The Tip Region of a Near-Surface Hydraulic Fracture

This paper investigates the tip region of a hydraulic fracture propagating near a free-surface via the related problem of the steady fluid-driven peeling of a thin elastic layer from a rigid substrate. The solution of this problem requires accounting for the existence of a fluid lag, as the pressure singularity that would otherwise exist at the crack tip is incompatible with the underlying linear beam theory governing the deflection of the thin layer. These considerations lead to the formulation of a nonlinear traveling wave problem with a free boundary, which is solved numerically. The scaled solution depends only on one number $K$⁠, which has the meaning of a dimensionless toughness. The asymptotic viscosity- and toughness-dominated regimes, respectively, corresponding to small and large $K$⁠, represent the end members of a family of solutions. It is shown that the far-field curvature can be interpreted as an apparent toughness, which is a universal function of $K$⁠. In the viscosity regime, the apparent toughness does not depend on $K$⁠, while in the toughness regime, it is equal to $K$⁠. By noting that the apparent toughness represents an intermediate asymptote for the layer curvature under certain conditions, the obtention of time-dependent solutions for propagating near-surface hydraulic fractures can be greatly simplified. Indeed, any such solutions can be constructed by a matched asymptotics approach, with the outer solution corresponding to a uniformly pressurized fracture and the inner solution to the tip solution derived in this paper.