The near-tip region of a fluid-driven fracture propagating in a permeable elastic solid

This paper is concerned with an analysis of the near-tip region of a fluid-driven fracture propagating in a permeable saturated rock. It focuses on the calculation of the pore fluid pressure in the tip cavity, the region corresponding to the lag between the front of the fracturing fluid and the fracture tip. In contrast to impermeable rocks where the tip cavity can be considered to be at zero pressure, the fluid pressure in the tip cavity is here unknown and not uniform as exchange of pore fluid between the cavity and the porous medium and flow of pore fluid within the cavity is taking place. Solution of the fluid pressure in the tip region requires therefore simultaneous consideration of fracture mechanics (for the aperture of the tip cavity), diffusion theory for the movement of fluid within the porous medium, and viscous flow along the crack. Construction of such a solution within the framework of some simplifying assumptions is the main objective of this paper. It is shown that the problem depends, in general, upon two numbers with the meaning of a permeability and a propagation velocity. For the asymptotic case of large propagation speed, these two numbers merge into a single parameter, while the solution becomes independent of the propagation velocity in the limit of small velocity. The particular case of large velocity is solved analytically, while both the general and the small velocity cases are computed numerically but with different techniques. The paper concludes with a comprehensive analysis of numerical results.