Blade-Shaped Hydraulic Fracture Driven by a Turbulent Fluid in an Impermeable Rock

Abstract

Water-driven hydraulic fractures with high flow rates are more common now than ever in the oil and gas industry. Although these fractures are small, the high injection rate and low viscosity of the water lead to high Reynolds numbers and potential turbulence in the fracture. This paper presents a semianalytical solution for a blade-shaped [Perkins-Kern-Nordgren (PKN)] geometry hydraulic fracture driven by a turbulent fluid in the limit of zero fluid leak-off to the formation. Turbulence in the PKN fracture is modeled using the Gaukler-Manning-Strickler parametrization, which relates the flow rate of the water to the pressure gradient along the fracture. The fully turbulent limit is considered with no transition region anywhere at any time, but the effect of fracture toughness on crack propagation is not considered. The key parameter in this relation is the Darcy-Weisbach friction factor for the roughness of the crack wall. Coupling this turbulence parametrization with conservation of mass allows a nonlinear partial differential equation (PDE) to be written for the crack width as a function of space and time. By way of a similarity ansatz, a semianalytical solution is obtained using an orthogonal polynomial series. Very rapid convergence is found by embedding the asymptotic behavior near the fracture tip into the polynomial series; a suitably accurate solution is obtained with two terms of the series. This closed-form solution facilitates clear comparisons between the results and parameters for laminar and turbulent hydraulic fractures. In particular, it resolves one of the well-known problems whereby calibration of models to data has difficulty simultaneously matching the hydraulic fracture length and wellbore pressure.