published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

This dissertation develops analytical expressions for the effective hydraulic conductivity Kₑ of a three-dimensional porous medium bounded by two parallel planes of infinite extent separated by a distance 2a. Head varies randomly along each boundary about a uniform mean value. The log hydraulic conductivity Y forms a homogeneous, statistically anisotropic random field having a variance σᵧ² and principal integral scales λ₁, λ₂, λ₃. Flow is uniform in the mean parallel to the principal coordinate χ₁. A solution is first derived for mildly nonuniform media with σᵧ² ≪ 1 via an approximate form of the 1993 residual flux theory by Neuman and Orr. It is then extended to strongly nonuniform media with arbitrarily large σᵧ² by invoking the Landau-Lifshitz conjecture as Kₑ = KG exp {σᵧ² [1/2 — (D + S)]} . Here, K(G) is the geometric mean of hydraulic conductivities and D and S are domain and surface integrals, respectively. Based on a rigorous limiting analysis we show that when the length scale ratio p = a / λ₁ → 0, Kₑ is equal to the arithmetic mean hydraulic conductivity K(A). This supports the theoretical finding of Neuman and Orr and the numerical result by Desbarats. When ρ → ∞ we obtain expressions for Kₑ that have been previously derived in the stochastic literature for infinite flow domains. For strongly anisotropic media with integral scale ratios ε₂ = λ₂ / λ₁ and ε₃ = λ₃ / λ₁ equal to each other and tending to zero or infinity ( ) i 0) we obtain the closed form solution Kₑ = K(G) exp {σᵧ²[exp(—p) — 0 .5]} . The latter reduces to K(A) when ρ → 0 and tends to the harmonic mean K(H) as ρ → ∞. One can think of the case ε₂ = ε₃ = 0 as mean flow along parallel channels having mutually uncorrelated hydraulic conductivities, and of the case ε₂ = ε₃ → ∞ as mean flow normal to layers having uniform hydraulic conductivities. For statistically isotropic media we show numerically that Kₑ equals K(A) when ρ = 0.01; when ρ ≥ 4, Kₑ = K(G) exp(σᵧ²/6) the three-dimensional infinite domain solution. Our results support the analytical finding of Rubin and Dagan, and predict and explain all related bounded domain numerical results. Finally, contrary to Dagan's assertion, we show that for small ρ boundary effects are extremely important; the absolute value of the surface integral S equals the value of the domain integral D.

Spatial variability of porous media often prevents precise physical characterization of the system. In order to model moisture and solute transport through this media, certain sacrifices in precision must be made. Physical characteristics of the system must be averaged over large scales, lumping the small scale variability into the large scale characterization. Although this precludes a precise definition of the small scale flow characteristics, parameterization is much more attainable. This study addresses methods for determining effective hydraulic conductivity of unsaturated porous media. Effective conductivity is used to describe the large scale behavior of the system. Different methods for calculating the effective conductivity are presented and compared. Results indicate that the unit mean gradient method produces good estimates of the effective conductivity and can be applied using limited field data. The zone of correlation of the hydraulic parameters can be used in experimental design to minimize the errors associated with estimation of the mean pressure. An inverse method for evaluating the optimum effective hydraulic parameters is presented. Results indicate the optimization procedure is more sensitive to wetting than to drying conditions. Because of interaction between the hydraulic parameters, concurrent optimization of more than two of the parameters based on soil pressure data alone is not advised. Anisotropy in an unsaturated soil was found to be a function of the profile mean soil pressure. Results indicate the effective conductivity for flow parallel to soil layering can be estimated from the arithmetic mean of the unsaturated conductivity values for each of the layers and is between the harmonic and geometric means of these data for flow perpendicular to the layering. Estimates of the effective unsaturated hydraulic conductivity obtained through stochastic analysis agreed well with simulation results. Deviations between the stochastic predictions and simulation results are larger when the variability of the soil profile is greater and begin to deviate significantly when the variance of ln K(ψ₀) exceeds 5.0 and the variance of a exceeds 0.02 1/cm².

It has been shown by Neuman [1990], Di Federico and Neuman [1997, 1998a,b] and Di Federico et al. [1999] that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework, which views hydraulic conductivity as a random fractal characterized by a power variogram. Any such random fractal field is statistically nonhomogeneous but possesses homogeneous spatial increments. When the field is statistically isotropic, it is associated with a power variogram γ(s) = Cs²ᴴ where C is a constant, s is separation distance, and If is a Hurst coefficient (0 < H< 1). If the field is Gaussian it constitutes fractional Brownian motion (fBm). The authors have shown that the power variogram of a statistically isotropic or anisotropic fractal field can be constructed as a weighted integral from zero to infinity of exponential or Gaussian vario grams of overlapping, homogeneous random fields (modes) having mutually uncorrelated increments and variance proportional to a power 2H of the integral (spatial correlation) scale. Low- and high-frequency cutoffs are related to length scales of the sampling window (domain) and data support (sample volume), respectively. Intermediate cutoffs account for lacunarity due to gaps in the multiscale hierarchy, created by a hiatus of modes associated with discrete ranges of scales. In this dissertation, I investigate the effects of domain and support scales on the multiscale properties of random fractal fields characterized by a power variogram using real and synthetic data. Neuman [1994] and Di Federico and Neuman [1997] have concluded empirically, on the basis of hydraulic conductivity data from many sites, that a finite window of length-scale L filters out (truncates) all modes having integral scales λ larger than λ = μL where μ ≃ 1/3. I confii in their finding computationally by generating truncated fBm realizations on a large grid, using various initial values of μ, and demonstrating that μ ≃ 1/3 for windows smaller than the original grid. My synthetic experiments also show that generating an fl3m realization on a finite grid using a truncated power variogram yields sample variograms that are more consistent with theory than those obtained when the realization is generated using a power variogram. Interpreting sample data from such a realization using wavelet analysis yields more reliable estimates of the Hurst coefficient than those obtained when one employs variogram analysis. Di Federico et al. [1997] developed expressions for the equivalent hydraulic conductivity of a box-shaped support volume, embedded in a log-hydraulic conductivity field characterized by a power variogram, under the action of a mean uniform hydraulic gradient. I demonstrate that their expression and empirically derived value of μ ≃ 1/3 are consistent with a pronounced permeability scale effect observed in unsaturated fractured tuff at the Apache Leap Research Site (ALRS) near Superior, Arizona. I then investigate the compatibility of single-hole air permeability data, obtained at the ALRS on a nominal support scale of about 1 m, with various scaling models including fBm, fGn (fractional Gaussian noise), fLm (fractional Lévy motion), bfLm (bounded fractional Lévy motion) and UM (Universal Multifractals). I find that the data have a Lévy-like distribution at small lags but become Gaussian as the lag increases (corresponding to bfLm). Though this implies multiple scaling, it is not consistent with the UM model, which considers a unique distribution. If one nevertheless applies a UM model to the data, one obtains a very small codimension which suggests that multiple scaling is of minor consequence (applying the UM model to permeability rather than log-permeability data yields a larger codimension but is otherwise not consistent with these data). Variogram and resealed range analyses of the log-permeability data yield comparable estimates of the Hurst coefficient. Resealed range analysis shows that the data are not compatible with an fGn model. I conclude that the data are represented most closely by a truncated fBm model.

Thesis (MScEng)--University of Stellenbosch, 2002 / ENGLISH ABSTRACT: A Representative Unit Cell (RUC) model for flow through two-dimensional porous media is presented and applied to two industrial related problems. The first application is to that of cross-flow in tube banks. Both staggered and square (inline) configurations are investigated and the model results are compared to experimental data. The second application is to flow through a stack in a timber-drying kiln. The RUC model is applied to the anisotropic timber stack ends and the centre part is modelled with a standard duct flow solution. The results of the models applied to a timber stack are compared to experimental data obtained from model tests undertaken in a wind tunnel. The results of the RUC and duct flow models are found to be in excellent agreement with the data of the experimental models. These models may be used to optimize kiln designs. / AFRIKAANSE OPSOMMING: 'n Verteenwoordigende Eenheid Sel (VES) model vir vloei deur twee-dimensionele poreuse media word weergegee en toegepas op twee industriële toepassings. Die eerste toepassing is op dwarsvloei deur banke van buise. Beide gestapelde en inlyn konfigurasies word ondersoek waarvan die model resulte met eksperimentele data vergelyk word. Die tweede toepassing is op vloei deur 'n stapel in 'n hout-droogoond. Die VES model word toegepas op die anisotropiese ente van houtstapels en die middelste seksie word gemodelleer deur 'n standaard kanaalvloei oplossing. Die resultate van die modelle toegepas op n 'houtstapel word vergelyk met eksperimentele data verkry uit model toetse wat in 'n wind-tonnel uitgevoer is. Die VES en kanaalvloei modelle se resultate stem uitstekend ooreen met die data van die eksperimentele modelle. Hierdie modelle kan gebruik word om die ontwerp van droogoonde te optimeer.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Fluid flow through porous media plays an important role in a variety of contexts of which filtration is one. Filtration efficiency of fibrous filters depends on the micro-structural characterization of these porous materials and is reflected in the permeability there-of. Compression of fibrous porous media has a significant effect on the permeability. Experimental data indicate that the permeability varies generally with more than an order of magnitude over the narrow porosity range in which the compression takes place. Relative to the amount of experimental studies regarding this phenomenon, there is a scarcity of geometric models in the literature that can account for the effect of compression on the permeability of a fibrous porous medium. Within the context of existing geometric porescale models based on rectangular geometry, a new model is presented and an existing model improved to predict the effect of one-dimensional compression in the streamwise direction. In addition, without compromising on a commitment to mathematical simplicity, empirical data of a non-woven fibrous porous medium was used to highlight the effect of model geometry on its predictive capability. Different mathematical expressions for the relationship between compression and porosity were considered. The permeability is expressed explicitly in terms of the fibre diameter and the compression fraction and implicitly in terms of the porosity. The porosity is incorporated through the relationship between the linear dimensions of the geometric model. The general applicability of the model(s) was validated by making use of data on airflow through a soft fibrous porous material as well as through glass and nylon fibres. The permeability predictions fall within the same order of magnitude as the experimental data. Given the mathematical simplicity of the model(s), the prediction capability is satisfactory. Attention is drawn to assumptions made and model restrictions within the analytical modelling procedure. A general predictive equation is presented for the permeability prediction in which a solid distribution factor is introduced. The proposed models serve as basis for further adaptation and refinement towards prediction capability. / AFRIKAANSE OPSOMMING: Vloei van vloeistowwe deur poreuse media speel ’n belangrike rol in ’n verskeidenheid kontekste waarvan filtrasie een is. Die filtrasie doeltreffendheid van vesel filters hang af van die mikro-strukturele karakterisering van hierdie poreuse materiale en word gereflekteer in die permeabiliteit. Kompressie van veselagtige poreuse media het ’n beduidende effek op die permeabiliteit. Eksperimentele data dui aan dat die verandering in permeabiliteit gewoonlik oor meer as ’n orde grootte strek oor die klein porositeitsinterval waarin die kompressie plaasvind. Relatief tot die aantal eksperimentele studies rakende hierdie verskynsel, is daar ’n tekort aan geometriese modelle in die literatuur wat die effek van kompressie op die permeabiliteit van veselagtige poreuse media in ag kan neem. Binne die konteks van bestaande geometriese kanaal-skaal modelle gebasseer op reghoekige geometrie, is ’n nuwe model voorgestel en ’n bestaande model verbeter om die effek van een-dimensionele kompressie in die stroomsgewyse rigting te voorspel. Sonder om die verbintenis tot wiskundige eenvoud prys te gee, is empiriese data van ’n nie-geweefde veselagtige poreuse medium gebruik om die effek van die geometrie van ’n model op sy voorspellingsvermo¨e uit te lig. Verskillende wiskundige uitdrukkings is oorweeg vir die verband tussen kompressie en porositeit. Die permeabiliteit is eksplisiet uitgedruk in terme van die veseldiameter en die kompressie breukdeel en implisiet in terme van die porositeit. Die porositeit is ge-inkorporeer deur die verhouding tussen die lineêre dimensies van die geometriese model. Die algemene toepaslikheid van die model(le) is gestaaf deur gebruik te maak van data oor lugvloei deur ’n sagte veselagtige poreuse materiaal sowel as deur glas en nylon vesels. Die voorspellings van die permeabiliteit val binne dieselfde groote orde as die eksperimentele data. Gegee die wiskundige eenvoud van die model(le), is die voorspellingsvermoë bevredigend. Aandag is gevestig op aannames wat gemaak is en modelbeperkings binne die analitiese modellerings prosedure. ’n Algemene voorspellingsvergelyking is voorgestel vir die voorspelling van die permeabiliteit waarin ’n vaste stof distribusie faktor geinkorporeer is. Die voorgestelde modelle dien as basis vir verdere aanpassing en verfyning van voorspellingsvermoë.

Under consideration is the finite-size scaling of two thermomechanical responses of random heterogeneous materials. Stochastic mechanics is applied here to the modeling of heterogeneous materials in order to construct the constitutive relations. Such relations (e.g. Hooke's Law in elasticity or Fourier's Law in heat transfer) are well-established under spatial homogeneity assumption of continuum mechanics, where the Representative Volume Element (RVE) is the fundamental concept. The key question is what is the size L of RVE? According to the separation of scales assumption, L must be bounded according to d&lt;L&lt;&lt;LMacro where d is the microscale (or average size of heterogeneity), and LMacro is the macroscale of a continuum mechanics problem. Statistically, for spatially ergodic heterogeneous materials, when the mesoscale is equal to or bigger than the scale of the RVE, the elements of the material can be considered homogenized. In order to attain the said homogenization, two conditions must be satisfied: (a) the microstructure's statistics must be spatially homogeneous and ergodic; and (b) the material's effective constitutive response must be the same under uniform boundary conditions of essential (Dirichlet) and natural (Neumann) types. / In the first part of this work, the finite-size scaling trend to RVE of the Darcy law for Stokesian flow is studied for the case of random porous media, without invoking any periodic structure assumptions, but only assuming the microstructure's statistics to be spatially homogeneous and ergodic. By analogy to the existing methodology in thermomechanics of solid random media, the Hill-Mandel condition for the Darcy flow velocity and pressure gradient fields was first formulated. Under uniform essential and natural boundary conditions, two variational principles are developed based on minimum potential energy and complementary energy. Then, the partitioning method was applied, leading to scale dependent hierarchies on effective (RVE level) permeability. The proof shows that the ensemble average of permeability has an upper bound under essential boundary conditions and a lower bound under uniform natural boundary conditions. / To quantitatively assess the scaling convergence towards the RVE, these hierarchical trends were numerically obtained for various porosities of random disk systems, where the disk centers were generated by a planar Poisson process with inhibition. Overall, the results showed that the higher the density of random disks---or, equivalently, the narrower the micro-channels in the system---the smaller the size of RVE pertaining to the Darcy law. / In the second part of this work, the finite-size scaling of effective thermoelastic properties of random microstructures were considered from Statistical to Representative Volume Element (RVE). Similarly, under the assumption that the microstructure's statistics are spatially homogeneous and ergodic, the SVE is set-up on a mesoscale, i.e. any scale finite relative to the microstructural length scale. The Hill condition generalized to thermoelasticity dictates uniform essential and natural boundary conditions, which, with the help of two variational principles, led to scale dependent hierarchies of mesoscale bounds on effective (RVE level) properties: thermal expansion strain coefficient and stress coefficient, effective stiffness, and specific heats. Due to the presence of a non-quadratic term in the energy formulas, the mesoscale bounds for the thermal expansion are more complicated than those for the stiffness tensor and the heat capacity. To quantitatively assess the scaling trend towards the RVE, the hierarchies are computed for a planar matrix-inclusion composite, with inclusions (of circular disk shape) located at points of a planar, hard-core Poisson point field. Overall, while the RVE is attained exactly on scales infinitely large relative to microscale, depending on the microstructural parameters, the random fluctuations in the SVE response become very weak on scales an order of magnitude larger than the microscale, thus already approximating the RVE. / Based on the above studies, further work on homogenization of heterogeneous materials is outlined at the end of the thesis. / Keywords: Representative Volume Element (RVE), heterogeneous media, permeability, thermal expansion, mesoscale, microstructure.

No description available.

Thesis (MScEng (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2006. / This work entails the improvement of an existing three dimensional pore-scale model. Stagnant zones are included, the closure of the volume averaged pressure gradient is improved and an improved calculation of pore-scale averages, using the RUC, is done for the model to be a more realistic representative of the REV and thus of the foamlike material. Both the Darcy and the Forchheimer regimes are modelled and a general momentum transport equation is derived by means of an asymptotic matching technique. The RUC model is also extended to cover non-Newtonian flow. Since metallic foams are generally of porosities greater than 90%, emphasis is put on the accurate prediction of permeability for these porosities. In order to improve permeability predictions for these high porosity cases an adaptation to the RUC model was considered, whereby rectangular prisms were replaced by cylinders. Although this adaptation appears to give more accurate permeabilities at very high porosities, its implementation in a generalised model seems impractical. The prediction of the characteristic RUC side length is discussed and results of both the cylindrical strand model and the square strand model are compared to experimental work.