Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock

Abstract

The propagation of a two-dimensional fluid-driven fracture in impermeable rock is considered. The fluid flow in the fracture is laminar. By applying lubrication theory a partial differential equation relating the half-width of the fracture to the fluid pressure is derived. To close the model the PKN formulation is adopted in which the fluid pressure is proportional to the half-width of the fracture. By considering a linear combination of the Lie point symmetries of the resulting non-linear diffusion equation the boundary value problem is expressed in a form appropriate for a similarity solution. The boundary value problem is reformulated as two initial value problems which are readily solved numerically. The similarity solution describes a preexisting fracture since both the total volume and length of the fracture are initially finite and non-zero. Applications in which the rate of fluid injection into the fracture and the pressure at the fracture entry are independent of time are considered. (2000). 74F10, 76D08, 74R15. Mathematics Subject Classification

Mathematical Model of a Hyperbolic Hydraulic Fracture with Tortuosity

“…Hydraulic fracturing has many applications. In mining, high pressure water is used to break rocks instead of explosives which leave small rock fragments in the air that could damage the lungs of miners [1].…”

“…We will only comment briefly on the application of Lie point symmetries to hydraulic fracturing, because that approach will be used in this paper. Lie point symmetry analysis was first applied to hydraulic fracturing by Fitt, Mason and Moss who considered the propagation of a fracture in impermeable rock [1]. Fareo and Mason analyzed the propagation of a hydraulic fracture in permeable rock with fluid leak-off into the rock mass and investigated the effect of leak-off on the rate of propagation of a fracture [10].…”