Propagation of a fluid-driven fracture with fracture length dependent apparent toughness

We investigate the growth of a hydraulic fracture assuming a power-law dependence of material toughness with fracture length for plane strain and radial geometries. Such a toughness fracture length dependence models in a simple manner a toughening mechanism for rocks. We develop an efficient numerical method for the hydraulic fracture growth problem combining Gauss-Chebyshev quadrature and Barycentric Lagrange interpolation techniques. Scaling and numerical results demonstrate that the transition from the viscosity to the toughness dominated regime occurs earlier. The toughness dominated regime always governs growth at large time for both geometries. In all cases, larger net pressure and shorter length are obtained. The solution is very well approximated by the existing constant toughness solutions using the instantaneous value of toughness. If the apparent fracture toughness saturates beyond a given length scale, the solution transitions back to the constant toughness solutions.