Fracture Geometry Modeling and Analysis of Layer-Crossing Conditions in Continental Shale Considering Apparent Fracture Toughness

Vertical viscous resistance and bedding deformation jointly influence hydraulic fracture propagation in continental shale. A coupled pseudo-three-dimensional (P3D) framework incorporates apparent fracture toughness (effective front toughness accounting for vertical viscous resistance) and tangential bedding compliance. Near-tip aperture matching yields an implicit toughness equation with a unique nonnegative root. A layered elastic weak form, fluid conservation, and front updates couple pressure, aperture, and interfacial displacement. A vertical propagation pressure relation is derived at a prescribed height for perfectly bonded conditions with symmetric stress barriers. Existing end-of-treatment data for stage 8 of a Sichuan Basin well are reanalyzed. The modified scheme yields an average height of 55.6 m, within the interpreted 52–58 m range, and a half-length of 178 m, deviating 3.49% from the interpreted 172 m. Increasing prescribed additional toughness from 0.1 to 0.7 MPa·m1/2 reduces average height and half-length by 38.62% and 22.22%, respectively, while maximum tangential interfacial displacement increases from 0.41 to 1.26 mm. The framework clarifies the distinct roles and coupling of propagation resistance and elastic bedding deformation, providing theoretical and case-based support for fracture height control, geometry analysis, and assessment of layer-crossing conditions.