3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item Analysis of fracture growth models for hydraulic fracturing of shale gas deposits in the karoo, South Africa(2019) Tumureebire, Patrick AtwinePrediction of fracture dimensions during propagation of a hydraulically induced fracture for well stimulation is essential for the design of a stimulation treatment. This study seeks to better understand the mechanisms of hydraulic fracturing through computational modelling of the fracture growth up to a specific time in MATLAB. The computations were based on three existing theories of fracture propagation: the Khristianovitch, Geertsma and de Klerk (KGD) model, Perkins and Kern model (PKN) and Pseudo 3D model. Owing to the absence of raw data for the Whitehill formation in the Karoo, analogous shale rock from the United States of America was used as a basis for the study. The MATLAB computations were thus performed based on the following rock properties: Shear modulus = 1.466 x 105Psi; Drained Poisson’s ratio = 0.2; Fluid Viscosity = 1cp; Pumping rate = 62.5bbl/min; In-situ stress = 3200Psi; Wellbore radius = 0.2ft; Passed time = 0.3min. This report documents the differences in height, length, width, and pressure, predicted by the 2D and 3D models for the same set of input parameters. It is shown that the growth of the fracture for the 2D models yield much shorter lengths than the 3D model. It is also seen that the wellbore pressure predicted by the PKN model, in contrast to the KGD model, increases to 2.564 x 105 psi. as the fracture length increases. The pressure predicted by the P3D model increases to a peak of 1.411 x 10 6 psi at t = 0.24sec before declining to a final 7.709 x 105 psi. Though the report proposed an understanding of the mechanisms of hydraulic fracturing in the Karoo, and even obtained solutions, it is limited to simulation models without application to field data.Item The propagation of a linear hydraulic fracture with tortuosity and fluid leak-off at the fluid-rock interface(2018) Mabasa, RishileIn this work, the propagation of a pre-existing hydraulic fracture in permeable rock is investigated. Apartiallyopentortuousfracturewithleak-offisreplacedbyatwo-dimensional symmetric model fracture with a modified Reynolds’ flow law to account for the effect of asperities on the fluid flow. The model is closed by considering a linear crack law, which considers the presence of touching asperities, and a Perkins-Kern and Nordgren (PKN) approximation, which relates the half-width of the model fracture to the normal stress at the fracture walls. The result is a nonlinear diffusion equation that accounts for leak-off atthefluid-rockinterfaceasaresultoftherock’spermeability. The leak-off velocity is not specified a priori and its functional form is determined by calculating Lie point symmetries of the governing non-linear diffusion equation for a model fracture which leads to a group invariant solution of the half-width of the fracture and the leak-off velocity respectively. We consider different forms of the Lie point symmetry of the governing equations by taking some of the constants in the generator to be zero. This leads to three cases of the group invariant solution, namely the general case, thetravelingwavesolutionandtheexponentialsolution. TwoexactanalyticalsolutionsareobtainedasaresultofassociationofaLiepointsymmetry with a conserved vector. However, the constant volume working condition is not furtherinvestigatedastherequirementsforittoholdarenotsufficientforthemodelconsidered. Other operating conditions at the fracture entry are also obtained by analysing the different properties of the partially open fracture. Numerical solutions of the halfwidthandtheleak-offdeptharecomputed. Thepropagationofalinearhydraulicfracture withtortuosityforthreecasesoftheLiepointsymmetryarefullyanalysedbynumerically solvingforthehalf-widthandthelengthofthefracture. Thederivationandanalysisofthewidthaveragedfluidvelocityleadstothederivation of an approximate analytical solution for the half-width of the model fracture which will then be compared to the numerical solution obtained. The approximate solution could prove useful when analytical solutions are unattainable or when numerical solutions are difficult to compute. The effects of leak-off and no leak-off in tortuous hydraulic fracture are compared to gain insight on the effect porosity has on the characteristics of the model fracture with tortuosity