Discrete and Three-Parameter Models of Hydraulic Fracture Crack Growth

An infinite elastic medium with a planar crack is considered. The crack is subjected to the pressure of fluid injected at a point on the crack surface. Description of the crack growth is based on the lubrication equation (balance of the injected fluid and the crack volume), the equation for crack opening caused by fluid pressure on the crack surface, the Poiseuille equation related local fluid flux with crack opening and pressure gradient, and the 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 for a constant crack size, jump to a new size defined by the fracture criterion, and filling the new crack configuration by the fluid. First, an isotropic medium with a penny-shaped crack is considered. Dependencies of the crack radius, opening, and pressure distributions on the crack surface on time, fluid viscosity, and fracture toughness of the medium are studied. It is shown that for small fluid viscosity and low injection rates, the pressure distribution can be approximated by a three-parameter model that simplifies substantially the numerical solution. Then, the threeparameter model is applied to the case of heterogeneous media; in this case, the crack shape may be noncircular in the process of hydraulic fracture. Examples of hydraulic fracture crack growth in layered media are presented. Key-Words: Fracture mechanics, hydraulic fracture, penny-shape crack, crack in heterogeneous media.