On the transient non-Fickian dispersion theory

The Fickian dispersion equation is the basic relationship used to describe the nonconvective mass flux of a solute in a porous medium. This equation prescribes a linear relationship between the dispersive mass flux and the concentration gradient. An important characteristic of the Fickian relationship is that it is independent of the history of dispersion (e.g. the time rate of change of the dispersion flux). Also, the dispersivities are supposed to be medium constants and invariant with temporal and spatial scales of observation. It is believed that in general these restrictions do not hold. A number of authors have proposed various alternative relationships. For example, differential equations have been employed that prescribe a relationship between the dispersion flux and its time and space derivatives. Also, stochastic theories result in integro-differential equations in which dispersion tensor grow asymptotically with time or distance. In this work, three different approaches, which lead to three different non-Fickian equations with a transient character, are discussed and their primary features and differences are highlighted. It is shown that an effective dispersion tensor defined in the framework of the transient non-Fickian theory, grows asymptotically with time and distance; a result which also follows from stochastic theories. Next, principles of continuum mechanics are employed to provide a solid theoretical basis for the non-Fickian transient dispersion theory. The equation of motion of a solute in a porous medium is used to provide a rigorous derivation of various dispersion relationships valid under different conditions. Under various simplifying assumptions, the generalized theory is found to agree with the conventional Fickian theory as well as several other non-Fickian relationships found in the literature. Moreover, it is shown that for nonconservative solutes, the traditional dispersion tensor is affected by the rate of mass exchange of the solute.Also with National Institute of Public Health and Environmental Protection (RIVM), PO Box 1; 3720BA Bilthoven, The Netherlands     

Estimation of Macrodispersion by Different Approximation Methods for Flow and Transport in Randomly Heterogeneous Media

We present two- and three-dimensional calculations for the longitudinal and transverse macrodispersion coefficient for conservative solutes derived by particle tracking in a velocity field which is based on the linearized flow equation. The simulations were performed upto 5000 correlation lengths in order to reach the asymptotic regime. We used a simulation method which does not need any grid and therefore allows simulations of very large transport times and distances.Our findings are compared with results obtained from linearized transport, from Corrsin's Conjecture and from renormalization group methods. All calculations are performed with and without local dispersion. The variance of the logarithm of the hydraulic conductivity field was chosen to be one to investigate realistic model cases.While in two dimensions the linear transport approximation seems to be very good even for this high variance of the logarithmic hydraulic conductivity, in three dimensions renormalization group results are closer to the numerical calculations. Here Dagan's theory and the theory of Gelhar and Axness underestimate the transverse macrodispersion by far. Corrsin's Conjecture always overestimates the transverse dispersion. Local dispersion does not significantly influence the asymptotic behavior of the various approximations examined for two-dimensional and three-dimensional calculations.     

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.     

Method of homogenization applied to dispersion in porous media

The theory of homogenization which is a rigorous method of averaging by multiple scale expansions, is applied here to the transport of a solute in a porous medium. The main assumption is that the matrix has a periodic pore structure on the local scale. Starting from the pores with the Navier-Stokes equations for the fluid motion and the usual convective-diffusion equation for the solute, we give an alternative derivation of the three-dimensional macroscale dispersion tensor for solute concentration. The original result was first found by Brenner by extending Brownian motion theory. The method of homogenization is an expedient approach based on conventional continuum equations and the technique of multiple-scale expansions, and can be extended to more complex media involving three or more contrasting scales with periodicity in every but the largest scale.     

The Role of Probabilistic Approaches to Transport Theory in Heterogeneous Media

A physical picture of contaminant transport in highly heterogeneous porous media is presented. In any specific formation the associated governing transport equation is valid at any time and space scale. Furthermore, the advective and dispersive contributions are inextricably combined. The ensemble average of the basic transport equation is equivalent to a continuous time random walk (CTRW). The connection between the CTRW transport equation, in a limiting case and the familiar advection–dispersion equation (ADE) is derived. The CTRW theory is applied to the results of laboratory experiments, field observations, and simulations of random fracture networks. All of these results manifest dominant non-Gaussian features in the transport, over different scales, which are accounted for quantitatively by the theory. The key parameter controlling the entire shape of the contaminant plume evolution and breakthrough curves is advanced as a more useful characterization of the transport than the dispersion tensor, which is based on moments of the plume. The role of probabilistic approaches, such as CTRW, is appraised in the context of the interplay of spatial scales and levels of uncertainty. We then discuss a hybrid approach, which uses knowledge of non-stationary aspects of a field site on a larger spatial scale (trends) with a probabilistic treatment of unresolved structure on a smaller scale (residues).     

Renormalization Group Method. Applications to Partial Differential Equations

Our aim in this article is to present a simplified form of the renormalization group (RG) method introduced by Chen, Goldenfeld, and Oono and to derive a rigorous study of the validity in time of the asymptotic solutions furnished by the RG method. We apply the renormalization group method to a slightly compressible fluid equation and to the Swift–Hohenberg equation.     

Analytical Solutions for Macrodispersion in a 3D Heterogeneous Porous Medium with Random Hydraulic Conductivity and Dispersivity

A stochastic analysis of macrodispersion for conservative solute transport in three-dimension (3D) heterogeneous statistically isotropic and anisotropic porous media when both hydraulic conductivity and local dispersivity are random is presented. Analytical expressions of macrodispersivity are derived using Laplace and Fourier transforms. The effects of various parameters such as ratio of transverse to longitudinal local dispersivity, correlation length ratio, correlation coefficient and direction of flow on asymptotic macrodispersion are studied. The behaviour of growth of macrodispersivity in preasymptotic stage is also shown in this paper. The variation in local dispersion coefficient causes change in transverse macrodispersivity. The consideration of random dispersivity along with random hydraulic conductivity indicates that the total dispersion is affected and important in the case when the hydraulic conductivity and dispersivity are correlated. It is observed that the pre-asymptotic behavior of the macrodispersivity is not sensitive to the choice of spectral density functions.     

A new two-point deformation tensor and its relation to the classical kinematical framework and the stress concept

Starting from the issue of what is the correct form for a Legendre transformation of the strain energy in terms of Eulerian and two-point tensor variables we introduce a new two-point deformation tensor, namely H=(F−F^−T)/2, as a possible deformation measure involving points in two distinct configurations. The Lie derivative of H is work conjugate to the first Piola–Kirchhoff stress tensor P. The deformation measure H leads to straightforward manipulations within a two-point setting such as the derivation of the virtual work equation and its linearization required for finite element implementation. The manipulations are analogous to those used for the Lagrangian and Eulerian frameworks. It is also shown that the Legendre transformation in terms of two-point tensors and spatial tensors require Lie derivatives. As an illustrative example we propose a simple Saint Venant–Kirchhoff type of a strain-energy function in terms of H. The constitutive model leads to physically meaningful results also for the large compressive strain domain, which is not the case for the classical Saint Venant–Kirchhoff material.     

Characteristic cones of the equations in the nonlinear theory of elasticity

The roots of the equation for the characteristic normals for two systems of differential equations in the nonlinear theory of elasticity are investigated. The first model is constructed using a thermodynamic identity. The second is a very simple hypoelastic model (the deviator of the stress-rate tensor is proportional to the deviator of the strain-rate tensor). It is shown that the roots of the equations for the normals to the characteristics for the second model are the same as the first-order terms in the expansion of the roots of the first model with respect to the strain-tensor deviator.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 126–132, May–June, 1974.The author is grateful to S. K. Godunov for discussions.     

The Smoluchowski equation for colloidal suspensions developed and analyzed through the GENERIC formalism

The Smoluchowski equation (SE) and the mechanical stress tensor for the over-damped dynamics of colloidal particles is derived directly at the pair distribution level starting from a thermodynamic basis using the general equations for equilibrium non-equilibrium reversible and irreversible coupling (GENERIC) formalism. Within the GENERIC formalism, the effect of the non-trivial convection due to hydrodynamic interactions is incorporated for the first time. The method generates a thermodynamically valid set of transport equations for the colloidal dispersion, thus properly identifying the extra stress due to the presence of the colloids. The derivation connects a formal entropy expansion to the many-body terms that arise in both the transport equation and the stress tensor, thus unifying their origin and providing a systematic path forward for improvement in the theory. The analysis identifies the thermodynamically valid stress expression, thus clarifying a long-standing problem in the literature that arises when separate derivations are performed for the transport equations and the stress tensor. The results of previous investigators are analyzed within this framework. Comparison with alternate methods of deriving the many-body Smoluchowski equation provide new insight into the nature of the many-body terms.     

Theory of Short-Term Microdamageability for a Homogeneous Material under Physically Nonlinear Deformation

The structural theory of short-term damageability is generalized to the case of physically nonlinear deformation of an undamaged material. The stochastic elasticity equations for a porous medium whose skeleton deforms nonlinearly are used. The failure criterion for a microvolume of the material is assumed to be in the Huber–Mises form. The microdamage balance equation for a physically nonlinear material is derived. This equation and the macrostress–macrostrain relation for a porous physically nonlinear material constitute a closed-form system describing the coupled processes of physically nonlinear deformation and microdamage. An algorithm is constructed for computing microdamage–macrostrain relationships and plotting deformation curves. Such curves are plotted for the case of uniaxial tension     

Solute dispersion in a pulsating flow

The one-dimensional model proposed by Taylor [1] of the dispersion of soluble matter describes approximately the distribution of the solute concentration averaged over the tube section in Poiseuille flow. Aris [2] obtained more accurately the effective diffusion coefficient in Taylor's model and solved the problem for the general case of steady flow in a channel of arbitrary section. Many papers have been published in the meanwhile devoted to particular applications of this theory (for example, [3–5]). Various dispersion models have been constructed [6–8] that make the Taylor—Aris model more accurate at small times and agree with it at large times. The acceleration of the mixing of the solute considered in these models in the presence of the simultaneous influence of molecular diffusion and convective transport also operates in unsteady flows. In particular, the presence of velocity pulsations influences the growth of the dispersion even if the mean flow velocity is equal to zero at every point of the flow. In the present paper, the Taylor—Aris theory is extended to the case of laminar flows with periodically varying flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 24–30, September–October, 1982.     

Nonlinear surface waves in media with complex rheology

Weak nonlinear waves in a generalized viscoelastic medium with internal oscillators are considered. The rheological relations contain higher time derivatives of the stresses and strains as well as their tensor products. The method of expansion in a small parameter with the introduction of slow time and a running space coordinate is employed. The first approximation gives wave velocities and relations between the parameters equivalent to the results of an acoustic analysis at elastic wave fronts [1]. The second approximation leads to an evolution equation for the displacement velocity. For this a Fourier-Laplace double integral transformation is used. Reversion to the inverse transforms of the unknown functions leads to an integrodifferential evolution equation, which contains a Hubert transform and is a generalization of the Benjamin-Ono equation of deep water theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 95–103, September–October, 1990.     

Dispersion de Taylor généralisée à un fluide à propriétés physiques variablesGeneralized Taylor dispersion for heterogeneous fluid

The investigation of non-reactive miscible solute dispersion in a vertical Hele–Shaw cell is considered. An asymptotic method is used to extend Taylor model to the case of the fluid density, the dynamic viscosity and the molecular diffusion coefficient are solute concentration-dependent. It is demonstrated that the averaged variables over the gap are governed by a convection–dispersion equation in which the dispersion tensor is concentration-dependent. To cite this article: C. Felder et al., C. R. Mecanique 332 (2004).     

An Approach to Transport in Heterogeneous Porous Media Using the Truncated Temporal Moment Equations: Theory and Numerical Validation

In the last decade, the characterization of transport in porous media has benefited largely from numerical advances in applied mathematics and from the increasing power of computers. However, the resolution of a transport problem often remains cumbersome, mostly because of the time-dependence of the equations and the numerical stability constraints imposed by their discretization. To avoid these difficulties, another approach is proposed based on the calculation of the temporal moments of a curve of concentration versus time. The transformation into the Laplace domain of the transport equations makes it possible to develop partial derivative equations for the calculation of complete moments or truncated moments between two finite times, and for any point of a bounded domain. The temporal moment equations are stationary equations, independent of time, and with weaker constraints on their stability and diffusion errors compared to the classical advection–dispersion equation, even with simple discrete numerical schemes. Following the complete theoretical development of these equations, they are compared firstly with analytical solutions for simple cases of transport and secondly with a well-performing transport model for advective–dispersive transport in a heterogeneous medium with rate-limited mass transfer between the free water and an immobile phase. Temporal moment equations have a common parametrization with transport equations in terms of their parameters and their spatial distribution on a grid of discretization. Therefore, they can be used to replace the transport equations and thus accelerate the achievement of studies in which a large number of simulations must be carried out, such as the inverse problem conditioned with transport data or for forecasting pollution hazards.     

Shear wave dispersion and attenuation in periodic systems of alternating solid and viscous fluid layers

The attenuation and dispersion of elastic waves in fluid-saturated rocks due to the viscosity of the pore fluid is investigated using an idealized exactly solvable example of a system of alternating solid and viscous fluid layers. Waves in periodic layered systems at low frequencies are studied using an asymptotic analysis of Rytov’s exact dispersion equations. Since the wavelength of shear waves in fluids (viscous skin depth) is much smaller than the wavelength of shear or compressional waves in solids, the presence of viscous fluid layers necessitates the inclusion of higher terms in the long-wavelength asymptotic expansion. This expansion allows for the derivation of explicit analytical expressions for the attenuation and dispersion of shear waves, with the directions of propagation and of particle motion being in the bedding plane. The attenuation (dispersion) is controlled by the parameter which represents the ratio of Biot’s characteristic frequency to the viscoelastic characteristic frequency. If Biot’s characteristic frequency is small compared with the viscoelastic characteristic frequency, the solution is identical to that derived from an anisotropic version of the Frenkel–Biot theory of poroelasticity. In the opposite case when Biot’s characteristic frequency is greater than the viscoelastic characteristic frequency, the attenuation/dispersion is dominated by the classical viscoelastic absorption due to the shear stiffening effect of the viscous fluid layers. The product of these two characteristic frequencies is equal to the squared resonant frequency of the layered system, times a dimensionless proportionality constant of the order 1. This explains why the visco-elastic and poroelastic mechanisms are usually treated separately in the context of macroscopic (effective medium) theories, as these theories imply that frequency is small compared to the resonant (scattering) frequency of individual pores.     

Macrodispersivity and Large-scale Hydrogeologic Variability

Although groundwater velocities vary over a wide range of spatial scales it is generally only feasible to model the largest variations explicitly. Smaller-scale velocity variability must be accounted for indirectly, usually by increasing the magnitude of the dispersivity tensor (i.e. by introducing a so-called macrodispersivity). Most macrodispersion theories tacitly assume that a macrodispersivity tensor which works well when there is only small-scale velocity variability will also work well when there is larger-scale variability. We analyze this assumption in a high resolution numerical experiment which simulates solute transport through a two-scale velocity field. Our results confirm that a transport model which uses an appropriately adjusted macrodispersivity can reproduce the large-scale features of a solute plume when the velocity varies only over small scales. However, if the velocity field includes both small and large-scale components, the macrodispersivity term does not appear to be able to capture all of the effects of small-scale variability. In this case the predicted plume is more well mixed and consistently underestimates peak solute concentrations at all times. We believe that this result can be best explained by scale interactions resulting from the nonlinear transformation from velocity to concentration. However, additional analysis will be required to test this hypothesis.     

On the general equation and the general solution in problems for plastodynamics with rigid-plastic material

This work is the continuation of the discussion of refs. [1–2]. We discuss the dynamics problems of ideal rigid — plastic material in the flow theory of plasticity in this paper. From introduction of the theory of functions of complex variable under Dirac-Pauli representation we can obtain a group of the so-called general equations (i.e. have two scalar equations) expressed by the stream function and the theoretical ratio. In this paper we also testify that the equation of evolution for time in plastodynamics problems is neither dissipative nor disperive, and the eigen-equation in plastodynamics problems is a stationary Schrödinger equation, in which we take partial tensor of stress-increment as eigenfunctions and take theoretical ratio as eigenvalues. Thus, we turn nonlinear plastodynamics problems into the solution of linear stationary Schrödinger equation, and from this we can obtain the general solution of plastodynamics problems with rigid-plastic material.     

Role of local and nonlocal interactions in the formation of the developed turbulence regime

The successful use of the renormalization group method for calculating the universal constants of developed turbulence has provoked a discussion on the extent to which the results obtained correspond to the ideas of Kolmogorov's theory of localization of the intermodel couplings, since the computational procedure employed was based on consideration of the essentially nonlocal direct effect of the small-scale on the large-scale modes. Within the framework of a field-theory approach, it is shown that the use of the renormalization group method in conjunction with the -expansion in fact means taking into account the local and filtering out the nonlocal intermodel interactions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 36–42, July–August, 1990.     

A Note on Rivlin’s Identities and Their Extension

An analytic method is developed for the derivation of a series of tensor identities that include Rivlins identities as a special case. The derivation is based on taking the derivatives of the Cayley–Hamilton equation. The identities generated involve multiple tensors on the n-dimensional vector space. Mathematics Subject Classifications (2000) 15A24, 74A99.Guansuo Dui: Present address: Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing, 100044, P. R. China.