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

This paper compares a semi-analytical self-similar solution of the problem of a hydraulically driven fracture with results obtained using the numerica l model Loramec. The problem under consideration is a hydraulic fracture pr opagating in an infinite impermeable elastic medium under plane strain cond itions. The fracture is driven by an incompressible Newtonian fluid injecte d, 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 modelin g of the near tip processes. The semi-analytical solution is built on the a ssumptions that the fracture is completely filled by the injection fluid an d that the solid has zero toughness, while the numerical model explicitly a ccounts for the existence of a priori unknown lag between the fluid and cra ck front. It is shown that the numerical results exhibit self-similarity; i n particular the predicted power law dependence on time of the net pressure , aperture and fracture length is well observed in the numerical results. A lso, a very good agreement between the self-similar and the numerical solut ion is observed under conditions of 'small' toughness. The results of this study actually suggest that the self-similar zero toughness solution is a g ood approximation to cases where the rock has a non-zero fracture toughness and a fluid lag develops, provided that the ratio theta of the rate of ene rgy dissipation in the solid over the viscous dissipation in the fluid is l ess than 10(-2). (C) 1999 Elsevier Science Ltd. All rights reserved.