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 numerical model Loramec[ The problem under consideration is a hydraulic fracture propagating in an in_nite impermeable elastic medium under plane strain conditions[ The fracture is driven by an incompressible Newtonian ~uid injected\ at a constant rate\ from a source located at the center of the fracture[ There are some di}erences 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 _lled by the injection ~uid and that the solid has zero toughness\ while the numerical model explicitly accounts for the existence of a priori unknown lag between the ~uid 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 ~uid lag develops\ provided that the ratio u of the rate of energy dissipation in the solid over the viscous dissipation in the ~uid is less than 09