Turbulent hydraulic fracturing described by Prandtl's mixing length
Date
2016-09-19
Authors
Newman, Despina
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Abstract
The problem of turbulent hydraulic fracturing is considered. Despite it being
a known phenomenon, limited mathematical literature exists in this field.
Prandtl’s mixing length model is utilised to describe the eddy viscosity and
a mathematical model is developed for two distinct cases: turbulence where
the kinematic viscosity is sufficiently small to be neglected and the case
where it is not. These models allow for the examination of the fluid’s behaviour
and its effect on the fracture’s evolution through time. The Lie point
symmetries of both cases are obtained, and a wide range of analytical and
numerical solutions are explored. Solutions of physical significance are calculated
and discussed, and approximate solutions are constructed for ease of
fracture estimation. The non-classical symmetries of these equations are also
investigated. It was found that the incorporation of the kinematic viscosity
within the modelling process was important and necessary.
Description
A dissertation submitted to the Faculty of Science, University of
the Witwatersrand, Johannesburg, South Africa, in fulfilment of
the requirements for the degree of Master of Science. 21 March 2016.