ETD Collection

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    Propagation of a hydraulic fracture with tortuosity : linear and hyperbolic crack laws
    (2016) Kgatle, Mankabo Rahab Reshoketswe
    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.
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    Group theoretical and compatibility approaches to some nonlinear PDEs arising in the study of non-Newtonian fluid mechanics
    (2015-05-06) Aziz, Taha
    This thesis is primarily concerned with the analysis of some nonlinear problems arising in the study of non-Newtonian fluid mechanics by employing group theoretic and compatibility approaches. It is well known now that many manufacturing processes in industry involve non-Newtonian fluids. Examples of such fluids include polymer solutions and melts, paints, blood, ketchup, pharmaceuticals and many others. The mathematical and physical behaviour of non-Newtonian fluids is intermediate between that of purely viscous fluid and that of a perfectly elastic solid. These fluids cannot be described by the classical Navier–Stokes theory. Striking manifestations of non-Newtonian fluids have been observed experimentally such as the Weissenberg or rod-climbing effect, extrudate swell or vortex growth in a contraction flow. Due to diverse physical structure of non-Newtonian fluids, many constitutive equations have been developed mainly under the classification of differential type, rate type and integral type. Amongst the many non-Newtonian fluid models, the fluids of differential type have received much attention in order to explain features such as normal stress effects, rod climbing, shear thinning and shear thickening. Most physical phenomena dealing with the study of non-Newtonian fluids are modelled in the form of nonlinear partial differential equations (PDEs). It is easier to solve a linear problem due to its extensive study as well due to
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    Group invariant solutions for the unsteady magnetohydrodynamic flow of a fourth grade fluid in a porous medium
    (2014-07-18) Carrim, Abdul Hamid
    The e ects of non-Newtonian uids are investigated by means of two appropri- ate models studying a third and fourth grade uid respectively. The geometry of both these models is described by the unsteady unidirectional ow of an in-compressible uid over an in nite at rigid plate within a porous medium. The uid is electrically conducting in the presence of a uniform applied magnetic eld that occurs in the normal direction to the ow. The classical Lie symmetry approach is undertaken in order to construct group invariant solutions to the governing higher-order non-linear partial dif-ferential equations. A three-dimensional Lie algebra is acquired for both uid ow problems. In each case, the invariant solution corresponding to the non-travelling wave type is considered to be the most signi cant solution for the uid ow model under investigation since it directly incorporates the magnetic eld term. A numerical solution to the governing partial di erential equation is produced and a comparison is made with the results obtained from the analytical ap-proach. Finally, a graphical analysis is carried out with the purpose of observing the e ects of the emerging physical parameters. In particular, a study is carried out to examine the in uences of the magnetic eld parameter and the non-Newtonian fluid parameters.