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Gas-driven fracture propagation

Description: A one-dimensional gas-flow drives a wedge-shaped fracture into a linearly elastic, impermeable half-space which is in uniform compression, sigma/sub infinity/, at infinity. Under a constant driving pressure, p/sub 0/, the fracture/flow system accelerates through a sequence of three self-similar asymptotic regimes (laminar, turbulent, inviscid) in which the fracture grows like an elementary function of time (exponential, near-unity power, and linear; respectively). In each regime, the transport equations are reducible under a separation-of-variables transformation. The integro-differential equations which describe the viscous flows are solved by iterative shooting-methods using expansion techniques to accommodate a zero-pressure singularity at the leading edge of the flow. These numerical results are complemented by an asymptotic analysis for large pressure ratio (N = p/sub 0//sigma/sub infinity/ ..-->.. infinity) which exploits the disparity between the fracture-length and penetration-length of the flow. The considered prototypic problem has geologic applications: containment evaluation of underground nuclear tests, explosive stimulation of oil and gas wells, and explosive permeability-enhancement prior to in-situ combustion of coal or oil-shale.
Date: October 1, 1981
Creator: Nilson, R.H.
Partner: UNT Libraries Government Documents Department

Transient gas or liquid flow along a preexisting or hydraulically-induced fracture in a permeable medium

Description: Similarity solutions are derived for the trasient two-dimensional flow of a gas or liquid along an isolated fracture in a permeable medium. The driving pressure at the fracture inlet is constant, and the confining stress is uniform. Two different cases are considered, pre-existing fractures with uniform aperture as well as hydraulic fractures with a variable aperture proportional to the local overpressure (fluid pressure less confining stress). The evolution of the pressure distribution is described by a set of four asymptotic solutions, each having a self-similar form. At early times the flow in the fracture is turbulent, and Darcian seepage losses into the porous surroundings are negligible. At late times the flow in the fracture is laminar, and seepage losses become a dominant consideration. At intermediate times there are two alternative asymptotes, depending upon physical parameters. The mathematical model also describe the flow along a fracture which is fulled with high-permeability porous material as well as the flow in an assemblage of porous blocks. 19 refs., 10 figs.
Date: May 1, 1985
Creator: Nilson, R.H. & Morrison, F.A. Jr.
Partner: UNT Libraries Government Documents Department

Transient, radial temperature distribution in a porous medium during fluid injection

Description: Analytical and numerical solutions are presented for the transient, radial temperature distribution in a porous medium which is subjected to a constant-rate injection of an incompressible fluid from a wellbore. The formulation includes energy transfer by conduction and convection, and the Danckwerts boundary condition is applied at the finite-radius wellbore. At late times, the numerical solutions approach a self-similar form which can be described in terms of the incomplete Gamma function. In typical petroleum and geothermal applications, convergence to the asymptotic similarity solutions occurs on a time scale of roughly one hour. The results are generally applicable to a broad range of convection-diffusion phenomena which are best described in radial coordinates.
Date: January 1, 1982
Creator: Dunn, J.C. & Nilson, R.H.
Partner: UNT Libraries Government Documents Department

Flow of cavity gas along fractures, cable bundles, and grouted cables

Description: Analytical and numerical solutions are presented for gas flow along isolated fractures in permeable media. Two different cases are considered: preexisting fractures with a uniform aperture, as well as hydraulically-driven fractures with a variable aperture which is proportional to the local overpressure (local fluid pressure less confining stress). Results for these two geometries are compared with one another and with results for wedge-shaped and penny-shaped fracture geometries. Application to underground nuclear testing is illustrated by a number of examples including: flow along shock-induced and hydraulically-induced fractures, seepage along single electrical cables surrounded by grout, and flow along a bundle of electrical cables surrounded by granular stemming material. 15 refs., 13 figs.
Date: January 1, 1985
Creator: Nilson, R.H. & Morrison, F.A. Jr.
Partner: UNT Libraries Government Documents Department

Contaminant transport during atmospheric pumping of a nuclear chimney: Progress report

Description: Cyclical variations in barometric pressure cause an oscillatory up-and-down motion of gases within the chimney produced by an underground nuclear test. Analytical and experimental modeling of this atmospheric pumping mechanism has been undertaken to better understand and to quantify the associated rates of cavity gas migration toward the earth's surface and the probable rate of release to the atmosphere. Three different types of models are being investigated: (1) homogeneous porous medium; (2) fractured medium with impermeable matrix blocks; and (3) double-porosity media consisting of fracture networks among porous matrix blocks. A primary purpose is to understand how the oscillatory character of the atmospheric pumping process might significantly enhance the contaminant transport in any or all of the three classes of media. This preliminary report describes some of the analytical, numerical, and experimental work which have been completed.
Date: September 1, 1986
Creator: Nilson, R.H. & Peterson, E.W.
Partner: UNT Libraries Government Documents Department

Band spreading in two-dimensional microchannel turns for electrophoretic or electroosmotic species transport

Description: Analytical and numerical methods are employed to investigate species transport by electrophoretic or electroosmotic motion in the curved geometry of a two-dimensional turn. Closed-form analytical solutions describing the turn-induced diffusive and dispersive spreading of a species band are presented for both the low and high Peclet number limits. The authors find that the spreading due to dispersion is proportional to the product of the turn included angle and the Peclet number at low Peclet numbers. It is proportional to the square of the included angle and independent of the Peclet number when the Peclet number is large. A composite solution applicable to all Peclet numbers is constructed from these limiting behaviors. Numerical solutions for species transport in a turn are also presented over a wide range of the included angle and the mean turn radius. Based on comparisons between the analytical and numerical results, the authors find that the analytical solutions provide very good estimates of both dispersive and diffusive spreading provided that the mean turn radius exceeds the channel width. These new solutions also agree well with data from a previous study. Optimum conditions minimizing total spreading in a turn are presented and discussed.
Date: March 1, 2000
Creator: Griffiths, S. K. & Nilson, R. H.
Partner: UNT Libraries Government Documents Department

Electroosmotic fluid motion and late-time solute transport at non-negligible zeta potentials

Description: Analytical and numerical methods are employed to determine the electric potential, fluid velocity and late-time solute distribution for electroosmotic flow in a tube and channel when the zeta potential is not small. The electric potential and fluid velocity are in general obtained by numerical means. In addition, new analytical solutions are presented for the velocity in a tube and channel in the extremes of large and small Debye layer thickness. The electroosmotic fluid velocity is used to analyze late-time transport of a neutral non-reacting solute. Zeroth and first-order solutions describing axial variation of the solute concentration are determined analytically. The resulting expressions contain eigenvalues representing the dispersion and skewness of the axial concentration profiles. These eigenvalues and the functions describing transverse variation of the concentration field are determined numerically using a shooting technique. Results are presented for both tube and channel geometries over a wide range of the normalized Debye layer thickness and zeta potential. Simple analytical approximations to the eigenvalues are also provided for the limiting cases of large and small values of the Debye layer thickness. The methodology developed here for electroosmotic flow is also applied to the Taylor problem of late-time transport and dispersion in pressure-driven flows.
Date: December 1, 1999
Creator: Griffiths, S. K. & Nilson, R. H.
Partner: UNT Libraries Government Documents Department

Hydrodynamic dispersion of a neutral non-reacting solute in electroosmotic flow

Description: Analytical methods are employed to determine the axial dispersion of a neutral non-reacting solute in an incompressible electroosmotic flow. In contrast to previous approaches, the dispersion is obtained here by solving the time-dependent diffusion-advection equation in transformed spatial and temporal coordinates to obtain the two-dimensional late-time concentration field. The coefficient of dispersion arises as a separation eigenvalue, and its value is obtained as a necessary condition for satisfying all of the required boundary conditions. Solutions based on the Debye-Huckel approximation are presented for both a circular tube and a channel of infinite width. These results recover the well-known solutions for dispersion in pressure-driven flows when the Debye length is very large. In this limit, the axial dispersion is proportional to the square of the Peclet number based on the characteristic transverse dimension of the tube or channel. In the tilt of very small Debye lengths, the authors find that the dispersion varies as the square of the Peclet number based on the Debye length. Simple approximations to the coefficient of dispersion as a function of the Debye length and Peclet number are also presented.
Date: June 1, 1999
Creator: Griffiths, S. K. & Nilson, R. H.
Partner: UNT Libraries Government Documents Department

Condensation pressures in small pores: An analytical model based on density functional theory

Description: Adsorption and condensation are critical to many applications of porous materials including filtration, separation, and the storage of gases. Integral methods are used to derive an analytical expression describing fluid condensation pressures in slit pores bounded by parallel plane walls. To obtain this result, the governing equations of Density Functional Theory (DFT) are integrated across the pore width assuming that fluid densities within adsorbed layers are spatially uniform. The thickness, density, and energy of these layers are expressed as composite functions constructed from asymptotic limits applicable to small and large pores. By equating the total energy of the adsorbed layers to that of a liquid-full pore, the authors arrive at a closed-form expression for the condensation pressure in terms of the pore size, surface tension, and Lennard-Jones parameters of the adsorbent and adsorbate molecules. The resulting equation reduces to the Kelvin equation in the large-pore limit. It further reproduces the condensation pressures computed by means of the full DFT equations for all pore sizes in which phase transitions are abrupt. Finally, in the limit of extremely small pores, for which phase transitions may be smooth and continuous, this simple analytical expression provides a good approximation to the apparent condensation pressure indicated by the steepest portion of the adsorption isotherm computed via DFT.
Date: February 1, 1999
Creator: Nilson, R. H. & Griffiths, S. K.
Partner: UNT Libraries Government Documents Department

A locally analytic density functional theory describing adsorption and condensation in microporous materials

Description: The fluid density distribution within microscopic pores is determined by solving integral equations relating to the local chemical potential to the Van der Waals attractions and hard sphere repulsions of surrounding material. To avoid resolving the density distribution on sub-molecular scales, the governing equations are averaged over zones of molecular size using analytic functions to represent local density variations within each zone. These local density profiles range form singularities to uniform distributions depending on the local variation of the potential field. Sample calculations indicate that this integral approach yields results in very good agreement with those based on traditional density functional theory (DFT), while reducing computing times by factors of 10{sup 3} to 10{sup 4} for one- dimensional geometries.
Date: February 1, 1997
Creator: Nilson, R.H. & Griffiths, S.K.
Partner: UNT Libraries Government Documents Department

In situ bioremediation: A network model of diffusion and flow in granular porous media

Description: In situ bioremediation is a potentially expedient, permanent and cost- effective means of waste site decontamination. However, permeability reductions due to the transport and deposition of native fines or due to excessive microorganism populations may severely inhibit the injection of supplemental oxygen in the contamination zone. To help understand this phenomenon, we have developed a micro-mechanical network model of flow, diffusion and particle transport in granular porous materials. The model differs from most similar models in that the network is defined by particle positions in a numerically-generated particle array. The model is thus widely applicable to computing effective transport properties for both ordered and realistic random porous media. A laboratory-scale apparatus to measure permeability reductions has also been designed, built and tested.
Date: April 1997
Creator: Griffiths, S. K.; Nilson, R. H. & Bradshaw, R. W.
Partner: UNT Libraries Government Documents Department

Optimum conditions for composites fiber coating by chemical vapor infiltration

Description: A combined analytical and numerical method is employed to optimize process conditions for composites fiber coating by chemical vapor infiltration (CVI). For a first-order deposition reaction, the optimum pressure yielding the maximum deposition rate at a preform center is obtained in closed form and is found to depend only on the activation energy of the deposition reaction, the characteristic pore size, and properties of the reactant and product gases. It does not depend on the preform specific surface area, effective diffusivity or preform thickness, nor on the gas-phase yield of the deposition reaction. Further, this optimum pressure is unaltered by the additional constraint of a prescribed deposition uniformity. Optimum temperatures are obtained using an analytical expression for the optimum value along with numerical solutions to the governing transport equations. These solutions account for both diffusive and advective transport, as well as both ordinary and Knudsen diffusion. Sample calculations are presented for coating preform fibers with boron nitride.
Date: April 1997
Creator: Griffiths, S. K. & Nilson, R. H.
Partner: UNT Libraries Government Documents Department

Modeling electrodeposition for LIGA microdevice fabrication

Description: To better understand and to help optimize the electroforming portion of the LIGA process, we have developed one and two-dimensional numerical models describing electrode-position of metal into high aspect-ratio molds. The one-dimensional model addresses dissociation, diffusion, electromigration, and deposition of multiple ion species. The two-dimensional model is limited to a single species, but includes transport induced by forced flow of electrolyte outside the mold and by buoyancy associated with metal ion depletion within the mold. To guide model development and to validate these models, we have also conducted a series of laboratory experiments using a sulfamate bath to deposit nickel in cylindrical molds having aspect ratios up to twenty-five. The experimental results indicate that current densities well in excess of the diffusion-limited currents may still yield metal deposits of acceptable morphology. However, the numerical models demonstrate that such large ion fluxes cannot be sustained by convection within the mold resulting from flow across the mold top. Instead, calculations suggest that the observed enhancement of transport probably results from natural convection within the molds, and that buoyancy-driven flows may be critical to metal ion transport even in micron-scale features having very large aspect ratios. Taking advantage of this enhanced ion transport may allow order-of-magnitude reductions in electroforming times for LIGA microdevice fabrication. 42 refs., 14 figs., 1 tab.
Date: February 1, 1998
Creator: Griffiths, S.K.; Nilson, R.H. & Bradshaw, R.W.
Partner: UNT Libraries Government Documents Department

Deposition uniformity, particle nucleation and the optimum conditions for CVD in multi-wafer furnaces

Description: A second-order perturbation solution describing the radial transport of a reactive species and concurrent deposition on wafer surfaces is derived for use in optimizing CVD process conditions. The result is applicable to a variety of deposition reactions and accounts for both diffusive and advective transport, as well as both ordinary and Knudsen diffusion. Based on the first-order approximation, the deposition rate is maximized subject to a constraint on the radial uniformity of the deposition rate. For a fixed reactant mole fraction, the optimum pressure and optimum temperature are obtained using the method of Lagrange multipliers. This yields a weak one-sided maximum; deposition rates fall as pressures are reduced but remain nearly constant at all pressures above the optimum value. The deposition rate is also maximized subject to dual constraints on the uniformity and particle nucleation rate. In this case, the optimum pressure, optimum temperature and optimum reactant fraction are similarly obtained, and the resulting maximum deposition rate is well defined. These results are also applicable to CVI processes used in composites manufacturing.
Date: June 1, 1996
Creator: Griffiths, S.K. & Nilson, R.H.
Partner: UNT Libraries Government Documents Department

Conditions for similitude between the fluid velocity and electric field in electroosmotic flow

Description: Electroosmotic flow is fluid motion driven by an electric field acting on the net fluid charge produced by charge separation at a fluid-solid interface. Under many conditions of practical interest, the resulting fluid velocity is proportional to the local electric field, and the constant of proportionality is everywhere the same. Here the authors show that the main conditions necessary for this similitude are a steady electric field, uniform fluid and electric properties, an electric Debye layer that is thin compared to any physical dimension, and fluid velocities on all inlet and outlet boundaries that satisfy the Helmholtz-Smoluchowski relation normally applicable to fluid-solid boundaries. Under these conditions, the velocity field can be determined directly from the Laplace equation governing the electric potential, without solving either the continuity or momentum equations. Three important consequences of these conditions are that the fluid motion is everywhere irrotational, that fluid velocities in two-dimensional channels bounded by parallel planes are independent of the channel depth, and that such flows exhibit no dependence on the Reynolds number.
Date: April 1, 1999
Creator: Cummings, E. B.; Griffiths, S. K.; Nilson, R. H. & Paul, P. H.
Partner: UNT Libraries Government Documents Department