Propagation of a penny-shaped hydraulic fracture parallel to a free, surface, with application to inducing rock mass caving for mining

摘要

In this paper, the problem of a penny-shaped hydraulic fracture propagating parallel to the free-surface of an elastic half-space is studied. The fracture is driven by an incompressible Newtonian fluid injected at a constant rate. The flow of viscous fluid in the fracture is governed by the lubrication equation, while the crack opening and the fluid pressure are related by singular integral equations. We construct two asymptotic solutions based on the assumption that the energy expended in the creation of new fracture surfaces is either small or large compared to the energy dissipated in viscous flow. One important outcome of the analysis is to show that the asymptotic solutions, when properly scaled, depend only on the dimensionless parameter R, the ratio of the fracture radius over the distance from the fracture to the free-surface. The scaled solutions can thus be tabulated and the dependence of the solution on time can be retrieved for specific parameters, through simple scaling and by solving an implicit equation.