Shearing and opening of a pre-existing discontinuity in response to fluid injection

We investigate the fluid-driven growth of a shear crack along a frictional discontinuity and its transition to hydraulic fracturing (sometimes referred to as hydraulic jacking) under plane-strain conditions. We focus on the case of a constant friction coefficient and account for the permeability changes associated with the fracture opening. By combining the scaling analysis and numerical simulations, we examine the evolution of both the shear and opening fronts as a function of the hydro-mechanical properties of the pre-existing discontinuity, in-situ stress state, and the fluid injection conditions. Further, we derive an approximate analytical solution for the relation between the positions of the slip and opening fronts at large times. We notably show that the ratio between the slip and opening fronts converges to a constant value at late times which only depends on the ratio between the shear stress and shear strength acting initially along the discontinuity. We compare this approximate solution against numerical simulations and demonstrate its usage to serve as a benchmark solution in the development of coupled hydro-mechanical numerical solvers for frictional fluid-driven fractures.