Evgenii Kanin, Dmitry Garagash, Andrei Osiptsov 

This work is devoted to an analysis of the near-tip region of a hydraulic fracture driven by slickwater in a permeable saturated rock. We consider a steady-state problem of a semi-infinite fracture propagating with constant velocity. The host rock is elastic and homogeneous, and fracture propagates according to linear elastic fracture mechanics. The fluid exchange between the fracture and reservoir is governed by Carters law. The distinguishing feature of the model is an account for the transition of the flow regime inside the crack channel from laminar to turbulent moving away from the fracture front. The main objective is to analyse the influence of the leak-off process on the laminar-to-turbulent transition and, thus, potential prominence of turbulent flow effects. Hydraulic fracturing fluid is water with polymeric additives (slickwater). These additives reduce viscous friction resulting in the decrease of energy consumption required for pumping. Compared to water, the slickwater exhibits significantly delayed transition to the turbulent regime described by the maximum drag reduction asymptote (Virk, 1975). The system of governing equations, which consists of elasticity equation, propagation condition, the continuity equation for viscous incompressible Newtonian fluid, and Poiseuilles law modified for the turbulent flow regime, is solved for the fracture aperture and fluid pressure along the fracture as a function of problem parameters. We find out that the leak-off process enhances the turbulent flow effects by shifting the transition between laminar and turbulent flow regimes closer to the fracture front, as compared to the zero-leak-off case (Lecampion & Zia, 2019), resulting in a broader region of the fracture hosting turbulent flow. Consequently, in the permeable reservoir case, the transition to turbulent flow can be realised at a distance from the front smaller than the typical field hydraulic fracture size (10 - 100 meters). We compare the fracture width profiles with the impermeable rock case and reveal that the fracture volume increases when leak-off occurs. We analyse the problem parametric space where five limiting regimes are identified: toughness, laminar-viscosity and -leak-off, turbulent-viscosity and -leak-off. We derive analytical expressions for the fracture width and pressure profiles in the turbulent-leak-off regime while others have been established previously. By comparing the limiting solutions with the general numerical solution, we can define their applicability domains and corresponding solution regime maps. The toughness and turbulent-viscosity regimes approximate the general solution in the near- and far-fields, while the other three limiting cases can emerge in the intermediate field.