Discrete model of hydraulic fracture crack propagation in homogeneous isotropic elastic media

An infinite homogeneous and isotropic elastic medium with a penny shape crack is considered. The crack is subjected to the pressure of fluid injected in the crack center. Description of the crack growth is based on the lubrication equation (balance of the injected fluid and the crack volume), equation for crack opening caused by fluid pressure on the crack surface, the Poiseullie equation related local fluid flux with the crack opening and pressure gradient, and classical criterion of crack propagation of linear fracture mechanics. The crack growth is simulated by a discrete process consisting of three basic stages: increasing the crack volume by a constant crack size, crack jump to a new size defined by the fracture criterion, and filling the appeared crack volume by the fluid. It is shown that the model results a reasonable dependence of the crack size on the time as well as the pressure distribution of fluid on the crack surface. Comparisons with the solutions of hydraulic fracture problems existing in the literature are presented.