An approximate solution for a plane strain hydraulic fracture that accounts for fracture toughness, fluid viscosity, and leak-off

The goal of this paper is to develop an approximate solution for a propagating plane strain hydraulic fracture, whose behavior is determined by a combined interplay of fluid viscosity, fracture toughness, and fluid leak-off. The approximation is constructed by assuming that the fracture behavior is primarily determined by the three-process (viscosity, toughness, and leak-off) multiscale tip asymptotics and the global fluid volume balance. First, the limiting regimes of propagation of the solution are considered, that can be reduced to an explicit form. Thereafter, applicability regions of the limiting solutions are investigated and transitions from one limiting solution to another are analyzed. To quantify the error of the constructed approximate solution, its predictions are compared to a reference numerical solution. Results indicate that the approximation is able to predict hydraulic fracture parameters for all limiting and transition regimes with an error of under one percent. Consequently, this development can be used to obtain a rapid solution for a plane strain hydraulic fracture with leak-off, which can be used for quick estimations of fracture geometry or as a reference solution to evaluate accuracy of more advanced hydraulic fracture simulators.