatoric split to describe tension-compression asymmetric degradation in anisotropic solids. Directional fracture resistance is represented by an anisotropic crack-surface density functional, while fluid pressure is governed by anisotropic Biot poroelasticity with damage-dependent Darcy flow. A bounded non-linear solution scheme enforces the physical bounds and irreversibility of the phase-field variable. Numerical examples verify the isotropic limit against the toughness-dominated Khristianovich-Geertsma-de Klerk and penny-shaped hydraulic crack solution and demonstrate anisotropy-driven fracture deflection, orientation-dependent pressure evolution, and directionally controlled crack patterns. Simulation results indicate that the present model can simulate hydraulic-fracture growth in both two- and three-dimensional anisotropic porous media, including anisotropy-driven fracture deflection and direction-dependent pressure evolution.