Comparison of toughness propagation criteria for blade-like and pseudo-3D hydraulic fractures

The goal of this study is to compare and evaluate the accuracy of different approaches to incorporating the effect of lateral fracture toughness into reduced models for blade-like and pseudo-3D hydraulic fractures. The following three methods are used for the comparison: (i) a classical model with a plane strain (or local) elasticity assumption and a pressure boundary condition calculated based on energetic considerations, (ii) a classical model with local elasticity and pressure boundary condition originating from “stitching” a radial fracture tip to the rest of the fracture, and (iii) a novel model with non-local elasticity and a boundary condition at the tip that is consistent with the linear elastic fracture mechanics propagation criterion. Predictions of all three approaches are compared to a reference solution calculated using a fully planar hydraulic fracturing simulator. The results indicate that the reduced model with non-local elasticity is able to provide an accurate approximation for a wide range of fracture toughness values. The models that feature the local elasticity assumption are able to provide reasonably accurate results for moderate values of fracture toughness, while they become less accurate for blade-like geometries and significantly less accurate (and in some cases unstable) for the pseudo-3D geometry for large values of the fracture toughness.