Tip region of a hydraulic fracture driven by a laminar-to-turbulent fluid flow

The focus of this study is to analyse the tip region of a hydraulic fracture, for which a fluid flow inside the crack transitions from the laminar to the turbulent regime away from the tip. To tackle the problem, a phenomenological formula for flow in pipes has been adapted to describe flow in a fracture through the concept of a hydraulic diameter. The selected model is able to capture laminar, turbulent and transition regimes of the flow. The near-tip region of a hydraulic fracture is analysed by focusing on steady propagation of a semi-infinite hydraulic fracture with leak-off, for which the aforementioned phenomenological formula for the fluid flow is utilized. First, the distance from the tip within which a laminar solution applies is estimated. Then, expressions for asymptotic solutions that are associated with fully developed turbulent flow inside the semi-infinite hydraulic fracture are derived. Finally, the laminar zone size and the asymptotic solutions are compared with the numerical solution, where the latter captures all regimes of the fluid flow.