Dispersion Tensor Research Articles

Dispersion of mass by two-dimensional homogeneous and incompressible Çinlar flows

We consider the dispersion of passive tracers in stationary, homogeneous and incompressible Çinlar flows on the plane. The associated velocity field is generated by the superposition of eddies of various size, arrival time and location which form a Poisson point process. Our focus is on the dispersion of a tracer cloud, which is measured through the variance of its centroid and the mean of the dispersion tensor. We also study single particle dispersion and particle pair separations in conjunction with dispersion. Monte Carlo simulations of all these measures first establish the relation of the dispersion to the parameters of the flow model. Second, the physical predictions on the behavior of these measures with respect to time as well as their relationship to each other are confirmed.

Volume averaging for determining the effective dispersion tensor: Closure using periodic unit cells and comparison with ensemble averaging

In this work, we use the method of volume averaging to determine the effective dispersion tensor for a heterogeneous porous medium; closure for the averaged equation is obtained by solution of a concentration deviation equation over a periodic unit cell. Our purpose is to show how the method of volume averaging with closure can be rectified with the results obtained by other upscaling methods under particular conditions. Although this rectification is something that is generally believed to be true, there has been very little research that explores this issue explicitly. We show that under certain limiting (but mild) assumptions, the closure problem provides a Fourier series solution for the effective dispersion tensor. When second‐order spatial stationarity is imposed on the velocity field, the method yields a simple Fourier series that converges to an integral form in the limit as the period of the unit cell approaches infinity. This limiting result is identical to the quasi‐Fickian forms that have been developed previously via ensemble averaging by Deng et al. [1993] and recently by Fiori and Dagan [2000] except in the definition of the averaging operation. As a second objective we have conducted a numerical study to evaluate the influence of the size of the averaging volume on the effective dispersion tensor and its volume averaged statistics. This second objective is complimentary in many ways to recent research reported by Rubin et al. [1999] (via ensemble averaging) and by Wang and Kitanidis [1999] (via volume averaging) on the block‐averaged effective dispersion tensor. The variability of the effective dispersion tensor from realization to realization is assessed by computing the volume‐averaged effective dispersion tensor for an ensemble of finite fields with the same (ensemble) statistics. Ensembles were generated using three different sizes of unit cells. All three unit cell sizes yield similar results for the value of the mean effective dispersion tensor. However, the coefficient of variation depends strongly upon the size of the unit cell, and our results are consistent with those developed by Fiori [1998] from the ensemble averaging perspective. This implies that in applications the actual value of the effective dispersion tensor may be significantly different than expected on the basis of unconditioned hydraulic conductivity statistics, and this variation should be considered when applying macrodispersion to real‐world systems.

Stochastic analysis of transverse dispersion in density‐coupled transport in aquifers

Spectral perturbation techniques have been used previously to derive integral expressions for dispersive mixing in concentration‐dependent transport in three‐dimensional, heterogeneous porous media, where fluid density and viscosity are functions of solute concentration. Whereas earlier work focused on evaluating longitudinal dispersivity in isotropic media and incorporating the result in a mean one‐dimensional transport model, the emphasis of this paper is on evaluation of the complete dispersion tensor, including the more general case of anisotropic media. Approximate analytic expressions for all components of the macroscopic dispersivity tensor are derived, and the tensor is shown to be asymmetric. The tensor is separated into its symmetric and antisymmetric parts, where the symmetric part is used to calculate the principal components and principal directions of dispersivity, and the antisymmetric part of the tensor is shown to modify the velocity of the solute body compared to that of the background fluid. An example set of numerical simulations incorporating the tensor illustrates the effect of density‐coupled dispersivity on a sinking plume in an aquifer. The simulations show that the effective transverse vertical spreading in a sinking plume to be significantly greater than would be predicted by a standard density‐coupled transport model that does not incorporate the coupling in the dispersivity tensor.

Lateral diffusivity and Lagrangian scales in the Pacific Ocean as derived from drifter data

The Global Drifter Program/Surface Velocity Program data set is used to derive maps of the lateral diffusivity and Lagrangian scales of velocity, length and time across the whole Pacific. For the lateral diffusivity, the minor principal component of both the Davis' diffusivity tensor and the half growth rate of the single‐particle dispersion tensor were calculated in 5° × 5° bins from drifters for the period 1979–1999. Estimates of the lateral diffusivity, K, obtained by means of the minor principal component approach, are shown to correspond well with other authors' estimates for the Pacific. The highest values of K (2–3 104 m2/s) are found in the eastern equatorial Pacific, while the lowest ones (2–3 103 m2/s) are typical for forty‐odd degrees in latitude northward of the subpolar fronts in both hemispheres and for the subtropics in the eastern part of the South Pacific. High values of K (about 1 × 104 m2/s) are also observed in the Kusoshio/Kuroshio Extension area and in the equatorial Pacific along the whole length of the equator. There are two approximately symmetrical tongues of high K value in the western Pacific centered around 21°N and 24°S, respectively. In general, K has a tendency to increase westward (eastward) in the midlatitudes (equatorial region) and toward the equator in both hemispheres. In the midlatitudes, the Lagrangian length scale L is found to be approximately equal to the first‐mode baroclinic radius of deformation, Ri, while the Rossby number defined as Ro = Tearth/T sinφ, where T is the Lagrangian timescale, Tearth is the period of the Earth's rotation, and φ is the latitude, is less than unity, implying that the quasigeostrophic mesoscale eddies are the main mechanism for lateral mixing. The lateral diffusivity in the midlatitudes is suggested to be parameterized as K = V2Ri, where V2 is the Lagrangian velocity scale inherent to a mesoscale eddy field. In the low latitudes, L becomes less than Ri, and the condition Ro ≤ 1 is no longer valid, which implies the change of the eddy‐controlled mechanism of lateral mixing by another mechanism.

Strong spatial dispersion in wire media in the very large wavelength limit

It is found that there exist composite media that exhibit strong spatial dispersion even in the very large wavelength limit. This follows from the study of lattices of ideally conducting parallel thin wires (wire media). In fact, our analysis reveals that the description of this medium by means of a local dispersive uniaxial dielectric tensor is not complete, leading to unphysical results for the propagation of electromagnetic waves at any frequencies. Since non--local constitutive relations have been usually considered in the past as a second order approximation, meaningful in the short wavelength limit, the aforementioned result presents a relevant theoretical interest. In addition, since such wire media have been recently used as a constituent of some discrete artificial media (or metamaterials), the reported results open the question of the relevance of the spatial dispersion in the characterization of these artificial media.

Dispersion on kriged hydraulic conductivity fields

Geostatistical interpolation of log conductivity measurements leads to velocity fields lacking small‐scale variability whereas large‐scale features are covered well. The unresolved, small‐scale velocity fluctuations cause underestimation of macrodispersion, dilution, and mixing in subsequent transport calculations. The same holds when the best linear unbiased estimate of geostatistical inverse methods is used in flow and transport simulations. Considering a regular grid of measurement points, we approximate the lack of dispersion, to which we refer as correction dispersion, by applying first‐order theory to the conditional covariance of the kriged log conductivity field. The nonstationarity of the conditional covariance is removed by averaging it over space. We distinguish between macrodispersion, describing the rate of change in the second spatial moments of a large plume, and effective dispersion of a plume introduced as a point source, parameterizing dilution and mixing. Depending on the objective, we either add a correction macrodispersion tensor or a correction effective dispersion tensor to the true local dispersion tensor in transport calculations on the smoothed log conductivity field. The numerical application to a two‐dimensional periodic domain shows excellent agreement in the one‐ and two‐particle moments between simulations on the highly resolved field and those on the interpolated one using corrected values for the local dispersion coefficients.

A 3D-PTV two-projection study of pre-asymptotic dispersion in porous media which are heterogeneous on the bench scale

Velocity fluctuations over evolving scales of motion, on the scale of observation, often lead to anomalous dispersion of conservative tracers in heterogeneous porous media. Recent theories of anomalous dispersion lead to space-time non-local constitutive models for the flux of concentration. We review one such model, which has its foundations in non-equilibrium statistical mechanics. The basic premise is that knowledge of the evolution of the self-part of the intermediate scattering function, G , is all that is required to model the phenomena of interest. We review the basic integro-partial-differential equation that G satisfies and solve the inverse problem to obtain the kernels, and then use these to describe the wave-vector and frequency dependent dispersive process. Subsequently we use this information to study the transition from anomalous to Fickian regime. We also make use of the finite size Lyapunov exponent in the description of the dispersive process. Two-camera, three-dimensional particle tracking velocimetry experiments are undertaken to study dispersion within matched-index porous media. Particle trajectories, mean square displacements, velocity covariance’s, intermediate scattering functions, classical dispersion tensors, wave-vector and frequency dependent generalized dispersion tensors, and the finite-size Lyapunov exponents are obtained. Comparisons are made in the small frequency and small wave vector limits to obtain the transition from pre-asymptotic to asymptotic dispersion.

A novel three‐dimensional lattice Boltzmann model for solute transport in variably saturated porous media

This paper presents a lattice Boltzmann method for the advection and dispersion of solute in three‐dimensional variably saturated porous media. The proposed method is based on the BGK model and discretizes the particle velocity space with a cuboid lattice in which the particles move in 19 directions and 7 speeds. In the proposed model a directionally dependent relaxation time is introduced to produce the second‐order dispersion tensor, and a modification of the equilibrium distribution functions is given to model the solute in porous media where the volumetric water content varies over space and time. The concentration is calculated from a weighted summation of the particle distribution functions to ensure that the BGK collision does not result in mass change. The accuracy and consistency of the proposed model are verified against benchmark problems and the finite difference solution of solute transport in an unsaturated heterogeneous soil. The results show that the proposed model conserves mass perfectly and gives efficient and accurate solutions for both advection‐dominated and dispersion‐dominated problems.

Effect of the galactic halo modeling on the DAMA-NaI annual modulation result: An extended analysis of the data for weakly interacting massive particles with a purely spin-independent coupling

The DAMA-NaI Collaboration has observed a $4\ensuremath{\sigma}$ C.L. model-independent effect investigating the annual modulation signature in the counting rate of an NaI(Tl) setup (total exposure of 57986 kg day) and the implications of this effect have been studied under different model-dependent assumptions. In this paper we extend one of the previous analyses, the case of a WIMP with a purely spin-independent coupling, by discussing in detail the implications on the results of the uncertainties on the dark matter galactic velocity distribution. We study in a systematic way possible departures from the isothermal sphere model, which is the parametrization usually adopted to describe the halo. We specifically consider modifications arising from various matter density profiles, effects due to anisotropies of the velocity dispersion tensor and rotation of the galactic halo. The hypothesis of WIMP annual modulation, already favored in the previous analysis using an isothermal sphere, is confirmed in all the investigated scenarios, and the effects of the different halo models on the determination of the allowed maximum-likelihood region in the WIMP mass and WIMP-nucleon cross section are derived and discussed.

New form of dispersion tensor for axisymmetric porous media with implementation in particle tracking

A general form of the dispersion tensor is derived for axisymmetric porous media involving four dispersivity coefficients corresponding to longitudinal and transverse dispersion in horizontal and vertical directions, defined as perpendicular and parallel to the axis of symmetry, respectively. The general form of the dispersion tensor provides for distinct vertical and horizontal longitudinal dispersivity values. Transverse dispersion is isotropic for flow parallel to the symmetry axis and anisotropic for flow perpendicular to the symmetry axis with distinct horizontal and vertical transverse dispersivities. The new form of the dispersion tensor is applied to several examples involving axisymmetric media utilizing particle tracking techniques and compared to the tensor proposed by Burnett and Frind [1987]. It is demonstrated that for the case of spatially variable flow the drift term ∇ · (ϕD)/ϕ must generally be included in the particle tracking algorithm to obtain accurate results.

NMR investigations of correlations between longitudinal and transverse displacements in flow through random structured media

Abstract Numerical simulations and experimental results employing two-dimensional pulsed field gradient nuclear magnetic resonance (PFG-NMR) for a fluid undergoing either self-diffusion or flow under a constant external pressure gradient are discussed and particle displacement statistics in orthogonal directions are compared for different sample geometries. While the spreading of the probability density of displacements (propagator) both along the flow axis (Z) and perpendicular to it (X) is described in recent literature, and average quantities such as the diagonal elements of the dispersion tensor are being investigated routinely, no results are available which quantitatively link the interdependence of displacements in orthogonal directions. The same is true for self-diffusion which also produces correlations between displacements in the presence of obstacles. PFG-NMR provides a tool which allows simultaneous encoding of particle motion in arbitrary directions in order to perform an analysis of the correlation between these quantities. The dependence of the second moment of transverse displacements on the longitudinal displacement, integrated over varying time scales being accessible by the NMR experiment, are chosen as one possible way to describe this connection. Results obtained for random or partially ordered porous systems are compared to model situations in which the correlation can be described analytically. The findings include power-law behaviour of the form 〈X(Z)2〉∝Zγ with spreading exponents γ being indicative for the properties of the pore space in interaction with the flowing liquid.

Real Porous Media: Local Geometry and Transports

Our recent progress in the analysis and reconstruction of real porous media are surveyed and some emphasis is put on the two-field and on the grain reconstruction methods. Then, after a short summary of our previous analyses of transports, two recent developments are presented, namely, the determination of the resistivity index and of the dispersion tensor in multiphase flow.

Averaged solute transport during solidification of a binary mixture: Active dispersion in dendritic structures

The macroscopic conservation equations governing solute transport during solidification of binary alloys are derived using an averaging procedure in the context of macroscale nonequilibrium. Special attention is focused on the derivation of the associated closure problems, leading to the determination of the effective dispersion tensor and macroscopic interphase coefficients that characterize active dispersion phenomena. These closure problems are solved numerically using schematic structures and digitized images of real columnar dendritic structures observed experimentally during solidification of succinonitrile-4 wt pct acetone. The influence of the geometry and dispersion on the effective solute-properties transport is analyzed, and comparison with passive dispersion is provided. The theoretical and numerical results indicate, first, that tortuosity effects are small, and this is related to the impact of the boundary condition at the interface between the two phases, and, second, that dispersion becomes very important only at very large Peclet numbers.

Groundwater Hydraulics and Pollutant Transport

3R55. Groundwater Hydraulics and Pollutant Transport. - RJ Charbeneau (Dept of Civil Eng, Env and Water Resources Eng Program, Univ of Texas, Austin TX). Prentice Hall, Upper Saddle River NJ. 2000. 593 pp. ISBN 0-13-975616-7. $105.00.Reviewed by SA Sherif (Dept of Mech Eng, Univ of Florida, 228 MEB, PO Box 116300, Gainesville FL 32611-6300).This book is intended for use as a textbook for upper level undergraduate and graduate courses in groundwater hydrology, groundwater hydraulics, and mass transport of subsurface contaminants. It is also intended to serve as a reference for practicing hydrologists, hydrogeologists, and environmental engineers. The book contains nine chapters, nine appendices, 460 references grouped at the end of the book, and an index. Chapter 1 is an introduction to groundwater hydrology with topics dealing with porous media, distribution of subsurface water, porosity and related properties of soil, subsurface hydrologic cycle, and hydrogeologic formations. Chapter 2 presents a discussion of Darcy’s Law, continuity relations for flow in porous media, and groundwater management models. Chapter 3, entitled Groundwater and Well Hydraulics, addresses topics such as steady and transient flow to a well in an ideal confined aquifer, pumping tests, slug tests, well tests, multiple well problems, potential flow for stratified aquifers, the interface in coastal aquifers, and other transient flow problems. Chapter 4, entitled The Vadose Zone and Groundwater Recharge, presents detailed discussions of soil water in the vadose zone, soil water characteristic curve, Darcy’s Law and Richard’s Equation, measurement of soil properties, infiltration models, redistribution of soil water, evaporation and desorption models, evaporation from a shallow water table, and water balance and groundwater recharge. Chapter 5 deals with sources of subsurface contamination, mass transport processes, the general continuity equation, solute partitioning, degradation losses of soil and groundwater contaminants, and simplified forms of the continuity equation. Chapter 6 entitled Solute Transport by Advection, includes topics such as advection transport, potential theory, potential and stream functions, some applications of potential theory, residence time distribution theory, standard flow patterns, and evaluating the environmental consequences of groundwater contamination. Chapter 7 is entitled Solute Transport by Diffusion and deals with Fick’s laws, molecular diffusion coefficients, diffusion in porous media, diffusion in multiphase systems, some applications of the diffusion equation, and volatilization losses of soil contaminants. Chapter 8 presents detailed discussions of one-dimensional flow and column experiments, radial flow from a well, transverse dispersion, the mechanical dispersion tensor, moments of the transport equation, analytical models of chemical spills and contaminant plumes, numerical simulation of solute transport, nonideal flow in porous media, and subsurface mass transport through the vadose zone. Chapter 9 deals with multiphase flow and hydrocarbon recovery. This includes topics such as capillary trapping and residual saturation, NAPL behavior in fractured media, monitoring of free-product petroleum hydrocarbon, NAPL infiltration in the vadose zone, screening models for fate/transport of organic chemicals in soil and groundwater, soil-vapor extraction systems, and free product recovery of petroleum hydrocarbon liquids. Many of the appendices have spreadsheet modules for topics discussed in the book. Examples include modules for calculating well functions, slug test well function, LNAPL distribution, and LNAPL recovery analysis. Groundwater Hydraulics and Pollutant Transport is very well written and organized, contains topics appropriate for its stated title and objectives, has numerous examples and chapter problems, and has very clear illustrations and diagrams. The author is obviously very knowledgeable in the subject matter and is able to use his skills in teaching the subject over many years to convey his ideas clearly to the reader. This reviewer recommends that the book be acquired by academic and research libraries as well as practicing hydrologists, hydrogeologists, and environmental engineers.

Relationships between diffusion tensor andq‐space MRI†

Fundamental relationships between diffusion tensor (DT) and 3D q-space MRI are derived which establish conditions when these two complementary MR methods are equivalent. It is shown that the displacement distribution measured by q-space MRI in both the large displacement (i.e., large r) and the long-wavelength (i.e., small q) limits is the same 3D Gaussian displacement distribution assumed in DT-MRI. In these limiting cases, q-space MR yields a dispersion tensor that is identical to the effective DT, D, measured in DT-MRI. An experiment is then proposed to measure D using q-space methods. These findings establish that the effective DT, measured in DT-MRI, characterizes molecule motions on a coarse length-scale. Finally, the feasibility of and requirements for performing 3D q-space MRI on a clinical scanner are considered.

Transient dispersion in porous media: A comparison between exact and approximate solutions in a case study

Three methods are proposed for studying solute dispersion over short periods of time in the case study of a one-dimensional flow: the non-local method, the spatial moments method, and the volume averaging method distinguishing two zones using both rigorous closure and classical closure. The non-local method (exact) uses the Green function. Its application in calculating the generalized dispersion tensor and solving the macroscopic problem, is a cumbersome task. The methods of moments and volume averaging distinguishing two zones using rigorous closure can be used to find a correct description in terms of the spatial moments of the solute concentration distribution up to the second order. Nevertheless, the rigorous closure cannot, in general, be applied to periodic media. Volume averaging with classical closure gives coefficients which are different from those obtained by the method of moments. Surprisingly, the macroscopic solution is very similar to the exact one, in some of the tested cases.

Solute Concentration Statistics in Heterogeneous Aquifers for Finite Péclet Values

A Lagrangian framework is used for analysing the concentration fields associated with transport of nonreactive solutes in heterogeneous aquifers. This is related to two components: advection by the random velocity field v(x) and pore-scale dispersion, characterized by the dispersion tensor Dd; the relative effect of the two components is quantified by the Peclet number. The principal aim of this paper is to define the probability density function (pdf) of a nonreactive solute concentration and its relevant moments >C< and σ2c as sampled on finite detection volumes. This problem could be relevant in technical applications such as risk analysis, field monitoring and pollution control. A method to compute the concentration statistical moments and pdf is developed in the paper on the basis of the reverse formulation widely adopted to study solute dispersion in turbulent flows. The main advantages of this approach are: (i) a closed form solution for concentration mean and variance is attained, in case of small size of the sampling volume; (ii) a numerically efficient estimate of the concentration pdf can be derived. The relative effects of injection and sampling volume size and Peclet number on concentration statistics are assessed. The analysis points out that the concentration pdf can be reasonably fitted by the beta function. These results are suitable to be employed in practical applications, when the estimate of probability related to concentration thresholds is required.

Passive dispersion in dendritic structures

In order to improve mass transport description in solidification modeling, this study deals with the determination of the solute dispersion tensor in columnar dendritic mushy zone. The closure problem associated with the derivation of the macroscopic species conservation equation, using the volume averaging method, is solved numerically. Using schematic structures and digitized images of real dendritic structures observed experimentally during solidification of succinonitrile-4 wt.% acetone, the influence of tortuosity and microscopic dispersion on the determination of the effective solute dispersion coefficients is represented in terms of the Péclet number.

Estimation of local scale dispersion from local breakthrough curves during a tracer test in a heterogeneous aquifer: the Lagrangian approach

The local scale dispersion tensor, D d, is a controlling parameter for the dilution of concentrations in a solute plume that is displaced by groundwater flow in a heterogeneous aquifer. In this paper, we estimate the local scale dispersion from time series or breakthrough curves, BTCs, of Br − concentrations that were measured at several points in a fluvial aquifer during a natural gradient tracer test at Krauthausen. Locally measured BTCs were characterized by equivalent convection dispersion parameters: equivalent velocity, v eq( x ) and expected equivalent dispersivity, 〈 λ eq( x )〉. A Lagrangian framework was used to approximately predict these equivalent parameters in terms of the spatial covariance of log e transformed conductivity and the local scale dispersion coefficient. The approximate Lagrangian theory illustrates that 〈 λ eq( x )〉 increases with increasing travel distance and is much larger than the local scale dispersivity, λ d. A sensitivity analysis indicates that 〈 λ eq( x )〉 is predominantly determined by the transverse component of the local scale dispersion and by the correlation scale of the hydraulic conductivity in the transverse to flow direction whereas it is relatively insensitive to the longitudinal component of the local scale dispersion. By comparing predicted 〈 λ eq( x )〉 for a range of D d values with 〈 λ eq( x )〉 obtained from locally measured BTCs, the transverse component of D d, D dT, was estimated. The estimated transverse local scale dispersivity, λ dT= D dT/ U 1 ( U 1=mean advection velocity) is in the order of 10 1–10 2 mm, which is relatively large but realistic for the fluvial gravel sediments at Krauthausen.

Instability of a current-carrying finite-beta collisional plasma.

The microinstability of a cross-field current-carrying plasma in which the electron collisions are important on the time scale of the oscillations and can be modeled with a Bhatnagar-Gross-Krook operator is studied using linearized kinetic theory under conditions of finite electron beta. The finiteness of beta allows for coupling between electrostatic and electromagnetic modes and necessitates dealing with the entire dispersion tensor. Fundamental features of the resulting instability are identified and contrasted with those found in previous studies of the lower hybrid current-driven instability in which either collisions or finite-beta effects were neglected. As beta increases, collisions play a more important role in destabilization, alter the character and extent of electromagnetic coupling, shift the instability to more perpendicular modes, and lead to a recapturing of some of the fluidlike properties the modes have in the electrostatic limit in contrast with their highly kinetic character in the collisionless limit.