3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item Hydraulic fracture with Darcy and non-Darcy flow in a porous medium(2017) Nchabeleng, Mathibele WillyThis research is concerned with the analysis of a two-dimensional Newtonian fluid-driven fracture in a permeable rock. The fluid flow in the fracture is laminar and the fracture is driven by the injection of a Newtonian fluid into it. Most of the problems in litera- ture involving fluid flow in permeable rock formation have been modeled with the use of Darcy's law. It is however known that Darcy's model breaks down for flows involv- ing high fluid velocity, such as the flow in a porous rock formation during hydraulic fracturing. The Forchheimer flow model is used to describe the non-Darcy fluid flow in the porous medium. The objective of this study is to investigate the problem of a fluid-driven fracture in a porous medium such that the flow in the porous medium is non-Darcy. Lubrication theory is applied to the system of partial di erential equations since the fracture that is considered is thin and its width slowly varies along its length. For this same reason, Perkins-Kern-Nordgren approximation is adopted. The theory of Lie group analysis of differential equations is used to solve the nonlinear coupled sys- tem of partial differential equations to obtain group invariant solutions for the fracture half-width, leak-o depth and length of the fracture. The strength of fluid leak-off at the fracture wall is classi ed into three forms, namely, weak, strong and moderate. A group invariant solution of the traveling wave form is obtained and an exact solution for the case in which there is weak fluid leak-off at the interface is found. A dimensionless parameter, F0, termed the Forchheimer number was obtained and investigated. Nu- merical results are obtained for both the case of Darcy and non-Darcy flow. Computer generated graphs are used to illustrate the analytical and numerical results.Item Shock wave interaction with porous compressible foams(2016) Atkins, Mark DTwo foams, a polyether (density 32.5 kg/m3) and a polyester (density 38 kg/m3) foam were tested in a shock tube to analyze the interaction of a normal shock wave and a compressible porous material. The foam specimens were placed in the shock tube test section! the foam being bounded by two steel walis, two glass windows and a solid back plate. The compression chamber of the shock tube was pressurized and the diaphragm separating the compression chamber and the expansion chamber was ruptured, thus producing a normal shock wave which travels down the shock tube and strikes the foam. Piezoelectric pressure transducers 'vvera used to record the pressure before, alongside and behind the foam. A complete set of schlieren photographs, recording the interaction of the incident shock wave and the porous material were taken for each foam. A method ,tortracking the path of particles of foam (path photographs) was developed. Combining the information obtained from the pressure records, schlieren photographs and path photographs a complete picture of the shock wave foam interaction was developed. All the gas waves were identified and analyzed, A foam wave (velocity 90 m/s) travelling through the skeleton of the material was discovered. A physical model was developed to explain the high pressure recorded behind the foam. This model is based upon the foam being compressed and forming an almost solid piston, thus forcing the trapped gas into a diminishing volume and creating a high pressure behind the foam. The theoretical analyses of Monti (30), Gel'fand (20) and IBvozdeva (22) were analyz.ed and compared. The general finding was that for the range of incident mach numbers 1.~i1 to 1.46 Monti's analysis under predicts the reflected Mach number by 3 % and Gel'fand's analysis over predicts the reflected Mach number by 6 %. The coefficient of pressure increase (the ratio of the maximum pressure recorded behind the foam to the equivalent pressure recorded during ~he reflection of a shock wave from a solid wall) as predicted iJy Gvoz.deva's ane.lysisfor the polyether foam lies wjthin the scatter of the experimental results. However for the polyester foam Gvozdeva's analysis under predicts the coefficient of pressure increase by 15%.