Fractals are a relatively recent development in mathematics that show promise as a foundation for models of complex systems like natural porous media. One important issue that has not been thoroughly explored is the affect of different algorithms commonly used to generate random fractal porous media on their properties and processes within them. The heterogeneous method can lead to large, uncontrolled variations in porosity. It is proposed that use of the homogeneous algorithm might lead to more reproducible applications. Computer codes that will make it easier for researchers to experiment with fractal models are provided. In Chapter 2, the application of percolation theory and fractal modeling to porous media are combined to investigate percolation in prefractal porous media. Percolation thresholds are estimated for the pore space of homogeneous random 2-dimensional prefractals as a function of the fractal scale invariance ratio b and iteration level i. Percolation in prefractals occurs through large pores connected by small pores. The thresholds increased beyond the 0.5927 porosity expected in Bernoulli (uncorrelated) networks. The thresholds increase with both b (a finite size effect) and i. The results allow the prediction of the onset of percolation in models of prefractal porous media. Only a limited range of parameters has been explored, but extrapolations allow the critical fractal dimension to be estimated for many b and i values. Extrapolation to infinite iterations suggests there may be a critical fractal dimension of the solid at which the pore space percolates. The extrapolated value is close to 1.89 -- the well-known fractal dimension of percolation clusters in 2-dimensional Bernoulli networks. The results of Chapters 1 and 2 are synthesized in an application to soil water retention in Chapter 3.

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The energy analysis of the wave motion in the Lamb’s problem for a poroelastic half-space in the framework of Biot’s theory is presented. The results for the energy velocity and quality factor of poroelastic waves are revisited. In the case of no dissipation the approach originally established for perfectly elastic media by Miller & Pursey is generalized herein to include poroelastic waves. Special cases of the resonant excitation of the Rayleigh wave and the absence of the Rayleigh wave beyond the cut-off frequency are discussed in detail. Directional diagrams for the volumetric waves are presented. A quantitative picture of the energy partition among the traveling waves is provided for several driving configurations. In the general case of dissipative media the analysis is based on the semi-analytic solution of the Lamb’s problem. In the near field, the surface load generates three wavetrains corresponding to the bulk modes. These wavetrains consist of waves which are longer and exhibit greater viscous attenuation than the corresponding volumetric modes, so that, P1, P2 and S modes emerge from the corresponding wavetrains at a certain distance from the source. For the far field, asymptotic expressions have been obtained and clearly indicate that it is only in the far field that the wave motion represents the superposition of the P1, P2, S and Rayleigh waves characterized by their corresponding wavelengths and attenuations. Moreover, these waves also exhibit geometric attenuation x^3/2 (similar to the waves in a perfectly elastic half-space). To analyze the energy partition the total input power supplied by the source is decomposed into the contributions associated with the wavetrains and the Rayleigh wave. These results provide the means for controlling the excitation of the various wave modes via changes to the driving configuration. Biot’s theory is a particular example of a non-conservative Lagrangian system with a Rayleigh dissipation function. The group velocities of poroelastic waves are complex and do not provide any information about the velocity of the energy transport. Moreover, in general the precise physical meaning of the complex group velocity is unclear. The analysis based on the detailed study of the coupled system of the damped Klein-Gordon equations (Biot’s theory yields such a formalism in the low frequency limit) suggests that both precise and approximate physical interpretations of the complex group velocity are possible. Moreover, these considerations further allow the derivation of exact closed form expressions for the energy velocity and Q factor for both longitudinal and shear poroelastic waves from energy principles. Most notably, the analysis of the resulting expressions reveals that the energy velocity of both longitudinal and shear waves equals (exceeds) the corresponding phase velocity in the case of the low (full) frequency range Biot’s theory. The exact expression for the Q factor contains an additive correction due to viscoelastic interphase interaction in the higher frequency range.

Any natural surface is in essence non-smooth, consisting of more or less regular roughness and/or mobile structures of different scales. From a fluid mechanics point of view, these natural surfaces offer better aerodynamic performances when they cover moving bodies, in terms of drag reduction, lift enhancement or control of boundary layer separation; this has been shown for boundary layer or wake flows around thick bodies. The numerical simulation of microscopic flows around "natural" surfaces is still out of reach today. Therefore, the goal of this thesis is to study the modeling of the apparent flow slip occurring on this kind of surfaces, modeled as a porous medium, applying Whitaker's volume averaging theory. This mathematical model makes it possible to capture details of the microstructure while preserving a satisfactory description of the physical phenomena which occur. The first chapter of this manuscript provides an overview of previous efforts to model these surfaces, detailing the most important results from the literature. The second chapter presents the mathematical derivation of the volume-averaged Navier-Stokes equations (VANS) in a porous medium. In the third chapter the flow stability at the interface between a free fluid and a porous medium, formed by a series of rigid cylinders, is studied. The presence of this porous layer is treated by including a drag term in the fluid equations. It is shown that the presence of this term reduces the rates of amplification of the Kelvin-Helmholtz instability over the whole range of wavenumbers, thus leading to an increase of the wavelength of the most amplified mode. In this same context, the difference between the isotropic model and a tensorial approach for the drag term has been evaluated, to determine the most consistent approach to study these flow instabilities. This has led to the conclusion that the model that uses the apparent permeability tensor is the most relevant one. In the following chapter, based on this last result, the apparent permeability tensor, based on over one hundred direct numerical simulations carried out over microscopic unit cells, has been identified for a three-dimensional porous medium consisting of rigid cylinders. In these configurations the tensor varies according to four parameters: the Reynolds number, the porosity and the direction of the average pressure gradient, defined by two Euler angles. This parameterization makes it possible to capture local three-dimensional effects. This database has been set up to create, based on a kriging-type approach, a behavioral metamodel for estimating all the components of the apparent permeability tensor. In the fifth chapter, simulations of the VANS equations are carried out on a macroscopic scale after the implementation of the metamodel, to get reasonable computing times. The validation of the macroscopic approach is performed on a closed cavity flow covered with a porous layer and a comparison with the results of a very accurate DNS, homogenized a posteriori, has shown a very good agreement and has demonstrated the relevance of the approach. The next step has been the study of the passive control of the separation of the flow past a hump which is placed on a porous wall, by the same macroscopic VANS approach. Finally, general conclusions and possible directions of research in the field are presented in the last chapter.

The lattice-gas model of sorption in disordered porous media is studied for a variety of settings, using existing, updated and newly developed numerical techniques. Firstly, we construct an efficient algorithm to calculate the exact partition function for small lattice-gas systems. The exact partition function is used for detailed analysis of the core features exhibited by such systems. We proceed to develop an interactive Monte Carlo (MC) simulation engine, that simulates sorption in a porous media sample and provides real-time visual data of the state space projection and the 3d view of the sample among other parameters of interest, as the external fields are manipulated. The use of such tool provides a more intuitive understanding of the system behaviour. The MC simulations are employed to study sorption in several porous solids: silica aerogel, Vycor glass and soil. We investigate how the phenomena depend on the microstructure of the original samples, how the behaviour varies with the external conditions, and how it is reflected in the paths that the system takes across its state space. Secondly, we develop two methods for estimation of the relative degeneracy (the number of microstates that have the same value of some macroscopic variables) in the systems that are too large to be handled exactly. The methods, based on a restricted infinite temperature sampling, obtain equidegenerate surfaces and the degeneracy gradient across the state space. Combined with the knowledge of an internal energy of a microstate, it enables us to construct the free energy map and thus the equilibrium probability distribution for the studied projection of the state space. Thirdly, the jump-walking Monte-Carlo algorithm is revisited and updated to study the equilibrium properties of systems exhibiting quasi-ergodicity. It is designed for a single processing thread as opposed to currently predominant algorithms for large parallel processing systems. The updated algorithm is tested on the Ising model and applied to the lattice-gas model for sorption in aerogel and Vycor glass at low temperatures, when dynamics of the system is significantly slowed down. It is demonstrated that the updated jump-walking simulations are able to produce equilibrium isotherms which are typically hidden by the hysteresis effect characteristic of the standard single-flip simulations. As a result, we answer the long standing question about the existence of the first-order phase transitions in Vycor. Finally, we investigate sorption in several distinct topology network representations of soil and aerogel samples and demonstrate that the recently developed analytical techniques for random networks can be used to achieve a qualitative understanding of the phenomena in real materials.

The work presented in this thesis has focused on the development and implementation of two-dimensional (2D) nuclear magnetic resonance (NMR) correlation techniques to unambiguously discriminate and characterise competitive adsorption and mass transfer processes in porous materials. This has primarily involved investigations on porous oxides; in particular silicas and aluminas commonly used as catalyst supports. The techniques used are demonstrated to be capable of acquiring information relevant to the performance of heterogeneous catalysts and adsorbents across the hierarchy of length scales relevant to industrial processes. The methodologies associated with the acquisition and processing of 2D NMR correlation data were first established through the development of analytical models capable of simulating 2D signal attenuation data for T1-T2, D-T2 and T2-T2 experiments. Common artefacts were also discussed by means of experimental and simulated examples and, where appropriate, methods have been introduced for their prevention. The NMR relaxation behaviour of water saturating the pore space of silica was observed to correlate strongly with independent measurements of the activation energy of dehydroxylation, thus establishing NMR relaxometry as a tool for directly probing the surface energetics of silica surfaces. This interpretation of T1/T2 ratios differs from that in conventional applications of the technique which typically present the ratio as an indicator of surface-adsorbate interaction strength. Here, the T1/T2 ratios of three liquid probe molecules: ethanol, diethyl ether and cyclohexane, are used to investigate the influence of pore size and density of adsorption sites on relaxation behaviour. Competitive adsorption behaviour has been directly investigated through the acquisition of T1-T2 correlation data of binary liquid mixtures imbibed in various silica supports. These measurements, in combination with a newly developed model for the relaxation of multi-component mixtures, have provided a comprehensive assessment of the ability of this technique to quantify intra-pellet compositions and address competitive adsorption behaviour in porous media. With the aid of a random walk Monte-Carlo model to simulate transverse relaxation in packed bed reactors, T2-T2 relaxation exchange measurements have been used to investigate mass transfer across the fluid-solid boundary in a packed bed reactor filled with γ-alumina and liquid cyclohexane. These data were used to quantify the rate of exchange between intra- and inter-pellet environments at a number of flow rates. Exchange rates were then converted into the more convenient terms of mass transfer coefficients and compared against literature data using two separate dimensionless mass transfer analyses.

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abstract: This dissertation studies two outstanding microscale fluid mechanics problems: 1) mechanisms of gas production from the nanopores of shale; 2) enhanced mass flow rate in steady compressible gas flow through a micro-conduit. The dissertation starts with a study of a volumetric expansion driven drainage flow of a viscous compressible fluid from a small capillary and channel in the low Mach number limit. An analysis based on the linearized compressible Navier-Stokes equations with no-slip condition shows that fluid drainage is controlled by the slow decay of the acoustic wave inside the capillary and the no-slip flow exhibits a slip-like mass flow rate. Numerical simulations are also carried out for drainage from a small capillary to a reservoir or a contraction of finite size. By allowing the density wave to escape the capillary, two wave leakage mechanisms are identified, which are dependent on the capillary length to radius ratio, reservoir size and acoustic Reynolds number. Empirical functions are generated for an effective diffusive coefficient which allows simple calculations of the drainage rate using a diffusion model without the presence of the reservoir or contraction. In the second part of the dissertation, steady viscous compressible flow through a micro-conduit is studied using compressible Navier-Stokes equations with no-slip condition. The mathematical theory of Klainerman and Majda for low Mach number flow is employed to derive asymptotic equations in the limit of small Mach number. The overall flow, a combination of the Hagen-Poiseuille flow and a diffusive velocity shows a slip-like mass flow rate even through the overall velocity satisfies the no-slip condition. The result indicates that the classical formulation includes self-diffusion effect and it embeds the Extended Navier-Stokes equation theory (ENSE) without the need of introducing additional constitutive hypothesis or assuming slip on the boundary. Contrary to most ENSE publications, the predicted mass flow rate is still significantly below the measured data based on an extensive comparison with thirty-five experiments. / Dissertation/Thesis / Doctoral Dissertation Mechanical Engineering 2019