A comparison between a semi-analytical and a numerical solution of a two-dimensional hydraulic fracture

Abstract

This paper compares a semi-analytical self-similar solution of the problem of a hydraulically driven fracture with results obtained using the numerical model Loramec. The problem under consideration is a hydraulic fracture propagating in an infinite impermeable elastic medium under plane strain conditions. The fracture is driven by an incompressible Newtonian fluid injected, at a constant rate, from a source located at the center of the fracture. There are some differences between the two models in regard to the modeling of the near tip processes. The semi-analytical solution is built on the assumptions that the fracture is completely filled by the injection fluid and that the solid has zero toughness, while the numerical model explicitly accounts for the existence of a priori unknown lag between the fluid and crack front. It is shown that the numerical results exhibit self-similarity; in particular the predicted power law dependence on time of the net pressure, aperture and fracture length is well observed in the numerical results. Also, a very good agreement between the self-similar and the numerical solution is observed under conditions of `small' toughness. The results of this study actually suggest that the self-similar zero toughness solution is a good approximation to cases where the rock has a non-zero fracture toughness and a fluid lag develops, provided that the ratio of the rate of energy dissipation in the solid over the viscous dissipation in the fluid is less than 10 2. 1999 Elsevier Science Ltd. All rights reserved.