The critical energy release rate of the crack is always set as an inherent material property (constant value) in the phase-field method (PFM) to depict the compression-shear and mixed-mode failures of the shale rock. However, the recent experimental results about supercritical carbon dioxide (SC-CO₂) fracturing indicate that the degradation in strength of the porous medium is of significant importance because of the associated effect of fluid infiltration. To capture this tendency, a modified phase-field method is developed to fully consider the mixed-mode failure and shear strength degradation during the SC-CO₂ fracturing (SCF) process. Precisely, the finite element method (FEM) is used to discretize the solution space and staggered scheme is employed to solve the system of non-linear equations derived from the proposed PFM. A new critical energy release rate criterion that considers the pore pressure gradient is generated and a distinction between two different failure modes (Mode-I and Mode-II) is made through the spectral decomposition of strain energy. The model is validated against experimental results and depicts the crack branching behavior with a wide range of fluid viscosities and injection rates. Tension and shear failure modes alternately dominate the behavior of branching, followed by periodic fluctuation of the crack tip velocity and equivalent driving term, which is quantitatively depicted by the phase diagram.

在相场法（PFM）中，裂纹的临界能量释放率总是设置为固有的材料特性（常数值），以描述页岩的压剪和混合模式破坏。然而，最近关于超临界二氧化碳（SC-CO ₂ ）压裂的实验结果表明，由于流体渗透的相关影响，多孔介质的强度下降具有重要意义。为了捕捉这种趋势，开发了一种改进的相场方法，以充分考虑 SC-CO _{2过程中的混合模式失效和剪切强度退化。}压裂（SCF）工艺。确切地说，有限元法（FEM）用于离散解空间，并采用交错方案来求解从所提出的PFM导出的非线性方程组。产生了一个新的考虑孔隙压力梯度的临界能量释放率准则，并通过应变能的谱分解来区分两种不同的失效模式（模式 I 和模式 II）。该模型根据实验结果进行了验证，并描述了具有广泛流体粘度和注射率。张力和剪切破坏模式交替支配分支的行为，其次是裂纹尖端速度的周期性波动和等效驱动项，这由相图定量描述。