A thesis submitted to the Faculty of Science, University of the
Witwatersrand, Johannesburg, South Africa, in fulfilment of the requirements for
the degree of Doctor of Philosophy, 2013. / The aim of the thesis is to derive group invariant, exact, approximate analytical and numerical
solutions for a two-dimensional laminar, non-Newtonian pre-existing hydraulic fracture propagating
in impermeable and permeable elastic media. The fracture is driven by the injection
of an incompressible, viscous non-Newtonian fluid of power law rheology in which the fluid
viscosity depends on the magnitude of the shear rate and on the power law index n > 0. By
the application of lubrication theory, a nonlinear diffusion equation relating the half-width of
the fracture to the fluid pressure is obtained.
When the interface is permeable the nonlinear diffusion equation has a leak-off velocity
sink term. The half-width of the fracture and the net fluid pressure are linearly related through
the PKN approximation. A condition, in the form of a first order partial differential equation
for the leak-off velocity, is obtained for the nonlinear diffusion equation to have Lie point symmetries.
The general form of the leak-off velocity is derived. Using the Lie point symmetries
the problem is reduced to a boundary value problem for a second order ordinary differential
equation. The leak-off velocity is further specified by assuming that it is proportional to the
fracture half-width. Only fluid injection at the fracture entry is considered. This is the case of
practical importance in industry.
Two exact analytical solutions are derived. In the first solution there is no fluid injection
at the fracture entry while in the second solution the fluid velocity averaged over the width of
the fracture is constant along the length of the fracture. For other working conditions at the
fracture entry the problem is solved numerically by transforming the boundary value problem
to a pair of initial value problems. The numerical solution is matched to the asymptotic solution
at the fracture tip. Since the fracture is thin the fluid velocity averaged over the width
of the fracture is considered. For the two analytical solutions the ratio of the averaged fluid
velocity to the velocity of the fracture tip varies linearly along the fracture. For other working
conditions the variation is approximately linear. Using this observation approximate analytical
solutions are derived for the fracture half-width. The approximate analytical solutions are
compared with the numerical solutions and found to be accurate over a wide range of values
of the power-law index n and leak-off parameter β.
The conservation laws for the nonlinear diffusion equation are investigated. When there
is fluid leak-off conservation laws of two kinds are found which depend in which component
of the conserved vector the leak-off term is included. For a Newtonian fluid two conservation
laws of each kind are found. For a non-Newtonian fluid the second conservation law does
not exist. The behaviour of the solutions for shear thinning, Newtonian and shear thickening
fluids are qualitatively similar. The characteristic time depends on the properties of the fluid
which gives quantitative differences in the solution for shear thinning, Newtonian and shear
thickening fluids.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/12707 |
| Date | 02 May 2013 |
| Creators | Fareo, Adewunmi Gideon |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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