Evgenii Kanin, Dmitry Garagash, Andrei Osiptsov 

This chapter considers a model for a radial hydraulic fracture propagation in a permeable, linear elastic rock formation driven by a point source fluid injection. The linear elastic fracture mechanics theory controls the quasi-static propagation. The hydraulic fracturing fluid is slickwater -- pure water solution with polymeric additives which allow reducing the fluid flow friction in the wellbore and fracture in reservoir field applications. We focus on the possible transformation of the fluid flow regime inside the fracture channel from laminar to turbulent with distance from the fracture front. We assume that the turbulent friction of slickwater is described by the maximum drag reduction asymptote, while Carter's law governs the leak-off into the permeable rock. The solution is obtained numerically using the algorithm based on the Gauss-Chebyshev quadrature and Barycentric Lagrange interpolation techniques. We compute solution examples for typical field cases and demonstrate a significant impact of the turbulent flow regime during the initial few minutes of propagation, namely, shorter radius and wider maximum aperture than the laminar model provides. Moreover, we observe higher fluid pressure values at the wellbore within tens of minutes of the start of the injection. This leads to a larger hydraulic pumping power requirement than the laminar model predicts. We also find that the fluid leak-off into the permeable rock enhances the turbulent flow effect in the fracture when compared to the impermeable rock case. In order to analyze the parametric dependence of the general solution, we convert the governing equations into the dimensionless form. We perform an extensive exploration of the normalized solution in space of two non-dimensional parameters, leak-off and characteristic Reynolds numbers, and normalized time. Specifically, we determine the applicability domains of the limiting propagation regimes to frame the general solution, investigate the alterations of the crack characteristics depending on the governing parameters, and identify zones where the turbulent flow is important.