Group invariant solutions for a pre-existing fluid-driven fracture in permeable rock

Abstract

The propagation of a two-dimensional fluid-driven pre-existing fluid-filled fracture in permeable rock by the injection of a viscous, incompressible Newtonian fluid is considered. The fluid flow in the fracture is laminar. By the application of lubrication theory, a partial differential equation relating the half-width of the fracture to the fluid pressure and leak-off velocity is obtained. The leak-off velocity is an unspecified function whose form is derived from the similarity solution. The model is closed by the adoption of the PKN formulation in which the fluid pressure is proportional to the fracture half-width. The constant of proportionality depends on the material properties of the rock through its Young modulus and Poisson ratio . The group invariant solutions obtained describe hydraulic fracturing in a permeable rock. Results are also obtained for the case in which the rock is impermeable. Applications in which the rate of fluid injection into the fracture and the pressure at the fracture entry are independent of time are analysed. The limiting solution in which the fracture length and fracture half-width grow exponentially with time is derived. Approximate power law solutions for large values of time for the fracture length and volume are derived. Finally, the case in which the fluid is injected by a pump working at a constant rate is investigated. The results are illustrated by computer generated graphs.