The effects of non-inverse-square-root stress singularity on some crack propagation criteria in hydraulic fracturing

Numerous crack propagation criteria have been employed in numerical investigations of hydraulic fracturing. Some criteria rely on linear elastic fracture mechanics, which assume the presence of an inverse-square-root stress singularity and an elliptical crack profile. However, it has been observed that hydraulic fracturing exhibits a weaker stress singularity, which is governed by a dimensionless parameter known as apparent fracture toughness. This parameter represents the energy required to fracture the solid compared to the energy needed to overcome fluid friction. Abnormal stress singularity was found for the viscosity dominated regime where the apparent fracture toughness approaches zero from the positive side. Variations in results can occur when different criteria are employed, primarily due to the breakdown of the inverse-square-root stress singularity. In this paper, a mathematical model utilizing a straight slit crack was employed to investigate these discrepancies. Four commonly used criteria were implemented in the same mathematical model, and high-precision methods based on modified Chebyshev polynomials and implicit algorithms were employed for calculations. The transient crack dimensions and wellbore pressure were found to diverge for these criteria. Notably, the CTOD, CDM, and FPZ criteria were found to be challenging to apply in the viscosity-dominated regime where the apparent fracture toughness was less than 1.0. The underlying mechanisms were discussed based on crack tip asymptotics and crack opening behavior. This study provides valuable insights for selecting appropriate crack propagation criteria in hydraulic fracturing.