Similarity Solutions for Shallow Hydraulic Fracture in Impermeable Rock

ABSTRACT The paper deals with the plane strain problem ofa shallow fluid-driven fracture propagating parallel to a free surface in an impermeable elastic rock. For a shallow fracture, the aperture corresponds essentially to the deflection of the layer defined by the free-surface and the crack. This problem is thus governed by the Euler-Bernoulli beam equation and by the lubrication theory. We show that the solution generally depends on two independent time scales and that there exist 4 similarity solutions denoted as (Equation), (Equation), (Equation) and (Equation). Furthermore, we demonstrate that the parametric space of solution is a rectangle with the vertices corresponding to the similarity solutions. The (Equation) and (Equation)-solutions are two different early time solutions characterized by a small length of the fluid-infiltrated region compared to the fracture length (i.e., the fluid lag occupies nearly the whole fracture), while the (Equation) and (Equation) solutions are two large-time solutions characterized by a vanishingly small lag. The (Equation) and the (Equation) vertices are similarity solutions for the particular case when Gc = 0, while (Equation) and (Equation) are similarity solutions for the case σo = 0. Thus the (Equation)-edge and the (Equation)-edge represent two particular fracture evolutions, the first one when Gc = 0 and the second one when σo = 0. However, the large time solution for the general case Gc > 0 and σo > 0 has not yet been identified, as the numerical solutions conflict with the large time matched asymptotic solution. INTRODUCTION Various instances of fluid-driven fracture near a free surface can be found in nature and in geo-engineering. They include excavation of hard rocks (Young, 1999), cave in-ducement in mining (Jeffrey et al., 2000), contaminant spill remediation (Murdoch, 2002), and the formation of sills by magma rising from deep underground chambers (Bunger and Cruden, 2011; Michaut, 2011). Models of hydraulic fractures propagating near the surface of a homogeneous elastic semi-infinite domain can be formulated mathematically using elastic integral equations that relate the displacement discontinuity across the fracture to the net traction acting on the fracture faces, in combination with the nonlinear lubrication equation that governs the flow of viscous fluid inside the fracture (Zhang et al., 2002, 2005; Gordeliy and Detournay, 2011). The resulting system of integro-differential equations together with the boundary conditions at the injection point (typically a constant injection rate) and at the fracture tip (including the propagation criterion) is closed in regard to predicting the evolution of fracture geometry, fluid pressure, and aperture from given initial conditions.