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.