Mixture theory and unsaturated flow in swelling soils

The current state of knowledge on various aspects of mixture theory applied to unsaturated/saturated swelling soils is discussed. Two and three phase problems are studied. On the smallest scale (micro) the individual platelets and adsorbed (vicinal) water exist as separate phases. On the intermediate scale (meso) the platelets and vicinal water are homogenized to form a saturated particle where vicinal water and solid are overlaying continua. On the macroscale, bulk water, water vapor, and the mesoscale particles are homogenized resulting in three overlaying continua for solid, bulk water, vicinal water, and water vapor. Stress tensor formulations and Darcy’s laws are presented at the mesoscale and macroscales. A theoretical formulation for surface crusting is presented at the mesoscale. General viscoelastic mesoscale and macroscale models are discussed and related to classical approaches.     

Coupled Solvent and Heat Transport of a Mixture of Swelling Porous Particles and Fluids: Single Time-Scale Problem

A three-spatial scale, single time-scale model for both moisture and heat transport is developed for an unsaturated swelling porous media from first principles within a mixture theoretic framework. On the smallest (micro) scale, the system consists of macromolecules (clay particles, polymers, etc.) and a solvating liquid (vicinal fluid), each of which are viewed as individual phases or nonoverlapping continua occupying distinct regions of space and satisfying the classical field equations. These equations are homogenized forming overlaying continua on the intermediate (meso) scale via hybrid mixture theory (HMT). On the mesoscale the homogenized swelling particles consisting of the homogenized vicinal fluid and colloid are then mixed with two bulk phase fluids: the bulk solvent and its vapor. At this scale, there exists three nonoverlapping continua occupying distinct regions of space. On the largest (macro) scale the saturated homogenized particles, bulk liquid and vapor solvent, are again homogenized forming four overlaying continua: doubly homogenized vicinal fluid, doubly homogenized macromolecules, and singly homogenized bulk liquid and vapor phases. Two constitutive theories are developed, one at the mesoscale and the other at the macroscale. Both are developed via the Coleman and Noll method of exploiting the entropy inequality coupled with linearization about equilibrium. The macroscale constitutive theory does not rely upon the mesoscale theory as is common in other upscaling methods. The energy equation on either the mesoscale or macroscale generalizes de Vries classical theory of heat and moisture transport. The momentum balance allows for flow of fluid via volume fraction gradients, pressure gradients, external force fields, and temperature gradients.     

A Multiscale Theory of Swelling Porous Media: II. Dual Porosity Models for Consolidation of Clays Incorporating Physicochemical Effects

A three-scale theory of swelling clay soils is developed which incorporates physico-chemical effects and delayed adsorbed water flow during secondary consolidation. Following earlier work, at the microscale the clay platelets and adsorbed water (water between the platelets) are considered as distinct nonoverlaying continua. At the intermediate (meso) scale the clay platelets and the adsorbed water are homogenized in the spirit of hybrid mixture theory, so that, at the mesoscale they may be thought of as two overlaying continua, each having a well defined mass density. Within this framework the swelling pressure is defined thermodynamically and it is shown to govern the effect of physico-chemical forces in a modified Terzaghi's effective stress principle. A homogenization procedure is used to upscale the mesoscale mixture of clay particles and bulk water (water next to the swelling mesoscale particles) to the macroscale. The resultant model is of dual porosity type where the clay particles act as sources/sinks of water to the macroscale bulk phase flow. The dual porosity model can be reduced to a single porosity model with long term memory by using Green's functions. The resultant theory provides a rational basis for some viscoelastic models of secondary consolidation.     

Macroscopic Flow Potentials in Swelling Porous Media

In swelling porous media, the potential for flow is much more than pressure, and derivations for flow equations have yielded a variety of equations. In this article, we show that the macroscopic flow potentials are the electro-chemical potentials of the components of the fluid and that other forms of flow equations, such as those derived through mixture theory or homogenization, are a result of particular forms of the chemical potentials of the species. It is also shown that depending upon whether one is considering the pressure of a liquid in a reservoir in electro-chemical equilibrium with the swelling porous media, or the pressure of the vicinal liquid within the swelling porous media, a critical pressure gradient threshold exists or does not.     

Particle velocimetry inside Newtonian and non-Newtonian droplets impacting a hydrophobic surface

A particle velocimetry technique is described which enables the measurement of the fluid velocity inside impacting drops. Using high speed photography of 2 μm fluorescent tracer particles suspended in the fluid, the velocity field was measured as a function of time and radial position. The potential of the technique is illustrated using velocimetry measurements of drops of pure water and aqueous solutions of 200 ppm poly-(ethylene oxide) (PEO). Dilute solutions of PEO have been known for some time to suppress the rebound of water from hydrophobic surfaces. The dissipation has traditionally been attributed to an increased extensional viscosity as the polymers stretch in the extensional flow of the droplet. Our results enable us to infer that the extensional viscosity of PEO drops, during both the spreading and retraction phase, is similar to that of pure water. The data suggest that the true source of dissipation lies at the droplet edge. We also show, by analysing the spreading of water drops, that the Roisman-Yarin theory for a droplet spreading on a surface is valid in the bulk of the droplet prior to the final stages of spreading.     

煤层气在非饱和水流阶段的非定常渗流摄动解

煤层甲烷由煤层的割理裂隙系统流入生产井一般经历：单相水流、非饱和流和气、水两相饱和流三个阶段，在非饱和流阶段，储层压力降至临界解吸压力之后，储存在煤基质中的吸附气体少量被解吸出来形成互不连续的气泡并阻止水的流动，含气量尚未达到饱和程度。同时煤层甲烷运移包含渗流场、变形场和应力场的动态耦合过程。本文考虑渗流过程中水-气两相不溶混流体与固体耦合作用，建立了非饱和水流阶段非定常渗流问题的流固耦合数学模型，对该强非线性一维数学模型采用摄动法和积分变换法进行解析求解，并讨论了其压力动态特性，分析了压力随饱和度S及时间t变化的规律和气相及耦合作用的影响，这些研究对煤层气、石油和天然气的开采等地下工程领域具有一定的指导意义。     

Mesoscale modeling of microgel mechanics and kinetics through the swelling transition

The mechanics and swelling kinetics of polymeric microgels are simulated using a mesoscale computational model based on dissipative particle dynamics. Microgels are represented by a random elastic network submerged in an explicit viscous solvent. The model is used to probe the effect of different solvent conditions on the bulk modulus of the microgels. Comparison of the simulation results through the volume phase transition reveals favorable agreement with Flory-Rehner’s theory for polymeric gels. The model is also used to examine the microgel swelling kinetics, and is found to be in good agreement with Tanaka’s theory for spherical gels. The simulations show that, during the swelling process, the microgel maintains a nearly homogeneous structure, whereas deswelling is characterized by the formation of chain bundles and network coarsening.     

Epitaxy of Binary Compounds and Alloys

The present work aims at constructing a theoretical framework within which to address the issues of morphological instabilities (one-dimensional step bunching and two-dimensional step meandering), alloying, and phase segregation in binary systems in the context of (physical or chemical) vapor deposition. The length scale of interest, although nanoscopic, is sufficiently large that the steps on a vicinal surface can be viewed as smooth curves and, correspondingly, the theory is a continuum one. In a departure from theories inaugurated by Burton, Cabrera, and Frank [The growth of crystals and the equilibrium structure of their surfaces. Phil. Trans. Roy. Soc. A 243 (1951) 299–358] the steps are endowed with a thermodynamic structure whose main ingredients are a step free-energy density and edge species chemical potentials. Moreover, crystal anisotropy, with its altering of the dynamics of steps and the associated morphological instabilities, is accounted for – in a manner consistent with the second law – both in the thermodynamic and kinetic properties of terraces and, more importantly, of steps. Additionally, in contrast with most of the literature on the subject (cf. [J. Krug, Introduction to step dynamics and step instabilities. In: A. Voigt (ed.) Multiscale Modeling in Epitaxial Growth. Birkhäusser, Berlin (2005)]), adsorption–desorption along the steps, bulk atomic diffusion, and chemical reactions (both on the terraces and along the step edges) are incorporated and coupled to the other mechanisms, e.g., terrace adatom diffusion and step attachment–detachment kinetics, whose interplay governs the evolution of steps on vicinal surfaces. Importantly, aided by the concept of configurational forces for which a separate balance law is postulated Configurational Forces as Basic Concepts of Continuum Physics. Springer, Berlin Heidelberg New York (2000)]), the proposed theory allows the steps to evolve away from local equilibrium thus contributing to a general treatment of the dynamics of steps. Finally, a specialization to the epitaxy of binary compounds and alloys is afforded, yielding a generalization of the classical Gibbs–Thomson relation in the former and novel evolution equations for an individual step in the latter.     

A Fractal Model for Gas–Water Relative Permeability in Inorganic Shale with Nanoscale Pores

A reliable gas–water relative permeability model in shale is extremely important for the accurate numerical simulation of gas–water two-phase flow (e.g., fracturing fluid flowback) in gas-shale reservoirs, which has important implication for the economic development of gas-shale reservoir. A gas–water relative permeability model in inorganic shale with nanoscale pores at laboratory condition and reservoir condition was proposed based on the fractal scaling theory and modified non-slip boundary of continuity equation in the nanotube. The model not only considers the gas slippage in the entire Knudsen regime, multilayer sticking (near-wall high-viscosity water) and the quantified thickness of water film, but also combines the real gas effect and stress dependence effect. The presented model has been validated by various experiments data of sandstone with microscale pores and bulk shale with nanoscale pores. The results show that: (1) The Knudsen diffusion and slippage effects enhance the gas relative permeability dramatically; however, it is not obviously affected at high pressure. (2) The multilayer sticking effect and water film should not be neglected: the multilayer sticking would reduce the water relative permeability as well as slightly decrease gas relative permeability, and the film flow has a negative impact on both of the gas and water relative permeability. (3) The increased fractal dimension for pore size distribution or tortuosity would increase gas relative permeability but decrease the water relative permeability for a given saturation; however, the effect on relative permeability is not that notable. (4) The real gas effect is beneficial for the gas relative permeability, and the influence is considerable when the pressure is high enough and when the nanopores of bulk shale are mostly with smaller size. For the stress dependence, not like the intrinsic permeability, none of the gas or water relative permeability is sensitive to the net pressure and it can be ignored completely.     

A Model for Flow and Deformation in Unsaturated Swelling Porous Media

A thermomechanical theory for multiphase transport in unsaturated swelling porous media is developed on the basis of Hybrid Mixture Theory (saturated systems can also be modeled as a special case of this general theory). The aim is to comprehensively and non-empirically describe the effect of viscoelastic deformation on fluid transport (and vice versa) for swelling porous materials. Three phases are considered in the system: the swelling solid matrix s, liquid l, and air a. The Coleman–Noll procedure is used to obtain the restrictions on the form of the constitutive equations. The form of Darcy’s law for the fluid phase, which takes into account both Fickian and non-Fickian transport, is slightly different from the forms obtained by other researchers though all the terms have been included. When the fluid phases interact with the swelling solid porous matrix, deformation occurs. Viscoelastic large deformation of the solid matrix is investigated. A simple form of differential-integral equation is obtained for the fluid transport under isothermal conditions, which can be coupled with the deformation of the solid matrix to solve for transport in an unsaturated system. The modeling theory thus developed, which involves two-way coupling of the viscoelastic solid deformation and fluid transport, can be applied to study the processing of biopolymers, for example, soaking of foodstuffs and stress-crack predictions. Moreover, extension and modification of this modeling theory can be applied to study a vast variety of problems, such as drying of gels, consolidation of clays, drug delivery, and absorption of liquids in diapers.     

Pressure drop prediction with a modified frictional-kinetic model for alumina in bypass pneumatic conveying system

A new frictional-kinetic model is proposed and modified for pressure drop prediction of alumina in a bypass pneumatic conveying system. This new model is based on the conventional Johnson–Jackson frictional-kinetic model. The critical value of solids volume fraction and maximum packing limit are modified based on the fluidized bulk density and tapped bulk density, respectively. In addition, an offset solid volume fraction is introduced into the frictional pressure model as well as into the radial distribution functions which represents the correction factors to modify the probability of collisions between particles when solid phase becomes excessively dense. For the application of the model, computational fluid dynamics (CFD) simulations were conducted by using kinetic theory, conventional frictional-kinetic model and modified frictional-kinetic model. The simulation results were then compared with the experimental results. It was found that the modified frictional-kinetic model showed the largest improvement on pressure drop prediction results compared with results obtained from applying the kinetic theory and the conventional frictional-kinetic model, especially for denser flows with low air mass flow rates and high solid loading ratios (SLR). In addition, the solids volume investigation of CFD simulations shows a strong comparison to the actual flow conditions in the pipe, as transient slug type flow of alumina is observed.     

Mechanics and chemical thermodynamics of phase transition in temperature-sensitive hydrogels

This paper uses the thermodynamic data of aqueous solutions of uncrosslinked poly(N-isopropylacrylamide) (PNIPAM) to study the phase transition of PNIPAM hydrogels. At a low temperature, uncrosslinked PNIPAM can be dissolved in water and form a homogenous liquid solution. When the temperature is increased, the solution separates into two liquid phases with different concentrations of the polymer. Covalently crosslinked PNIPAM, however, does not dissolve in water, but can imbibe water and form a hydrogel. When the temperature is changed, the hydrogel undergoes a phase transition: the amount of water in the hydrogel in equilibrium changes with temperature discontinuously. While the aqueous solution is a liquid and cannot sustain any nonhydrostatic stress in equilibrium, the hydrogel is a solid and can sustain nonhydrostatic stress in equilibrium. The nonhydrostatic stress can markedly affect various aspects of the phase transition in the hydrogel. We adopt the Flory-Rehner model, and show that the interaction parameter as a function of temperature and concentration obtained from the PNIPAM-water solution can be used to analyze diverse phenomena associated with the phase transition of the PNIPAM hydrogel. We analyze free swelling, uniaxially and biaxially constrained swelling of a hydrogel, swelling of a core-shell structure, and coexistent phases in a rod. The analysis is related to available experimental observations. Also outlined is a general theory of coexistent phases undergoing inhomogeneous deformation.     

Generalized Forchheimer Equation for Two-Phase Flow Based on Hybrid Mixture Theory

In this paper, we derive a Forchheimer-type equation for two-phase flow through an isotropic porous medium using hybrid mixture theory. Hybrid mixture theory consists of classical mixture theory applied to a multiphase system with volume averaged equations. It applies to media in which the characteristic length of each phase is small relative to the extent of the mixture. The derivation of a Forchheimer equation for single phase flow has been obtained elsewhere. These results are extended to include multiphase swelling materials which have nonnegligible interfacial thermodynamic properties.     

Flow-induced nonequilibrium thermodynamics of lamellar semicrystalline polymers

In this work we present a first attempt to quantify the effect of flow deformation on the microstructure of semicrystalline polymers. This necessitates bridging the macroscopic flow length scale with the microscopic (segment) length scale of the semicrystalline structure. To achieve this connection we developed a hierarchical approach where a thermodynamically consistent macroscopic constitutive equation is interfaced with a microscopic lattice-based Monte Carlo (MC) simulation of the polymer chain conformation. We first illustrate this approach in a two-dimensional (2D) “toy” application where the 2D equivalent of a macroscopic constitutive equation based on reptation theory is applied to describe the chain deformation and extended free energy in the amorphous bulk phase. The values for the derivative of the free energy with respect to the mean segment orientation tensor, calculated for a planar extensional flow, are then used as an extended nonequilibrium thermodynamic forcing term. This is added in a traditional Metropolis Monte Carlo scheme, developed for a 2D lattice representation of a lamellar semicrystalline polymer, to drive the flow-induced microstructure. Significant flow-induced changes are calculated, steadily increasing as the Weissenberg number increases.We subsequently extend these ideas further in a much more realistic three-dimensional (3D) application where the information for the thermodynamics of the bulk amorphous phase under a uniaxial extensional flow is extracted from a macroscopic network model, such as that of Phan-Thien and Tanner (PTT), connecting the free energy to the second moment of the end-to-end distance of a multisegment chain. Through a series of 3D nonequilibrium Monte Carlo simulations of both the amorphous and the semicrystalline microscopic morphologies, it is shown that the interaction of the flow-induced deformation with the semicrystalline microstructure is nonlinear: the amorphous interlamellar structure changes significantly from its corresponding homogeneous bulk amorphous state, even far away from the crystalline interface. Our approach allows for a quantitative estimation of this effect on both thermodynamic quantities, like the extended microscopic free energy, as well as various statistics of the chain conformations.     

Constitutive unsaturated hydro-mechanical model based on modified mixture theory with consideration of hydration swelling

This paper has extended modified mixture theory with consideration of hydration swelling in unsaturated rock. By using non-equilibrium thermodynamics and Biot elasticity, a fully coupled formulation including hydration swelling term is derived. Standard arguments of non-equilibrium thermodynamics are used to derive the Darcy’s law for unsaturated flow. Helmholtz free energy has been used to give the relationship between the stress and pore pressure. The chemical potential of water in pore space and clay platelets has been included in the analysis of water sensitive materials such as shale. Finally, a simple numerical example has been presented for illustrative purpose, the results show that the swelling parameter has a strong influence on stress and strain.     

Elastic Waves in Swelling Porous Media

Time harmonic waves in a swelling porous elastic medium of infinite extent and consisting of solid, liquid and gas phases have been studied. Employing Eringen’s theory of swelling porous media, it has been shown that there exist three dilatational and two shear waves propagating with distinct velocities. The velocities of these waves are found to be frequency dependent and complex valued, showing that the waves are attenuating in nature. Here, the appearance of an additional shear wave is new and arises due to swelling phenomena of the medium, which disappears in the absence of swelling. The reflection phenomenon of an incident dilatational wave from a stress-free plane boundary of a porous elastic half-space has been investigated for two types of boundary surfaces: (i) surface having open pores and (ii) surface having sealed pores. Using appropriate boundary conditions for these boundary surfaces, the equations giving the reflection coefficients corresponding to various reflected waves are presented. Numerical computations are performed for a specific model consisting of sandstone, water and carbon dioxide as solid, liquid and gas phases, respectively, of the porous medium. The variations of phase speeds and their corresponding attenuation coefficients are depicted against frequency parameter for all the existing waves. The variations of reflection coefficients and corresponding energy ratios against the angle of incidence are also computed and depicted graphically. It has been shown that in a limiting case, Eringen’s theory of swelling porous media reduces to Tuncay and Corapcioglu theory of porous media containing two immiscible fluids. The various numerical results under these two theories have been compared graphically.     

Multicomponent, Multiphase Thermodynamics of Swelling Porous Media with Electroquasistatics: I. Macroscale Field Equations

A systematic development of the macroscopic field equations (conservation of mass, linear and angular momentum, energy, and Maxwell's equations) for a multiphase, multicomponent medium is presented. It is assumed that speeds involved are much slower than the speed of light and that the magnitude of the electric field significantly dominates over the magnetic field so that the electroquasistatic form of Maxwell's equations applies. A mixture formulation for each phase is averaged to obtain the macroscopic formulation. Species electric fields are considered, however it is assumed that it is the total electric field which contributes to the electrically induced forces and energy. The relationships between species and bulk phase variables and the macroscopic and microscopic variables are given explicitly. The resulting field equations are of relevance to many practical applications including, but not limited to, swelling clays (smectites), biopolymers, biological membranes, pulsed electrophoresis, and chromatography.     

Interaction of a shock wave with a loose dusty bulk layer

The interaction of a planar shock wave with a loose dusty bulk layer has been investigated both experimentally and numerically. Experiments were conducted in a shock tube. The incident shock wave velocity and particle diameters were measured with the use of pressure transducers and a Malvern particle sizer, respectively. The flow fields, induced by shock waves, of both gas and granular phase were visualized by means of shadowgraphs and pulsed X-ray radiography with trace particles added. In addition, a two-phase model for granular flow presented by Gidaspow is introduced and is extended to describe such a complex phenomenon. Based on the kinetic theory, such a two-phase model has the advantage of being able to clarify many physical concepts, like particulate viscosity, granular conductivity and solid pressure, and deduce the correlative constitutive equations of the solid phase. The AUSM scheme was employed for the numerical calculation. The flow field behind the shock wave was displayed numerically and agrees well with our corresponding experimental results.      

Water-lubricated transport of high-viscosity oil in horizontal pipes: The water holdup and pressure gradient

This paper has investigated the water holdup and the pressure gradient of water-lubricated transport of high-viscosity oil flow in horizontal pipes. Experimental results on the water holdup and the pressure gradient of water-lubricated high-viscosity oil two-phase flow in a horizontal 1 in. pipe were discussed. Models for the prediction of the water holdup and/or the pressure gradient of core flow or water-lubricated flow were reviewed and evaluated. It was found that the water holdup of the water-lubricated flow is not only closely related to the input water volume fraction but also the degree of the oil phase eccentricity which is attributed to the oil phase Froude number. This can explain the inconsistency of the experimental results with regard to the relationship between the water holdup and the input water volume fraction in the literature. The applicability of the existing empirical or mechanistic models of water-lubricated high-viscosity oil flow were discussed and demonstrated. A modified correlation to the water holdup correlation of Arney et al. (1993) which was shown to be exclusively applicable for concentric core flow was introduced for stable water-lubricated flow, including both concentric and eccentric core flows. This correlation was evaluated and a fair applicability was shown. The accuracy of different models for the prediction of the pressure gradient of water-lubricated transport of high-viscosity oil was demonstrated to be not high in general. This is closely associated with the difficulty in accurately accounting for the influence of oil fouling on the pressure gradient.