Turbulent hydraulic fracturing described by Prandtl's mixing length - Newman, Despina (My University, 2016)
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.
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.
Keywords: Turbulence; Hydraulic fracturing; Symmetry (Mathematics)