Two-phase flow in heterogeneous porous media: The method of large-scale averaging

The analysis of two-phase flow in porous media begins with the Stokes equations and an appropriate set of boundary conditions. Local volume averaging can then be used to produce the well known extension of Darcy's law for two-phase flow. In addition, a method of closure exists that can be used to predict the individual permeability tensors for each phase. For a heterogeneous porous medium, the local volume average closure problem becomes exceedingly complex and an alternate theoretical resolution of the problem is necessary. This is provided by the method of large-scale averaging which is used to average the Darcy-scale equations over a region that is large compared to the length scale of the heterogeneities. In this paper we present the derivation of the large-scale averaged continuity and momentum equations, and we develop a method of closure that can be used to predict the large-scale permeability tensors and the large-scale capillary pressure. The closure problem is limited by the principle of local mechanical equilibrium. This means that the local fluid distribution is determined by capillary pressure-saturation relations and is not constrained by the solution of an evolutionary transport equation. Special attention is given to the fact that both fluids can be trapped in regions where the saturation is equal to the irreducible saturation, in addition to being trapped in regions where the saturation is greater than the irreducible saturation. Theoretical results are given for stratified porous media and a two-dimensional model for a heterogeneous porous medium.     

Derivation of a Darcy’s Law for a Porous Medium Composed of Two Solid Phases Saturated by a Single- Phase Fluid: A Homogenization Approach

The objective of this article is to derive a macroscopic Darcy’s law for a fluid-saturated moving porous medium whose matrix is composed of two solid phases which are not in direct contact with each other (weakly coupled solid phases). An example of this composite medium is the case of a solid matrix, unfrozen water, and an ice matrix within the pore space. The macroscopic equations for this type of saturated porous material are obtained using two-space homogenization techniques from microscopic periodic structures. The pore size is assumed to be small compared to the macroscopic scale under consideration. At the microscopic scale the two weakly coupled solids are described by the linear elastic equations, and the fluid by the linearized Navier–Stokes equations with appropriate boundary conditions at the solid–fluid interfaces. The derived Darcy’s law contains three permeability tensors whose properties are analyzed. Also, a formal relation with a previous macroscopic fluid flow equation obtained using a phenomenological approach is given. Moreover, a constructive proof of the existence of the three permeability tensors allows for their explicit computation employing finite elements or analogous numerical procedures.     

Theoretical Analysis of Anion Exclusion and Diffusive Transport Through Platy-Clay Soils

Diffusive transport through geosynthetic clay liners and engineered compacted clay landfill liners is the primary mechanism for mass transport from well-engineered modern landfills. For this reason, accurate estimates of diffusion coefficients for clay soils are essential for the design of engineered liner systems. A long-standing theoretical problem is the role of anion exclusion on the estimation of diffusion coefficients for ionic solutes migrating through charged porous media. This paper describes the steady-state solution of a fully coupled set of transport equations modeling ion movement through a permanently charged platy-clay soil. The microscale analysis takes into account the actual diffusion coefficient for each ion species, ion-pairing (as required by electroneutrality of the solution), as well as anion exclusion and cation inclusion ,arising from the permanent charge on clay particles. To render the problem tractable, the theoretical analysis focuses on an extremely small two-dimensional unit cell in an ideal, saturated, two-phase porous medium. The analysis presented here is limited to a particular geometrical example, but this example is sufficiently general for characteristic behaviours of systems of this kind to be identified. Most importantly, new insight is gained into the mechanism of ion migration through a charged platy-clay soil. The numerical results obtained from this study show that the identification of macroscopic transport quantities such as effective diffusion coefficients and membrane potentials from diffusion cell tests using standard diffusion theory only hold for a specific system. While ion exclusion behaviours are often referred to in the literature, as far as the authors are aware there has been no previous detailed microscale analysis of their role in steady-state diffusion through a charged platy-clay soil.     

Some discussions on finite deformation of continuous media

Three problems are discussed in this paper:1.The physical significance of the characteristic tensorθ~ii of finitedeformation is discussed as a complement to the literature.2.The four characterietic tensors of finite deformation,introduced inthe.literature,ard analyzed and discussed more thoroughly.3.The representation of the general finite deformation through simple.loading process is not always possible,the condition for its realization beingthat the given finite deformation satisfies the equations of compatibility.It is pointed out in this paper that for the finite deformation set forth byL.I.Sedov in an illustrative example in[9],the equations of compatibilitycannot be satisfied even for k=1,henceforth for whatever value of k,thefinite deformation set forth by Sedov can not be represented through simpleloading process.     

A Representation Theorem for Material Tensors of?Weakly-Textured Polycrystals and?Its Applications in?Elasticity

Material tensors pertaining to polycrystalline aggregates should manifest also the influence of crystallographic texture on the material properties in question. In this paper we make use of tensors which form bases of irreducible representations of the rotation group and prove a representation theorem by which a given material tensor of a weakly-textured polycrystal is expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a set of undetermined material parameters. Once the irreducible basis tensors that appear in the formula are determined, the representation formula, which is valid for all texture and crystal symmetries, will delineate quantitatively the effect of crystallographic texture on the material tensor in question. We present an integral formula and an orthonormalization process which serve as the basis for a procedure to determine explicitly the irreducible basis tensors required in the representation formula. For applications we determine a set of irreducible basis tensors for the elasticity tensor and a set for fourth-order tensors that define constitutive equations in incompressible elasticity and Hill’s quadratic yield functions in plasticity. We show that orientation averaging of a tensor can be done easily if we have in hand a set of irreducible basis tensors for the decomposition of the tensor in question. As illustration we derive a formula, which is valid for all texture and crystal symmetries, for the elasticity tensor under the Voigt model.     

Generalized Darcy's Law and the Structure of the Phase and Relative Phase Permeabilities for Two-Phase Flows through Anisotropic Porous Media

For two-phase immiscible fluid flows a generalized Darcy's law is written in invariant tensor form for crystallographic point symmetry groups and anisotropic textures. The representation of the phase permeability coefficient tensors and the structure of the expressions for the relative phase permeabilities are analyzed for all symmetry groups. The relation between the phase and absolute permeability coefficient tensors is specified by a fourth-rank tensor with the external symmetry coinciding with external symmetry of the phase permeability tensors. It is shown that the external symmetry of the phase permeability coefficient tensors can differ from the external symmetry of the absolute permeability tensor. For triclinic and monoclinic symmetry groups it is shown that the phase permeability coefficient tensors may not be coaxial with each other and with the absolute permeability tensor; moreover, the directions of the principal axes of the phase permeability coefficient tensors can depend on the saturation.     

A structure-based model for transport in stably stratified homogeneous turbulent flows

We present an extension that allows a recently proposed structure-based model for turbulent scalar transport to account for buoyancy effects. The proposed model is based on a generalization of the Interactive Particle Representation Model (IPRM) and is accompanied by a four-equation transport model that provides the turbulence scales needed for the closure of the complete structure-based model (SBM). The structure tensors and their invariants are used to model the additional buoyancy terms that emerge in the four-equation transport equations. Model parameters are set by matching the asymptotic decay exponents in decaying turbulence. The validity of the model is considered for a large number of different types of stably stratified flows at different Richardson numbers (Ri), showing encouraging results. The complete structure-based model achieves fair agreement with LES and DNS predictions for vertical shear in the presence of vertical mean stratification, while the structure tensors are shown to be suitable for use as diagnostic tools for the morphology of highly anisotropic turbulent structures. Additionally, the proposed model is shown to be sensitive to the variation of the inclination angle θ between the direction of the mean velocity gradient and the orientation of the mean scalar gradient. Furthermore, the model correctly predicts that the evolution of the inverse shear parameter is insensitive to the choice of inclination angle, yielding a turbulent Prandtl number close to unity, in accordance with DNS results.     

Rarefied Pure Gas Transport in Non-isothermal Porous Media: Effective Transport Properties from Homogenization of the Kinetic Equation

Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This article develops a model based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described e.g., by a digitized 3D image. The homogenization procedure—neglecting the effect of gas density gradients on heat transfer through the solid—leads to macroscopic transfer relations, and to closure problems in for the obtention of effective properties. Coherence of the approach with previous literature on the subject is discussed. The asymptotic limits of the model (rarefied and continuum regimes) are also studied. One of the main results is that the effect of the geometry on thermal transpiration has to be described by a tensor, which is distinct from the permeability and Knudsen diffusion tensors.     

Relation between the stress and couple-stress tensors in the microcontinuum theory of elasticity

The stress tensor is expressed in terms of an arbitrary symmetric tensor field of second rank and the couple-stress tensor. The stress and couple-stress tensors are represented by arbitrary tensor fields satisfying the homogeneous equilibrium equations. These tensors are also given in the form of the expressions satisfying the inhomogeneous equilibrium equations used in the microcontinuum theory of elasticity. The stress tensor functions are considered.     

Phase equilibrium in non-fluids and non-fluid mixtures

A criterion of phase equilibrium for mixtures of materials with arbitrary symmetry (e.g. between solid and fluid or two solid mixture phases) is deduced using a rational thermodynamics approach. This criterion, known also as Maxwell relation, is expressed via the difference of chemical potential tensors (Eshelby tensors) on the singular surface dividing the bulk phases.The thermomechanical balance equations, the entropy inequality and the Maxwell relation for phase equilibrium are given first for the case of pure (one-constituent) materials of arbitrary symmetry and then for the case of mixtures (including chemically reacting ones) of arbitrary symmetry.In the special case of fluids it is shown that the chemical potential tensors reduce to the classical scalar chemical potentials and the Maxwell relations to the classical thermodynamic criterion for the phase equilibrium.     

n the fourth order tensor valued function of the stress in return map algorithm

针对各向同性材料,基于一组相互正交的基张量,建立了一套有 效的相关运算方法. 基张量中的两个分别是归一化的二阶单位张量和偏应力张量,另一个则 使用应力的各向同性二阶张量值函数经过归一化构造所得,三者共主轴. 根据张量函数表示 定理,本构方程和返回映射算法中所涉及到的应力的二阶、四阶张量值函数及其逆都由这组 基所表示. 推演结果表明：这些张量之间的运算,表现为对应系数矩阵之间的简单 关系. 其中,四阶张量求逆归结为对应的3\times3系数矩阵求逆,它对二阶张量的变换 则表现为该矩阵对3times 1列阵的变换. 最后,对这些变换关系应用于返回映 射算法的迭代格式进行了相关讨论.     

Elements of a finite strain-gradient thermomechanical theory for material growth and remodeling

A second-gradient theory in finite strains is proposed to deal with the phenomena of material growth and remodeling, as happens in biomechanics, on account of mass transport and morphogenetic species. It involves first-order and second-order transplants (local structural rearrangements) and two material connections on the material manifold. It is shown that the evolution of these structural changes or “material inhomogeneities” is governed by Eshelby-like stress and hyperstress tensors. A thermodynamically admissible set of constitutive equations is proposed. The complexity due to the finite-strain gradient theory is a necessity in order to accommodate mass exchanges and diffusion of species.     

Characterizing the Gas Permeability of Natural and Synthetic Materials

The objective of this article is to propose an experimental method to compare the gas permeability of all the different materials used as gas barrier, such as compacted clay liners or geomembranes. This method is based on the falling pressure experiment, allowing the determination of a single coefficient whatever the material tested. This coefficient is the time constant τ, which is obtained by analytical solutions of the simplified equations describing the transport of gas through the material. The domain of validity is specified for porous media, thanks to a numerical solution of the complete equations system. Two applications are presented, showing the applicability of the method on compacted clay liners and on high density polyethylene geomembranes.     

EHD equations and transport coefficients in a strong electric field

The methods of the kinetics theory are used to obtain a closed system of equations describing the behavior of a multicomponent, partially ionized gaseous mixture in an electromagnetic field in which the space charge is significant. A criterion is presented that makes it possible to separate the problem of finding the magnetic field from that of finding the other defining parameters. Expressions are obtained for the viscosity stress tensors and the thermal and diffusive fluxes; the transport coefficients are calculated in the strong electric field. The relations for the friction force and the energy exchange between components during particle collisions are analyzed. The equations for a mixture consisting of neutral particles and charged particles of a single sign are discussed in detail. The dimensionless EHD criteria are written and analyzed. Possible simplifications of the system of equations are examined, and various forms of Ohm's law are discussed. Weak discontinuities in EHD are analyzed. The equations of EHD under various assumptions have also been considered in several studies^* and in [1–3].     

Thermodynamics of continuous media with intrinsic rotation and magnetoelectric coupling

The thermodynamics of an electrically charged, multicomponent continuous medium with intrinsic rotation is analysed in the presence of electromagnetic fields with a weak linear magnetoelectric coupling in the non-relativistic limit. Taking into account the chemical composition of the current densities and stress tensors yields scalar dissipation terms accounting for chemical reactivities and vectorial dissipation terms accounting for transport. Three equations characterising the continuous medium are derived: a thermostatic equilibrium equation, a reversible and an irreversible thermodynamic evolution equation. Explicit expressions for the temperature and the chemical potentials are derived in terms of the electromagnetic fields and the magnetoelectric coupling. The transport equations contain electromagnetic terms normally not included in a standard thermodynamic phenomenology.     

Articular cartilage with intra- and extrafibrillar waters – mass transfer and generalized diffusion

The present analysis complements the chemo-mechanical model of articular cartilage developed in Loret and Simões [Loret, B., Simões, F.M.F., 2004. Articular cartilage with intra- and extrafibrillar waters. A chemo-mechanical model. Mech. Mater. 36 (5–6), 511–541; Loret, B., Simões, F.M.F., 2005a. Mechanical effects of ionic replacements in articular cartilage. Part I – The constitutive model. Biomech. Model. Mechanobiol. 4 (2–3), 63–80. Part II – Simulations of successive substitutions of NaCl and CaCl₂, 81–99], where only equilibria were considered, and therefore time was absent. The focus here is, first, to present how transport phenomena are aggregated to the porous media framework, and, second, to detail the constitutive equations of these transports. Indeed, these equations are developed in the context of a three-phase multi-species electro-chemo-mechanical model that accounts for the effects of two water compartments, namely intrafibrillar water stored between collagen fibrils and extrafibrillar water covering the negatively charged proteoglycans. The electrolyte circulating the two fluid phases contains ions sodium Na⁺, calcium Ca²⁺ and chloride Cl⁻.Species diffuse within their phase. They transfer from one fluid phase to the other. The various sources of dissipation are built in a thermodynamic framework, segregated and decoupled via the Clausius–Duhem inequality.Linear and non-linear equations of mass transfer are proposed along an onsagerist approach.The generalized diffusion in the extrafibrillar compartment accounts for Darcy's law of seepage through the porous solid skeleton, Fick's law of ionic diffusion, and Ohm's law of electric flow. An original derivation of the constitutive equations of generalized diffusion is proposed. Indeed, the dissipation inequality is written in two forms, which are required to be equivalent. This approach has the advantage of delivering the general structure of the diffusion matrix. It also displays in explicit form the degrees of freedom for possible refinements. Simple assumptions, phrased in terms of entities that are standard in transport of porous media, allow to recover arrowhead diffusion matrices. Comparison with an earlier proposal is detailed.An osmotic coefficient is found to be hidden in the equations, and anomalous negative osmosis is observed to take place for both sodium chloride and calcium chloride electrolytes.Finally, an experimental setup to measure transport properties is analyzed. The model describes correctly the increase and leveling of the experimental diffusion coefficient, and no additional ad hoc constitutive assumptions are needed in contrast to some suggestions in the literature.The results are presented for sodium chloride NaCl and calcium chloride CaCl₂.     

Objective Tensor Rates and Applications in Formulation of Hyperelastic Relations

Following Ogden, a class of objective (Lagrangian and Eulerian) tensors is identified among the second-rank tensors characterizing continuum deformation, but a more general definition of objectivity than that used by Ogden is introduced. Time rates of tensors are determined using convective rates. Sufficient conditions of objectivity are obtained for convective rates of objective tensors. Objective convective rates of strain tensors are used to introduce pairs of symmetric stress and strain tensors conjugate in a generalized sense. The classical definitions of conjugate Lagrangian (after Hill) and Eulerian (after Xiao et al.) stress and strain tensors are particular cases of the definition of conjugacy of stress and strain tensors in the generalized sense used in the present paper. Pairs of objective stress and strain tensors conjugate in the generalized sense are used to formulate constitutive relations for a hyperelastic medium. A family of objective generalized strain tensors is introduced, which is broader than Hill’s family of strain tensors. The basic forms of the hyperelastic constitutive relations are obtained with the aid of pairs of Lagrangian stress and strain tensors conjugate after Hill (the strain tensors in these pairs belong to the family of generalized strain tensors). A method is presented for generating reduced forms of the constitutive relations with the aid of pairs of Lagrangian and Eulerian stress and strain tensors conjugate in the generalized sense which are obtained from pairs of Lagrangian tensors conjugate after Hill by mapping tensor fields on one configuration of a deformable body to tensor fields on another configuration.      

A control volume finite difference method for buoyant flow in three-dimensional curvilinear non-orthogonal co-ordinates

This paper presents a control-volume-based finite difference method in non-orthogonal curvilinear coordinates on a local basis in which the vectors and tensors are all based on the general curvilinear coordinates for buoyant flow calculations in arbitrary three-dimensional geometries. The governing equations are transformed from Cartesian co-ordinates into generalized curvilinear co-ordinates. After integrating the set of equations for the control volumes, the finite difference equations are then formulated by a proper treatment of the heat flux and stress tensors and by incorporating the QUICK scheme for the convective terms. The solution procedure then follows the one for three-dimensional Cartesian co-ordinates. Examples are given in problems of natural convection in such three-dimensional enclosures as parallelepiped enclosures and horizontal closed cylinders with differentially heated ends. In the latter case, important applications have been found in crystal growth by means of chemical vapour deposition in a cylindrical ampoule, in which uniform heat fluxes along the two ends are required in order to produce high-quality crystals. Special attention is given to the insertion of baffles in the cylinder to improve the recirculating flow patterns near the two ends.     

Piezoeffect in polar materials using moment theory

A method for describing the piezoelectric effect in a polar material is proposed based on the use of a composite particle model with seven degrees of freedom and a nonzero dipole moment. Based on micropolar theory, a system of equations is obtained which differs from the classical theory of piezoelectricity in the presence of additional terms. It is shown that under certain assumptions, the proposed system of equations becomes the classical system but the piezoelectric moduli depend strongly on the spontaneous polarization vector. It is shown that for anisotropic media with different symmetries, the structure of the third-rank piezoelectric tensors obtained using the proposed micropolar theory coincides with the structure of the tensors obtained using classical theory. For the media considered, dispersion relations are given and it is shown that in the proposed theory, unlike classical theory, the piezoelectric moduli are proportional to the spontaneous polarization.     

THE LINEAR BI-SPATIAL TENSOR EQUATIONφ_(ij)A~iXB~j= C

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