Blade-shaped (PKN) Hydraulic Fracture Driven By A Turbulent Fluid In An Impermeable Rock

TL;DR

This paper develops a semi-analytical model for blade-shaped hydraulic fractures driven by turbulent water flow, incorporating turbulence effects via a friction factor, and provides solutions that improve understanding of fracture behavior at high flow rates.

Contribution

It introduces a semi-analytical solution for turbulent hydraulic fractures in PKN geometry, integrating turbulence modeling with a polynomial series approach for rapid convergence.

Findings

Rapid convergence with two polynomial terms

Enhanced model calibration for fracture length and pressure

Comparison between laminar and turbulent fracture parameters

Abstract

High flow rate, water-driven hydraulic fractures are more common now than ever in the oil and gas industry. Although the fractures are small, the high injection rate and low viscosity of the water, lead to high Reynolds numbers and potentially turbulence in the fracture. Here we present a semi-analytical solution for a blade-shaped (PKN) geometry hydraulic fracture driven by a turbulent fluid in the limit of zero fluid leak-off to the formation. We model the turbulence in the PKN fracture using the Gaukler-Manning-Strickler parametrization, which relates the the flow rate of the water to the pressure gradient along the fracture. 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 us to write a nonlinear pde for the crack width as a function of space and time. By way of a similarity ansatz, we obtain a semi-analytical solution using an orthogonal polynomial series. Embedding the asymptotic behavior near the fracture tip into the polynomial series, we find very rapid convergence: 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.