Multiscale Tip Asymptotics of a Shallow Hydraulic Fracture With Fluid Lag

Abstract This study investigates the boundary layer solution in the tip region of shallow hydraulic fractures that propagate with a finite fluid lag. Owing to their proximity to the free surface, shallow hydraulic fractures develop under relatively low confining stress, conditions that promote the formation of a pronounced fluid lag. Simultaneously, stress and geometric asymmetries induce mixed-mode propagation, giving rise—under specific conditions—to a sliding zone at the crack tip. A scaling analysis shows that the tip region solution is governed by two dimensionless parameters: the dimensionless toughness K and the dimensionless stress S. The dependence of the fluid lag length and the extent of the sliding zone on these parameters is analyzed in detail. The resulting solution displays a multiscale asymptotic structure comprising tip, intermediate, and far-field regions, with the configuration of these regions varying systematically with K and S. The simultaneous presence of a fluid lag and a sliding zone substantially increases the complexity of the boundary layer behavior compared with cases where the lag is absent. Furthermore, elastic deformation of the substrate influences the far-field moment, necessitating the inclusion of a root rotation term in the far-field beam asymptotic representation. The dependences of both the far-field moment and the root rotation on the governing parameters K and S are shown to be highly nonlinear, reflecting the coupled effects of the fluid lag and the sliding zone on the overall solution structure.