A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2015. / The propagation of hydraulic fractures with tortuosity is investigated. Tortuosity is the
complicated fracture geometry that results from asperities at the
fluid-rock interface and,
if present, from contact regions. A tortuous hydraulic fracture can either be open without
contact regions or partially open with contact regions. We replace the tortuous hydraulic
fracture by a two-dimensional symmetric model fracture that accounts for tortuosity. A
modified Reynolds
flow law is used to model the tortuosity in the
flow due to surface
roughness at the fracture walls. In order to close the model, the linear and hyperbolic
crack laws which describe the presence of contact regions in a partially open fracture
are used. The Perkins-Kern-Nordgren approximation in which the normal stress at the
crack walls is proportional to the half-width of the symmetric model fracture is used. A
Lie point symmetry analysis of the resulting governing partial differential equations with
their corresponding boundary conditions is applied in order to derive group invariant solutions
for the half-width, volume and length of the fracture. For the linear hydraulic
fracture, three exact analytical solutions are derived. The operating conditions of two of
the exact analytical solutions are identified by two conservation laws. The exact analytical
solutions describe fractures propagating with constant speed, with constant volume and
with
fluid extracted at the fracture entry. The latter solution is the limiting solution of
fluid extraction solutions. During the
fluid extraction process,
fluid
flows in two directions,
one towards the fracture entry and the other towards the fracture tip. It is found
that for
fluid injection the width averaged
fluid velocity increases approximately linearly
along the length of the fracture. This leads to the derivation of approximate analytical
solutions for
fluid injection working conditions. Numerical solutions for
fluid injection
and extraction are computed. The hyperbolic hydraulic fracture is found to admit only
one working condition of
fluid injected at the fracture entry at a constant pressure. The
solution is obtained numerically. Approximate analytical solutions that agree well with
numerical results are derived. The constant pressure solutions of the linear and hyperbolic
hydraulic fracture are compared. While the hyperbolic hydraulic fracture model is
generally considered to be a more realistic model of a partially open fracture, it does not
give information about
fluid extraction. The linear hydraulic fracture model gives various
solutions for di erent working conditions at the fracture entry including
fluid extraction.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/19300 |
| Date | January 2016 |
| Creators | Kgatle, Mankabo Rahab Reshoketswe |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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