This research deals with an analysis of the problem of a fluid-driven fracture propagating through a poroelastic medium. Formulation of such model of an hydraulic fracture is at the cross-road of four classical disciplines of engineering mechanics: lubrication theory, filtration theory, fracture mechanics, and poroelasticity, which includes both elasticity and diffusion. The resulting mathematical model consists of a set of non-linear integro-differential history-dependent equations with singular behaviour at the moving fracture front.
The main contribution of this research is a detailed study of the large-scale 3D diffusion around the fracture and its associated poroelastic effects on fracture propagation. The study hinges on scaling and asymptotic analyses. To understand the behavior of the solution in the tip region, we study a semi-infinite fracture propagating at a constant velocity. We show that, in contrast to the classical case of the Carter's leak-off model (1D diffusion), the tip region of a finite fracture cannot, in general, be modeled by a semi-infinite fracture when 3D diffusion takes place. Moreover, 3D diffusion does not permit separation of the problem into two regions: the tip and the global fracture.
We restrict our study of the fracture propagation to an investigation of two limiting cases: zero viscosity and zero toughness. We show that large-scale 3D diffusion and its associated poroelastic effects can significantly affect the fracture evolution. In particular, we observe a significant increase of the net fracturing fluid pressure compared to the case of 1D diffusion due to the porous medium dilation. Another consequence of 3D diffusion is the possibility of fracture arrest. Indeed, the fracture stops propagating at large time, when the fracturing fluid injection rate is balanced by the leak-off rate at pressure below the critical propagation pressure.