Efficient numerical solution of the hydraulic fracture problem for planar cracks

An infinite isotropic elastic medium with a planar crack is considered. The crack is subjected to pressure of fluid injected inside the crack at a point of its surface. The fluid is viscous newtonian, incompressible, the medium is impermeable. Description of the crack growth is based on the lubrication equation (local balance of the injected fluid and the crack volume), the elasticity equation for crack opening caused by fluid pressure, Poiseulle equation related the fluid flux with crack opening and the pressure gradient, and the criterion of crack propagation of linear fracture mechanics. The crack growth is simulated by a series of discrete steps. Each step consists of three stages: increasing the crack volume by a constant crack size, crack jump to a new size defined by the fracture criterion, and filling the new crack configuration by the fluid presented in the crack. The problem is ill-posed and requires specific methods for numerical solution. The proposed method is based on an appropriate class of approximating functions for fluid pressure distributions on the crack surface and the theory of solution of ill-posed problems. Example of crack growth in a homogeneous and isotropic elastic medium is considered, influence of the fluid viscosity on the process of crack growth is studied.