Hydraulic Fracture

We consider a variation of Griffith's analysis of rupture, one which simulates the process of hydrofracturing, where fluid forced into a crack raises the fluid pressure until the crack begins to grow. Unlike that of Griffith, in this analysis fluid pressure drops as a hydrofracture grows. We find that growth of the fracture depends on the ratio of the compliances of the fluid and the fracture, a non-dimensional parameter called $\alpha_0$ here. The pressure needed to initiate a hydrofracture is found to be the same as that derived by Griffith. Once a fracture initiates, for relatively low values of the model parameter $\alpha_0$ ($\alpha_0 \leq 0.2$) the fracture advances spontaneously to a new radius which depends on the value of $\alpha_0$. For $\alpha_0 \leq 0.2$ further fluid injection is required to increase the fracture radius after spontaneous growth stops. For the case where $\alpha_0 > 0.2$ each increment of fracture growth requires injection of more fluid. For the extreme case where $\alpha_0 = 0$ our results are the same as Griffith's, i.e., a fracture once initiated grows without limit.