Numerical model for a penny-shaped hydraulic fracture driven by laminar/turbulent fluid in an impermeable rock

Abstract

As hydraulic fracturing at high injection rates with low viscosity fluids grows in popularity, so also there is a growing need to include not only the more common laminar fluid flow, but also the turbulent and transition flow regimes in numerical simulators. One common scenario is embodied in the behavior of a radial (penny-shaped) hydraulic fracture where flow is turbulent near the inlet, laminar near the tip, and in transition somewhere between. The main goal of this paper is to investigate the impact of this transition on hydraulic fracture growth through development and use of a numerical simulator for penny-shaped hydraulic fractures using the so-called drag reduction method to estimate the friction factor inside the crack for all relevant flow regimes. Upon solving this problem numerically for the case of zero toughness, comparing the results with fully laminar and fully turbulent asymptotic solutions shows that the early time behavior of radial hydraulic fractures is predominantly turbulent while large time behavior if predominantly laminar. The time scale associated with this transition determines the relevance of either limiting regime to practical cases, i.e. when the transition takes place in a small fraction of the total treatment time it suffices to approximate growth using the laminar asymptotic solution and when the transition requires are large time compared to the treatment time it suffices to approximate growth using the turbulent asymptotic solution.