Group Invariant Solution for the Hydraulic fracture of two closely spaced beams
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Date
2019
Authors
khoza, Confide
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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.
Description
A dissertation submitted to the Faculty of Science, University of the
Witwatersrand, Johannesburg, South Africa, in fulfilment of the requirements
for the degree ofMaster of Science.