Effect of fluid compressibility and borehole radius on the propagation of a fluid-driven fracture

This paper is concerned with the problem of a radial penny-shaped fracture transverse to a borehole with finite radius. It focuses on compressibility and borehole finite radius effects as possible candidates to explain the early unstable phase of the fracture growth observed during laboratory experiments conducted with low viscosity fluids. We assume that the fracture is driven in an impermeable elastic material by a compressible, inviscid fluid, for which the pressure gradient along the fracture is zero. A solution is proposed in terms of dimensionless fluid pressure, fracture radius and inlet opening. It is shown that the problem depends on one single evolution parameter S describing the transition of the solution between two asymptotic regimes in which the solution becomes self-similar. Compressibility effects control the solution at small S while the solution at large S is dominated by the material toughness. A built-in instability is identified in the system of equations describing the problem and a criterion is proposed to predict the occurrence of an unstable growth step after fracture breakdown. The possibility of an unstable growth depends on the initial flaw length, the fluid compressibility, the volume of fluid and the material elastic modulus. The finite borehole radius introduces one additional length scale to the formulation of the similar problem in an infinite medium.