Group Invariant Solution for the Hydraulic fracture of two closely spaced beams

Abstract

The hydraulic fracture of two closely spaced beams, due to an ultra-high pressure fluid which is injected into the space between the beams, is considered. The fluid is Newtonian and the flow is laminar. The objective of this study is to investigate how the beams deform when the fluid under high pressure is injected. The mathematical model consists of the fourth order time dependent Euler-Bernoulli beam equation and a nonlinear diffusion equation derived using lubrication theory and relating the half-width of the fracture to the difference in pressure between the fluid pressure and the normal compressive stress on the beam. The Lie point symmetries of this systemof two partial differential equations are derived and used to reduce the two partial differential equations and associated boundary conditions to a boundary value problemfor a system of ordinary differential equations. The case of finite bending moment at the fracture tip is considered. The asymptotic solution of the ordinary differential equations at the fracture tip are derived. A shooting method is used to derive the numerical solution. It is found that the beams tend to "buckle" during the hydraulic fracturing.