A linear hydraulic fracture with tortuosity and fluid–slip at the fluid–rock interface

A hydraulic fracture with tortuosity and fluid–slip at the fluid–rock interface is replaced by a two-dimensional symmetric model fracture. The mathematical model is formulated using a modified Reynolds flow law and a linear crack law. The Perkins-Kern-Nordgren (PKN) approximation is used to close the model. The fluid–slip velocity is introduced in the analysis of boundary conditions at the fracture walls. A two-dimensional model fracture is described by a second-order nonlinear diffusion equation. The fluid–slip velocity is not defined a priori but is determined in the process of obtaining the group invariant solution. The Lie point symmetry is used to derive the group invariant solution for the half-width and the fluid–slip velocity. A comparison of a fracture with zero fluid–slip velocity at the fractures walls and that of non-zero fluid–slip velocity at fracture walls gives a relation of the fluid–slip velocity and the half-width. An exact analytical solution of a fracture propagating with constant volume is derived. The right combination of the fluid–slip parameter r and the tortuosity parameter n is found to remove singularity at the fracture tip. Four working conditions at the fracture entry are considered and their solutions are obtained numerically. Introducing fluid–slip at the fracture walls by decreasing the slip parameter r from 1 to 0 for a fixed scaled time t decreases the half-with and the volume of a model fracture. For a fixed scaled time t, decreasing r results in a less tortuous fracture with a shorter length and a more tortuous fracture that is longer in length.