A dissertation submitted to the Faculty of Science,University of the
Witwatersrand, in fulfilment of the requirements for the
degree of Master of Science.
December 2016. / This research is concerned with the analysis of a two-dimensional Newtonian
fluid-driven
fracture in a permeable rock. The
fluid
flow in the fracture is laminar and the fracture
is driven by the injection of a Newtonian
fluid into it. Most of the problems in litera-
ture involving
fluid
flow in permeable rock formation have been modeled with the use
of Darcy's law. It is however known that Darcy's model breaks down for
flows involv-
ing high
fluid velocity, such as the
flow in a porous rock formation during hydraulic
fracturing. The Forchheimer
flow model is used to describe the non-Darcy
fluid
flow
in the porous medium. The objective of this study is to investigate the problem of a
fluid-driven fracture in a porous medium such that the
flow in the porous medium is
non-Darcy. Lubrication theory is applied to the system of partial di erential equations
since the fracture that is considered is thin and its width slowly varies along its length.
For this same reason, Perkins-Kern-Nordgren approximation is adopted. The theory of
Lie group analysis of differential equations is used to solve the nonlinear coupled sys-
tem of partial differential equations to obtain group invariant solutions for the fracture
half-width, leak-o depth and length of the fracture. The strength of
fluid leak-off at
the fracture wall is classi ed into three forms, namely, weak, strong and moderate. A
group invariant solution of the traveling wave form is obtained and an exact solution for
the case in which there is weak
fluid leak-off at the interface is found. A dimensionless
parameter, F0, termed the Forchheimer number was obtained and investigated. Nu-
merical results are obtained for both the case of Darcy and non-Darcy
flow. Computer
generated graphs are used to illustrate the analytical and numerical results. / MT2017
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/22740 |
| Date | January 2017 |
| Creators | Nchabeleng, Mathibele Willy |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | Online resource (x, 109 leaves), application/pdf |
Page generated in 0.0021 seconds