3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item Modelling of gas recovery from South African shale reservoirs(2019-09-09) Qwabe, Thembinkosi SabeloMathematical models for a triple-porosity continuum describing the distribution of pressure-the driving force for gas flow during gas production-were developed on the basis of the continuity equation from first principles, focusing on the South African shales of the Karoo Basin. The triple-porosity continuum incorporated the matrix, the natural fracture network, and the hydraulic fracture network; the matrix and the natural fracture network systems were considered two dimensional, while the hydraulic fracture system was considered one dimensional. The three developed mathematical models are in the form of the general diffusion or heat equation, thus the numerical Finite difference method (FDM) was employed in solving the triple-porosity model. MATLAB software was used to develop and solve the FDM simulation or algorithm of each system of the triple-porosity model; results for each system are presented and discussed in chapter 6 of the research report. The results generated agree with models previously developed and validated with their respective field data, substantiating the reliability of the model developed in this research report. Currently the South Africa’s Barnett shale does not have field data, hence the field data from literature was used to validate the developed model. The model is thus a guide onto describing the gas flow behaviour for the South African shale gas reservoirs, and not specifically for the Barnett shale.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 Modelling of gas recovery from unconventional reservoir shale gas(2019) Khari, AndileIt is believed that petroleum gas is stored as free gas in natural fractures, free gas in pores, adsorbed gas and dissolved gas in kerogen bulk in the shale. Hydraulic Fracking is used to promote free gas flow to production well, but adsorbed and dissolved gas is not recovered and usually ignored. This work looks into modifying existing gas flow model, by including mechanisms that can promote and contribute to shale gas production recovery during reservoir depletion. The developed model includes non-Darcy flow which is suitable for high-velocity gas flow. Model equation was simulated with the help of MATLAB to solve the partial derivative equation to achieve a shale gas production a 3D profile of pressure vs time vs distance. The model results were seen behaving similarly to available developed models, whereby pressure is initially increasing and then decrease overtime. The initial increase in pressure is due to the free gas available in natural fractures and micro-fracture in the matrix and is produced first causing pressure to increase. During production, overtime free gas in these natural fractures and micro-fracture gets depleted, causing pressure to decrease and approach critical desorption pressure over time. The free gas in the matrix nanopores feed these depleted fracture networks, the kerogen nanopores is in turn fed by adsorbed and dissolved gas on kerogen inside the nanopores surface which take place at later stage of production. The developed model shows similar gas recovery production behaviour with the laboratory results, which proves that the proposed model can be used to predict the production profileItem Modelling of gas recovery from South African shale reservoirs (focusing on the KWV-1 bore hole in the Eastern Cape Province)(2018) Makoloane, NkhabuThe main aim of the study was to develop mathematical flow model of the shale gas at the Karoo Basin of South Africa (SA). The model development incorporates three systems (phases) to form a triple continuum flow model, the phases include matrix (m), natural (NF) and hydraulic fracture (HF). The model was developed from the continuity equation, and the general equations were formed. (0.05������ ���� = 3.90087 × 10−15 ��2���� ����2 + 3.90087 × 10−15 ��2���� ����2 − 1.95043 × 10−16(20 × 106 − ������), 0.01 �������� ���� = 2.00 × 10−15(20 × 106 − ������) − 2.00 × 10−9(20 × 106 − ������) + �� ���� [7.80 × 10−5 �������� ���� ] + �� ���� [7.80 × 10−5 �������� ���� ] �� ���� [0.1248269 �������� ���� ] + 0.1248269(20 × 106 − ������)− 4.98 × 10−4 = �������� ���� The model was solved using numerical method technique known as Finite Difference Method (FDM). For each phase a computer program MATLAB was used to plot the pressure gradient. Hydraulic pressure gradient fractures propagate between the distance of 100m and 500m. The model was verified using the data of Barnett Shale. Sensitivity analysis was also performed on the hydraulic permeability, drainage radius and the initial pressure of the reservoir.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 tortuosityItem 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 Turbulent hydraulic fracturing described by Prandtl's mixing length(2016-09-19) Newman, DespinaThe 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.Item The Karoo hydraulic fracturing debate : accounting for future generations.(2012-07-12) Yale-Kearney, Robinn Y.The temporal complexities of anthropogenic Global Climate Change (GCC) force us to extend our moral deliberations beyond what appear to be straightforward, contemporary issues to include the interests of future generations. The Karoo hydraulic fracturing debate is a case in point. The ethical debate thus far has focused on the present-day environmental aspects of Shell’s limited exploratory drilling proposal using hydraulic fracturing technology; but the shale-gas reserves that are believed to underlie the Karoo could assist in mitigating South Africa’s significant carbon emissions, the main contributor to anthropogenic GCC. Thus, I argue that the actual ethical debate is whether to allow gas exploration over the Karoo or to disallow the entire possibility of exploiting any gas reserves that may have been found. A consequentialist weighing up of the respective potential harms to all of the morally-considerable interests involved, including future generations, makes clear that not only is allowing exploration of the Karoo the morally correct decision, but it is ethically obligatory to do so.