Propagation of a height contained hydraulic fracture in turbulent flow regimes

We investigate theoretically and numerically the impact of the transition from laminar to turbulent flow on the propagation of a height contained hydraulic fracture (i.e. PKN geometry). We account for the inertial terms in the balance of momentum and express the viscous wall shear stress via Fanning friction. The evolution of the friction factor with Reynolds number and the fracture relative roughness is obtained for a PKN fracture geometry from known relations for circular pipes using the concept of an equivalent laminar hydraulic diameter. From dimensional analysis, we show that inertial forces are always negligible. We also obtain the transition time scale between the turbulent rough propagation regime -valid at early time- to the turbulent smooth regime. This transition time-scale appears much larger than typical injection duration which confirms the dominant effect of fracture roughness in the turbulent regime. We derive a number of limiting solutions for hydraulic fracture propagation assuming that the flow occurs in a given regime over all the fracture extent: turbulent smooth or turbulent rough. We then solve numerically the complete laminar to turbulent transition as function of the Reynolds number at the fracture entrance. The fraction of the fracture exhibiting laminar flow shrinks to a boundary layer at the fracture tip as the entrance Reynolds number increases. Our numerical results notably indicate that the entrance Reynolds number must be at least equal to 10,000 for the fully turbulent rough solution to be valid. In practical applications, the entrance Reynolds number is typically lower than 5,000 such that the fracture propagation is influenced by the complete transition from laminar to turbulent flow. Using our numerical scheme, we tabulate the evolution of the fracture length and width at the fracture entrance for different values of the entrance Reynolds number covering the transition from the fully laminar to the fully turbulent rough regime. The effect of polymer friction reducing agents, which drastically modify the transition to turbulence, is also investigated semi-analytically and numerically. The height contained hydraulic fracture propagation (assuming maximum drag reduction) is only about 15% different from the fully laminar solution in the range of relevant practical entrance Reynolds number, compared to up to 40% difference without the addition of polymer friction reducers.