Statistical Method for Tracing Hydraulic Fracture Front Without Evaluation of the Normal

In numerical simulation of hydraulic fracture propagation, tangent component of the fluid velocity generally considered to be neglected near the crack front. Then Reynolds transport theorem yields that the limit of the particle velocity coincides with the vector of the front propagation speed. We use this fact in combination with the Poiseuille-type equation, which implies that the particle velocity is always collinear to pressure gradient. We show that this specific feature of the hydraulic fracture problem may serve to simplify tracing the front propagation. The latter may be traced without explicit evaluation of the normal to the front, which is needed in conventional applications of the theory of propagating interfaces. Numerical experiments confirm that, despite huge errors in pressure and even greater errors in its gradient, the propagation speed, statistically averaged over a distance of a mesh size, is found quite accurate. We conclude that suggested method may simplify numerical simulation of hydraulic fractures driven by Newtonian and non-Newtonian fluids.