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Thesis (Ph. D.)--University of Texas at Austin, 2005. / Supervisors: Steven L. Bryant and James W. Jennings Jr. Vita. Includes bibliographical references.

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published_or_final_version / Applied Geosciences / Master / Master of Science

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in fulfilment of the requirements for the degree of Doctor of Philosophy, 2013. / The aim of the thesis is to derive group invariant, exact, approximate analytical and numerical solutions for a two-dimensional laminar, non-Newtonian pre-existing hydraulic fracture propagating in impermeable and permeable elastic media. The fracture is driven by the injection of an incompressible, viscous non-Newtonian fluid of power law rheology in which the fluid viscosity depends on the magnitude of the shear rate and on the power law index n > 0. By the application of lubrication theory, a nonlinear diffusion equation relating the half-width of the fracture to the fluid pressure is obtained. When the interface is permeable the nonlinear diffusion equation has a leak-off velocity sink term. The half-width of the fracture and the net fluid pressure are linearly related through the PKN approximation. A condition, in the form of a first order partial differential equation for the leak-off velocity, is obtained for the nonlinear diffusion equation to have Lie point symmetries. The general form of the leak-off velocity is derived. Using the Lie point symmetries the problem is reduced to a boundary value problem for a second order ordinary differential equation. The leak-off velocity is further specified by assuming that it is proportional to the fracture half-width. Only fluid injection at the fracture entry is considered. This is the case of practical importance in industry. Two exact analytical solutions are derived. In the first solution there is no fluid injection at the fracture entry while in the second solution the fluid velocity averaged over the width of the fracture is constant along the length of the fracture. For other working conditions at the fracture entry the problem is solved numerically by transforming the boundary value problem to a pair of initial value problems. The numerical solution is matched to the asymptotic solution at the fracture tip. Since the fracture is thin the fluid velocity averaged over the width of the fracture is considered. For the two analytical solutions the ratio of the averaged fluid velocity to the velocity of the fracture tip varies linearly along the fracture. For other working conditions the variation is approximately linear. Using this observation approximate analytical solutions are derived for the fracture half-width. The approximate analytical solutions are compared with the numerical solutions and found to be accurate over a wide range of values of the power-law index n and leak-off parameter β. The conservation laws for the nonlinear diffusion equation are investigated. When there is fluid leak-off conservation laws of two kinds are found which depend in which component of the conserved vector the leak-off term is included. For a Newtonian fluid two conservation laws of each kind are found. For a non-Newtonian fluid the second conservation law does not exist. The behaviour of the solutions for shear thinning, Newtonian and shear thickening fluids are qualitatively similar. The characteristic time depends on the properties of the fluid which gives quantitative differences in the solution for shear thinning, Newtonian and shear thickening fluids.

A dissertation submitted to the Faculty of Science,University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. December 2016. / This 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. / MT2017

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This study has resulted in the discovery of an exact method for the theoretical formulation of the effects of intrinsic damping where the attenuation coefficient, a(v), is an arbitrary function of the frequency, v. Absorption-dispersion pairs are computed using numerical Hilbert transformation; approximate analytical expressions that require the selection of arbitrary constants and cutoff frequencies are no longer necessary. For constant Q, the dispersive body wave velocity, p(v), is found to be p(v) = (p(v_N)/(1+(1/2Q H(-v)/v)) where H denotes numerical Hilbert transformation, p(v) is the phase velocity at the frequency v, and p(v_N) is the phase velocity at Nyquist. From (1) it is possible to estimate Q in the time domain by measuring the amount of increase, ΔW, of the wavelet breadth after a traveltime, Q=(2Δ𝛕)/(𝝅ΔW) The inverse problem, i.e., the determination of Q and velocity is also investigated using singular value decomposition (SVD). The sparse matrices encountered in the acquisition of conventional reflection seismology data result in a system of linear equations of the form AX = B, with A the design matrix, X the solution vector, and B the data vector. The system of normal equations is AᵀAX = AᵀB where the least-squares estimate of X = X = V(1/S)UᵀB and the SVD of A is A = USVᵀ. A technique to improve the sparsity pattern prior to decomposition is described. From an application of equation (2) using reference reflections from shallower reflectors, crystalline rocks in South Carolina over the depth interval from about 5 km to 10 km yield values of Qin the range Q = 250 - 300. Non-standard recording geometries ( "Q-spreads") and vibroseis recording procedures are suggested to minimize matrix sparseness and increase the usable frequency bandwidth between zero and Nyquist. The direct detection of body wave dispersion by conventional vibroseis techniques may be useful to distinguish between those crustal volumes that are potentially seismogenic and those that are not. Such differences may be due to variations in fracture density and therefore water content in the crust. / Ph. D.