Hydraulic fracture crack propagation in an elastic medium with varying fracture toughness

Hydraulic fracture crack propagation in an isotropic elastic medium with varying fracture toughness is considered. The crack is subjected to the pressure of fluid injected at a point on the crack surface. The fluid is viscous Newtonian and incompressible, the medium is impermeable. The analysis of crack growth is based on the three-parameter model of pressure distribution on the crack surface. The model allows one to satisfy the condition at the point of fluid injection, the balance equation of the volume of injected fluid and the crack volume, and fracture criterion at the crack edge. The analysis of evolution of the crack boundary is based on an original method of fast numerical solution of crack problems. In this method, Gaussian approximating functions are used for discretization of the problem, and fast Fourier transform technique is applied for solution of the discretized equation. The method allows constructing the crack boundary at discrete time moments for media with varying fracture toughness and time dependent positive injection rate. Examples of hydraulic fracture crack propagation in the medium that consists of two half-spaces with different fracture toughnesses and a layer of the material with another fracture toughness in a homogeneous elastic medium are considered. Evolution of the crack boundaries in the process of fluid injection, time dependence of pressure distributions and crack openings are presented in graphic forms.