Effects of velocity-dependent apparent toughness on the pre- and post-shut-in growth of a hydraulic fracture

We investigate the growth of a plane-strain/radial hydraulic fracture in an infinite impermeable medium driven by a constant injection rate assuming that the apparent toughness scales with the decreasing fracture growth rate in a power-law relation. The viscosity dominated regime always governs the fracture growth at large time for the plane-strain geometry. For a radial hydraulic fracture, we report a transition from early-time fracture growth dominated by viscous fluid flow to large-time propagation dominated by fracture toughness. Such a transition results from an overall increase of the energy dissipation by fracture surface creation. After shut-in, both plane-strain and radial fractures propagate at a lower velocity with decreasing fracture toughness. The fracture growth then depends on the dimensionless toughness at the stop of fluid injection and transitions towards a self-similar pulse-viscosity solution when viscous fluid flow dominates the energy dissipation. The fracture arrests when fulfilling two conditions - the apparent toughness reaching its minimum, and viscous forces being negligible in the fluid flow. The fracture dimension at arrest is independent of the velocity-dependent power-law relation.