Modeling Turbulent Hydraulic Fracture Near a Free Surface

Motivated by observations of the subglacial drainage of water, we consider a hydraulic fracture problem in which the crack grows parallel to a free surface, subject to fully turbulent fluid flow. Using a hybrid Chebyshev/series-minimization numerical approach, we solve for the pressure profile, crack opening displacement, and crack growth rate for a crack that begins relatively short but eventually becomes long compared with the distance to the free surface. We plot nondimensionalized results for a variety of different times, corresponding with different fracture lengths, and find substantial differences when free-surface effects are important.