Macrokinetic model of the trapping process in two-phase fluid displacement in a porous medium

A microinhomogeneity-averaged model of the kinetics of the trapping process is proposed for a porous medium in which two fluids are mutually displaced. The traps are treated as a new hydrodynamic phase, and the trapping process as a phase transition. Kinetic relations for the average trapping process are obtained. The structure and quantitative values of the kinetic coefficients are obtained for a model of a porous medium in the form of a system of doublets. The dependence of the characteristic time of the process on the degree of inhomogeneity of the medium is investigated. A variant of the macroscopic model of the process of two-phase flow, in which the kinetic relations obtained are used as the closing relations, is proposed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 92–101, May–June, 1995.     

Upscaling of Diffusion–Reaction Phenomena by Homogenisation Technique: Possible Appearance of Morphogenesis

Transport in Porous Media - The main objective of this work is to describe reaction–diffusion of two species in a porous medium. We aim at finding the macroscopic model equivalent to the...     

Diffusion–Convection in a Porous Medium with Impervious Inclusions at Low Flow Rates

The aim of this paper is to develop a macroscopic model for the transport of a passive solute, by diffusion and convection, in a heterogeneous medium consisting of impervious solids periodically distributed in a porous matrix. In the porous part, the flow is described by Darcy's law. Attempt is made to derive the macroscopic equation governing the average concentration field in the equivalent macroscopic medium and the macroscopic transport parameters. The analysis is conducted in the case when convection and diffusion are of the same order of magnitude at the macroscopic level, that is, when the Péclet number is of order 1. The proposed macroscopic model is obtained using the homogenization method for periodic structures with a double scale asymptotic expansion, in which the small parameter is the ratio between the two characteristic lengths l (the period scale of the impervious bodies distribution) and L (the scale of the macroscopic sample). The macroscopic parameters, which characterize the multiporous medium, depend solely on the transport parameters in the porous matrix and on the geometry of the impervious inclusions without any phenomenological assumption. Numerical computations are performed using a finite element method for several geometries of the solid inclusions, in two- and three-dimensional cases.     

Kinetic models for dilute solutions of dumbbells in non-homogeneous flows revisited

We propose a two-fluid theory to model a dilute polymer solution assuming that it consists of two phases, polymer and solvent, with two distinct macroscopic velocities. The solvent phase velocity is governed by the macroscopic Navier–Stokes equations with the addition of a force term describing the interaction between the two phases. The polymer phase is described on the mesoscopic level using a dumbbell model and its macroscopic velocity is obtained through averaging. We start by writing down the full phase-space distribution function for the dumbbells and then obtain the inertialess limits for the Fokker–Planck equation and for the averaged friction force acting between the phases from a rigorous asymptotic analysis. The resulting equations are relevant to the modelling of strongly non-homogeneous flows, while the standard kinetic model is recovered in the locally homogeneous case.     

On the macroscopic dynamics induced by a model wave-particle collision operator

The macroscopic dynamics of a kinetic equation involving a model wave-particle collision operator of plasma physics is investigated. The Chapman-Enskog asymptotics is first considered in the framework of a hydrodynamic scaling. The obtained macroscopic model still involves a kinetic variable, the particle energy in the rest frame of the fluid, but shares similarities with the compressible Navier-Stokes equation of gas dynamics. Then a diffusive scaling is examined under the hypothesis of small perturbations of a global equilibrium. In this case, the macroscopic model couples the usual incompressible Navier-Stokes with a diffusion equation for the energy distribution function of the particles, constrained by an extended version of the Boussinesq relation. In both cases, the effect of a Lorentz force term is developed, in the perspective of plasma physical modelling. Received June 16, 1997     

Generalized Taylor-Aris moment analysis of the transport of sorbing solutes through porous media with spatially-periodic retardation factor

Taylor-Aris dispersion theory, as generalized by Brenner, is employed to investigate the macroscopic behavior of sorbing solute transport in a three-dimensional, hydraulically homogeneous porous medium under steady, unidirectional flow. The porous medium is considered to possess spatially periodic geochemical characteristics in all three directions, where the spatial periods define a rectangular parallelepiped or a unit-element. The spatially-variable geochemical parameters of the solid matrix are incorporated into the transport equation by a spatially-periodic distribution coefficient and consequently a spatially-periodic retardation factor. Expressions for the effective or large-time coefficients governing the macroscopic solute transport are derived for solute sorbing according to a linear equilibrium isotherm as well as for the case of a first-order kinetic sorption relationship. The results indicate that for the case of a chemical equilibrium sorption isotherm the longitudinal macrodispersion incorporates a second term that accounts for the eflect of averaging the distribution coefficient over the volume of a unit element. Furthermore, for the case of a kinetic sorption relation, the longitudinal macrodispersion expression includes a third term that accounts for the effect of the first-order sorption rate. Therefore, increased solute spreading is expected if the local chemical equilibrium assumption is not valid. The derived expressions of the apparent parameters governing the macroscopic solute transport under local equilibrium conditions agreed reasonably with the results of numerical computations using particle tracking techniques.     

Propagation and scattering of ultrasonic waves in polycrystals with arbitrary crystallite and macroscopic texture symmetries

A general ultrasonic attenuation model for a polycrystal with arbitrary macroscopic texture and triclinic ellipsoidal grains is described with proper accounting for the anisotropic Green’s function for the reference medium. The texture and the ellipsoidal grain frames in the model are independent and the wave propagation direction is arbitrary. The attenuation coefficients are obtained in the Born approximation accompanied by the Rayleigh and stochastic asymptotes. The scattering model displays statistical anisotropy due to two independent factors: (1) shape of the oriented grains and (2) preferred crystallographic orientation of the grains leading to macroscopic anisotropy of the homogenized reference medium. The model is applicable to most single phase polycrystalline materials that may occur as a result of thermomechanical manufacturing processes leading to different macrotextures and elongated-shaped grains. It predicts the strength of ultrasonic scattering and its dependence on frequency and propagation direction as a function of grain shape, grain crystallographic symmetry and macroscopic texture parameters and provides the texture-induced dependence of macroscopic ultrasonic velocity on propagation angle. It considers proper wave polarizations due to macroscopic anisotropy and scattering-induced transformations of waves with different polarizations. Competing effects of grain shape and texture on the attenuation are observed. In contrast to the macroscopically isotropic case, where in the stochastic regime the attenuation is highest in the direction of the longest ellipsoidal axis of the grain, the wave attenuation in the elongation direction may be suppressed or amplified by the texture with different effects on the quasilongitudinal and quasitransverse waves. The frequency behavior is also interestingly affected by texture: a hump in the total attenuation coefficient is found for the fast quasitransverse wave which is purely the result of macroscopic anisotropy and the existence of two quasitransverse waves; this hump is not observed in the macroscopically isotropic case. Striking differences of the texture effect on the directional dependences of the attenuation coefficients are found at low versus high frequencies.     

On the motion of a fluid-saturated porous solid

Two-parameter structure model of a porous solid is proposed as an approximation of a real porous structure and the macroscopic mass and momentum balance equations are derived for such a medium filled with liquid. The approach presented leads to the equations of motion for a fluid-saturated porous medium with coupling terms via cross-mass couplings. The linear form of these equations is equivalent to the well-known Biot equations.     

Turbulent Flow Around Fluid–Porous Interfaces Computed with a Diffusion-Jump Model for <Emphasis Type="Italic">k</Emphasis> and <Emphasis Type="Italic">ε</Emphasis> Transport Equations

Flow over vegetation and bottom of rivers can be characterized by some sort of porous structure of irregular surface through which a fluid permeates. Also, in engineering systems, one can have components that make use of a working fluid flowing over irregular layers of porous material. This article presents numerical solutions for such hybrid medium, considering here a channel partially filled with a flat porous layer saturated by a fluid flowing in turbulent regime. One unique set of transport equations is applied to both the regions. A diffusion-jump model for both the turbulent kinetic energy and its dissipation rate, across the interface, is presented and discussed upon. The discretization steps taken for numerically accommodating such model in the system of algebraic equations are presented. Numerical results show the effects of Reynolds number, porosity, and permeability on mean and turbulence fields. Results indicate that when negative values for the stress jump coefficient are applied, the peak of the turbulent kinetic energy distribution occurs at the macroscopic interface.     

What is behind the plastic strain rate?

The plastic strain rate plays a central role in macroscopic models on elasto-viscoplasticity. In order to discuss the concept behind this quantity, we propose, first, a kinetic toy model to describe the dynamics of sliding layers representative of plastic deformation of single crystalline metals. The dynamic variable is given by the distribution function of relative strains between adjacent layers, and the plastic strain rate emerges as the average hopping rate between energy wells. We demonstrate the behavior of this model under different deformations and how it captures the elastic-to-plastic transition. Second, the kinetic toy model is reduced to a closed evolution equation for the average of the relative strain, allowing us to make a direct link to macroscopic theories. It is shown that the constitutive relation for the plastic strain rate does not only depend on the stress, but also on the macroscopic applied deformation rate, contrary to common practice.     

A dual-scale coupled micro/macro segregation model for dendritic alloy solidification

The present article reports on the formulation, numerical implementation, and application of a single-domain coupled micro/macroscopic model for simulation of dendritic alloy solidification. Microscopic solutal non-equilibrium effects have been included in the macroscopic modeling of solidification by using a fixed grid dual scale numerical approach. Salient features of the present approach include a continuum model for conservation of mass, momentum, energy and species on the macroscopic scale, a microscopic solute redistribution model, and the solution procedure and auxiliary equations necessary for coupling between the two models. The coupling between macro and micro scale models is practically made possible by introducing an iterative micro/macro time step scheme. The local solidification rate is calculated by implicit iterations of macroscopic conservation equations and the microscopic solute redistribution model. The present model is capable to simulate eutectic reaction, local re-melting, and account for the inter-linkage between micro and macroscopic solute redistribution (micro and macrosegregation).     

Constitutive relationship of brittle rock subjected to dynamic uniaxial tensile loads with microcrack interaction effects

A micromechanical model is proposed to describe both stable and unstable damage evolution in microcrack-weakened brittle rock material subjected to dynamic uniaxial tensile loads. The basic idea of the present model is to classify the constitution relationship of rock material subjected to dynamic uniaxial tensile loads into four stages including some of the stages of linear elasticity, pre-peak nonlinear hardening, rapid stress drop, and strain softening, and to investigate their corresponding micromechanical damage mechanisms individually. Special attention is paid to the transition from structure rearrangements on microscale to the macroscopic inelastic strain, to the transition from distribution damage to localization of damage and the transition from homogeneous deformation to localization of deformation. The influence of all microcracks with different sizes and orientations are introduced into the constitutive relation by using the statistical average method. Effects of microcrack interaction on the complete stress-strain relation as well as the localization of damage for microcrack-weakened brittle rock material are analyzed by using effective medium method. Each microcrack is assumed to be embedded in an approximate effective medium that is weakened by uniformly distributed microcracks of the statistically-averaged length depending on the actual damage state. The elastic moduli of the approximate effective medium can be determined by using the dilute distribution method. Micromechanical kinetic equations for stable and unstable growth characterizing the ‘process domains’ of active microcracks are taken into account. These ‘process domains’ together with ‘open microcrack domains’ completely determine the integration domains of ensemble averaged constitutive equations relating macro-strain and macro-stress. Theoretical predictions have shown to consistent with the experimental results.     

A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity

This paper presents the development of a physical-based constitutive model for the representation of viscous effects in rubber-like materials. The proposed model originates from micromechanically motivated diffusion processes of the highly mobile polymer chains described within the formalism of Brownian motion. Following the basic assumption of accounting for the elastic and the viscous effects in rubber viscoelasticity by their representation through a separate elastic ground network and several viscous subnetworks, respectively, the kinetic theory of rubber elasticity is followed and extended to represent also the viscous contribution in this work. It is assumed that the stretch probability of certain chain segments within an individual viscous subnetwork evolves based on the movement of the chain endpoints described by the Smoluchowski equation extended in this work from non-interacting point particles in a viscous surrounding to flexible polymer chains. An equivalent tensorial representation for this evolution is chosen which allows for the closed form solution of the macroscopic free energy and the macroscopic viscous overstress based on a homogenization over the probability space of the introduced micro-objects. The resulting model of the viscous subnetwork is subsequently combined with the non-affine micro-sphere model and applied in homogeneous and non-homogeneous tests. Finally, the model capacity is outlined based on a comparison with in the literature available experimental data sets.     

Multiscale modeling of a fluid saturated medium with double porosity: Relevance to the compact bone

In this paper, we develop a model of a homogenized fluid-saturated deformable porous medium. To account for the double porosity the Biot model is considered at the mesoscale with a scale-dependent permeability in compartments representing the second-level porosity. This model is treated by the homogenization procedure based on the asymptotic analysis of periodic “microstructure”. When passing to the limit, auxiliary microscopic problems are introduced, which provide the corrector basis functions that are needed to compute the effective material parameters. The macroscopic problem describes the deformation-induced Darcy flow in the primary porosities whereas the microflow in the double porosity is responsible for the fading memory effects via the macroscopic poro-visco-elastic constitutive law. For the homogenization procedure, we use the periodic unfolding method. We discuss also the stress and flow recovery at multiple scales characterizing the heterogeneous material. The model is proposed as a theoretical basis to describe compact bone behavior on multiple scales.     

Elastic behaviour of a regular lattice for meso-scale modelling of solids

A modelling strategy is proposed to link the meso-scale mechanical response of a solid material to the macroscopic material behaviour. The model is based on a regular lattice of truncated octahedral cells, with sites at the cell centres linked by two sets of bonds. The relationship between the macroscopic elastic behaviour of the model and the elastic properties of the bonds is studied numerically. The results demonstrate that, in contrast to previously proposed lattice arrangements, any elastic properties of metallic or cementitious materials can be obtained by appropriate selection of the axial and the shear stiffness of the bonds. Discussion of the modelling approach includes the potential of the site-bond model to simulate the evolution of damage driven not only by mechanical deformation but also by processes that involve the interaction of different mechanisms.     

On the hyperbolicity of a model with 30 moments for ultrarelativistic gases

An exact macroscopic extended model for ultrarelativistic gases, with an arbitrary number of moments, is present in literature. Here we exploit equations determining wave speeds for the model with 30 independent fields. We find interesting results; for example, the whole system for their determination can be split in some independent subsystems, some wave speeds are expressed by square roots of rational numbers, but not all of them. Moreover these wave speeds for the macroscopic model are the same of those in the kinetic model.