Plane-strain propagation of a fluid-driven fracture during injection and shut-in: Asymptotics of large toughness

This paper considers the problem of plane-strain fluid-driven fracture propagating in an impermeable elastic medium under condition of large toughness or, equivalently, of low fracturing fluid viscosity. We construct an explicit solution for a fracture propagating in the toughness-dominated regime when the energy dissipated in the viscous fluid flow inside the fracture is negligibly small compared to the energy expended in fracturing the solid medium. The next order corrections in viscosity to this limiting solution are then derived, allowing the range of problem parameters corresponding to the toughness-dominated regime to be established. The first-order small viscosity (large toughness) solution is shown to provide an excellent approximation of the solution for the crack length in the wide range of the viscosity parameter. Furthermore, this solution, when combined with the first-order small-toughness solution of Garagash and Detournay [Journal of Applied Mechanics, 2005], provides a simple analytical approximation of the crack length solution in practically the entire range of viscosity (toughness). It is also shown that the established method of asymptotic expansion in small parameter is equally applicable to study other small effects (e.g., fluid inertia) on the otherwise toughness-dominated solution. A solution for the fracture evolution during shut-in (i.e., after fluid injection rate is suddenly stopped) is also obtained. This solution, which corresponds to a slowing fracture evolving towards the toughness-dominated steady state, draws attention to the possibility of substantial fracture growth after fluid injection is ceased especially under conditions when the fracture propagation during injection phase is dominated by viscous dissipation.