Blade-Shaped Hydraulic Fracture Driven by a Turbulent Fluid in an Impermeable Rock
Publication: Journal of Engineering Mechanics
Volume 143, Issue 11
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.
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Acknowledgments
Navid Zolfaghari and Andrew P. Bunger wish to acknowledge the support from the University of Pittsburgh Swanson School and Engineering and Center for Energy. Their discussions with Professor Jim Rice and the insights from two anonymous reviewers are also gratefully acknowledged.
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Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 143 • Issue 11 • November 2017
Copyright
©2017 American Society of Civil Engineers.
History
Received: Jul 19, 2016
Accepted: May 10, 2017
Published online: Sep 8, 2017
Published in print: Nov 1, 2017
Discussion open until: Feb 8, 2018
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