Hydraulic fracture numerical model free of explicit tip tracking

The paper addresses the problem of modeling a fracture propagation in linear elastic porous media driven by injection of non-Newtonian power-law fluid. The model involves the lubrication theory equation expressing the mass conservation of fluid and a nonlocal singular integral relation for the fracture aperture as a functional of the fluid pressure. We assume only the viscosity-dominated case in which the fracture toughness is neglected. This allows considering the fracture as an opened part of a preexisting closed fracture of larger length. The numerical method consists in solving equations of the model over the whole length of the preexisting fracture without distinguishing the tip region of the opened part. The problem is solved via the finite element method. The weak formulation of the solution allows pressure singularity with the required asymptotics near the fracture tip. Comparison of the numerical and the exact self-similar solutions in the case of a constant flow rate, zero fluid leakoff, and zero fracture toughness reveals the accuracy of the approximate solution to at least O(h1/2)O(h1/2), where h is the maximal linear size of the grid cells. As an illustration, we also demonstrate the numerical experiments with periodic fluid injection and variable fluid efficiency.