On the moving boundary conditions for a hydraulic fracture

This paper re-examines the boundary conditions at the moving front of a hydraulic fracture when the fluid front has coalesced with the crack edge. This practically important particular case is treated as the zero fluid lag limit of the general case when the two fronts are distinct. The limiting process shows what becomes of the two boundary conditions on the fluid front, a pressure condition and a Stefan condition, when the lag vanishes. On the one hand, the pressure condition disappears as the net pressure (the difference between the fluid pressure and the magnitude of the far-field stress normal to the fracture) becomes singular. On the other hand, the Stefan condition, which equates the front velocity to the average fluid velocity, transforms into a zero flux boundary condition at the front. As a consequence, the velocity of the coalesced front does not appear explicitly in the boundary conditions. However, the front velocity can still be extracted from the near-tip aperture field by a nonlinear asymptotic analysis. The paper concludes with a description of an algorithm to propagate the combined front, which explicitly uses the known multiscale asymptotics of the fracture aperture.