Macrodispersion Coefficient Research Articles

Solute transport in heterogeneous formations of bimodal conductivity distribution: 1. Theory

Transport of a conservative solute takes place in a formation made up from a matrix of conductivity K0 and porosity ϑ0 and inclusions of properties K, ϑ. For given inclusions shape, the system is characterized by the two parameters κ = K/K0 and the inclusions volume fraction n. In the past, approximate solutions of the flow and transport problems were obtained under the limit of low variability, i.e., κ − 1 ≪ 1, and arbitrary n [Rubin, 1995]. The present study aims at solving the problem under the opposite limit of a dilute system, i.e., n ≪ 1 and arbitrary κ. We are particularly interested in elongated inclusions (high length/thickness ratio) of high‐permeability contrast to the matrix. Such configurations are related to applications in which lenses or cracks are present in a medium of highly different conductivity (Figure 1). The basic procedure was developed by Eames and Bush [1999] for cylindrical or spherical inclusions, with no porosity contrast. They compute the macrodispersion coefficient, for advective transport past a large number of inclusions located at random. It is based on the solution for the distortion of a material surface of marked particles, moving past an individual inclusion in an unbounded domain and with uniform flow at infinity. In the present study we extend the approach to inclusions of arbitrary porosity and elliptical shape, characterized by the parameter e, the ratio between the small and large axes, with emphasis on e ≪ 1. We present the analytical solution of the flow problem and the procedure, requiring two quadratures, to calculate the macrodispersivity. Analytical solutions are obtained for two particular limits: κ ≪1 and κ ≃ 1. The latter is compared with the limit n ≪ 1 of the solution of Rubin [1995]. The theoretical results are applied to a few cases of hydrological interest [Lessoff and Dagan, this issue].

Theoretical basis for the measurement of local transverse dispersion in isotropic porous media

Spreading of a conservative solute in potential flow in a helical porous medium is dominated by the radial velocity distribution within the domain. Higher velocities at the inside of the helix compared to the outside leads to spreading that is balanced by transverse pore‐scale dispersion. The angular moments of the concentration distribution exhibit the typical behavior of Taylor‐Aris dispersion in which the rate of change of the second central moment is inversely proportional to pore‐scale transverse dispersion. We present numerical results for the angular moments over the entire time range and derive analytical expressions for the angular macrodispersion coefficient at the large time limit. We propose a method in which macrodispersion is measured in a helix filled with the porous material of interest in order to infer pore‐scale transverse dispersivity values that are difficult to determine with existing methods.

On the influence of local dispersion in solute transport through formations with evolving scales of heterogeneity

The aim of the present study is to determine the longitudinal dispersion coefficient DL for transport in formations of long‐range permeability fields by considering both large‐scale advection and local‐scale dispersion; the nonergodicity of the plume will be considered throughout the work. The scope is twofold: (1) to analyze the mutual role played by both the “macroscale” and the local scales of heterogeneity in determining the overall transport properties and (2) to check the validity of the results obtained in the past, in particular concerning the occurrence of anomalous transport. The results are obtained through the Lagrangean formulation of transport, by the means of a few simplifying assumptions. Two models of permeability K variations are considered: (1) a stationary Y = ln K with unbounded integral scale and (2) a formation of Y of stationary increments. For both cases, the longitudinal macrodispersion coefficient DL always grows with time when local‐scale dispersion is present, indicating that transport is always anomalous for the random fields examined. The results are in variance with those obtained in the past by considering nonergodic transport but neglecting the local‐scale dispersion [e.g., Dagan, 1994; Bellin et al., 1996] and in qualitative agreement to those obtained by adopting the ergodic assumption [e.g., Neuman, 1990; Glimm and Sharp, 1991], which, however, predicted higher rates of growth of DL with time. We conclude that the interplay between large‐scale, advective displacement and local‐scale dispersion has a fundamental impact on the occurrence of anomalous transport in long‐range correlated permeability random fields.

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.

An advective‐dispersive stream tube approach for the transfer of conservative‐tracer data to reactive transport

Conservative‐tracer data are used for the parameterization of mixing‐controlled reactive transport. Temporal moments of the tracer breakthrough curve integrated over the outflow boundary of the domain yield the average velocity and the path‐averaged macrodispersion coefficient. On the basis of this information alone, no distinction is possible between spreading and mixing of the tracer. Analyzing the temporal moments of breakthrough curves locally obtained at single points in the domain gives additional information about the dilution of the tracer. In an accompanying paper [Cirpka and Kitanidis, this issue] we derive an apparent Péclet number of mixing Pea from local temporal moments. Assuming that Pea is constant over the cross section of the outflow boundary, a corrected probability density function of arrival times is determined. By interpreting the spatially integrated breakthrough curve as the result of advective‐ dispersive transport in independent stream tubes with identical Péclet number but differing seepage velocity, it is possible to transfer results of conservative transport to the transport of interacting compounds for cases in which mixing of the compounds is influenced significantly by local‐scale dispersion. This is an improvement to the stochastic‐convective model of Simmons et al. [1995] for the transfer of integrated tracer data to reactive transport. The approach is applied to the hypothetical case of a bimolecular reaction in a heterogeneous two‐dimensional aquifer.

Characterization of mixing and dilution in heterogeneous aquifers by means of local temporal moments

Breakthrough curves of a conservative tracer in a heterogeneous two‐dimensional aquifer are analyzed by means of their temporal moments. The average velocity and the longitudinal macrodispersion coefficient of the equivalent one‐dimensional aquifer obtained through cross‐sectional averaging of concentration can be defined from the first and second central moments of a breakthrough curve integrated over the outflow boundary of the domain. On the basis of an integrated breakthrough curve, one cannot distinguish between actual solute dilution, which involves concentration reduction, and variability of arrival times among parts of the plume at different cross‐sectional positions. Analyzing the temporal moments of breakthrough curves at a “point” within the domain gives additional information about the dilution of the tracer. From these local first and second central moments an apparent seepage velocity υa and an apparent dispersivity of mixing αa can be derived. For short travel distances, αa equals the local‐scale longitudinal dispersivity. It increases with the travel distance but much more slowly than the macrodispersivity. At the large‐distance limit, αa may eventually reach the level of macrodispersivity. Lenses of high conductivity where groundwater flow converges are identified as regions of preferential enhanced mixing. The spatial distribution of these regions causes a high degree of variability of αa within a domain, indicating a high degree of uncertainty in the quantification of dilution at early stages. In an accompanying paper [Cirpka and Kitanidis, this issue] the results of conservative tracer transport are utilized for the study of mixing‐controlled reactive transport.

A note on transport of a pulse of nonlinearly reactive solute in a heterogeneous formation

Saturated flow takes place in geological formations of spatially variable permeability which is regarded as a stationary random space function of given statistical moments. The flow is assumed to be uniform in the mean and the Eulerian velocity field has stationary fluctuations. Water carries solutes that react according to the nonlinear equilibrium Freundlich isotherm. Neglecting pore scale dispersion (high Peclet number), we study the behavior of an initially finite pulse injection of constant concentration. Mean flux-averaged concentration is derived in a simple manner by using the previously determined solution of transport in a homogeneous one-dimensional medium and the Lagrangian methodology developed by Cvetkovic and Dagan [5] to model reactive transport in a three-dimensional flow field. The mean breakthrough curves are computed and the combined effect of reactive parameters and heterogeneity upon reduction of the concentration peak is investigated. Furthermore, with the aid of temporal moments, we determine equivalent reaction and macrodispersion coefficients pertinent to a homogeneous medium.

Analysis of macrodispersion through volume averaging: comparison with stochastic theory

Velocity variability at scales smaller than the size of a solute plume enhances the rate of spreading of the plume around its center of mass. Macroscopically, the rate of spreading can be quantified through macrodispersion coefficients, the determination of which has been the subject of stochastic theories. This work compares the results of a volume-averaging approach with those of the advection dominated large-time small-perturbation theory of Dagan [1982] and Gelhar and Axness [1983]. Consider transport of an ideal tracer in a porous medium with deterministic periodic velocity. Using the Taylor-Aris-Brenner method of moments, it has been previously demonstrated [Kitanidis, 1992] that when the plume spreads over an area much larger than the period, the volume-averaged concentration satisfies the advection-dispersion equation with constant coefficients that can be computed. Here, the volume-averaging analysis is extended to the case of stationary random velocities. Additionally, a perturbation method is applied to obtain explicit solutions for small-fluctuation cases, and the results are compared with those of the stochastic macrodispersion theory. It is shown that the method of moments, which uses spatial averaging, for sufficiently large volumes of averaging yields the same result as the stochastic theory, which is based on ensemble averaging. The result is of theoretical but also practical significance because the volume-averaging approach provides a potentially efficient way to compute macrodispersion coefficients. The method is applied to a simplified representation of the Borden aquifer.

A note on conservative transport in anisotropic, heterogeneous porous media in the presence of small‐amplitude transients

The late‐time macrodispersion coefficients are obtained for the case of flow in the presence of a small‐scale deterministic transient in a three‐dimensional anisotropic, heterogeneous medium. The transient is assumed to affect only the velocity component transverse to the mean flow direction and to take the form of a periodic function. For the case of a highly stratified medium, these late‐time macrodispersion coefficients behave largely as the standard coefficients used in the transport equation. Only in the event that the medium is isotropic is it probable that significant deviations from the standard coefficients would occur.

Transport and biodegradation of solutes in stratified aquifers under enhanced in situ bioremediation conditions

Understanding the effect of the heterogeneous subsurface environment upon fate and transport of biologically reactive solutes is necessary for effective field application of in situ bioremediation. We consider a scenario in which an electron acceptor is continuously input into an aquifer having initial uniform concentrations of an adsorbed pollutant and immobile microorganisms. Biodegradation is governed by dual Monod kinetics, and the aquifer is assumed to have vertical stratification of pore water velocity and retardation factor. We characterize solute transport behavior with a theoretical approach and derive a simple expression for the long‐term pollutant biodegradation rate. Our theoretical approach is based upon the observation that the balanced interactions between the biodegradation and mixing processes cause the solute and biomass concentration profiles to form traveling waves after a long elapsed time. The theoretical results indicate that the effective macrodispersion coefficients for the depth‐averaged solute fronts decrease to zero at large times and that the long‐term biodegradation rate is independent of longitudinal and transverse dispersion, the vertical variations of flow velocity and retardation factor, microbial growth kinetics, and initial biomass concentration. Numerical simulations verify the appearance of such asymptotic transport and biodegradation properties at large times but show that there is an early transient period which depends upon all these parameters and conditions. Our results demonstrate that transverse dispersion is the important process controlling the establishment of the asymptotic transport and biodegradation properties.

Renormalization group analysis of macrodispersion in a directed random flow

We apply “field theoretic” methods to the calculation of the effective diffusivity (macrodispersion coefficient) in a random flow. We show how this approach can be utilized to calculate a perturbation series for the effective diffusivity of conservative tracers in an incompressible velocity field with nonvanishing mean. The first‐order (i.e., “one‐loop”) approximation of this series coincides with classical results derived by Gelhar and Axness and Dagan. A renormalization group (RNG) approach is utilized, and the results are compared to the classical first‐order perturbation theory. For a moderate variability of the permeability the renormalized theory predicts only small corrections to the longitudinal dispersivity. However, the transverse dispersivity can be larger than that predicted by the first‐order perturbation theory by several orders of magnitude. We compare these values to the outcome of Monte Carlo simulations and find that the RNG predictions are in much better, though not perfect, accordance with the results of simulations. Moreover, the results are in good quantitative agreement with reported observations from the Borden Site field experiment.

Lagrangian Analysis of Transport in Heterogeneous Formations Under Transient Flow Conditions

The coupled effects of porous media heterogeneity and flow unsteadiness on transport are investigated. We address the problem using a stochastic‐Lagrangian approach. The log condictivity is modeled as a space random function; however, the flow unsteadiness is considered as deterministic, and it affects the magnitude and direction of the mean head gradient. Using a first‐order solution in terms of the log conductivity variance, we developed a three‐dimensional solution that includes the velocity covariances, the macrodispersion coefficients, and the displacement tensor, all as a function of the media heterogeneity and the unsteady mean flow parameters. The model is applied to the case of a periodic variation in the flow direction. We show that the effects of unsteadiness on the longitudinal spread are insignificant, while it affects strongly the transverse spread, in the form of an added constant dispersivity. Results obtained in previous studies are discussed and are shown to be particular cases of the present results.

Stochastic Analysis of Adsorbing Solute Transport in Two‐Dimensional Unsaturated Soils

Adsorbing solute transport in two‐dimensional heterogeneous unsaturated soil was studied by means of stochastic numerical simulations. Heterogeneities in the soil's hydraulic properties and in the adsorption isotherm were simulated using random fields having specified statistical structures. Macrodispersion was analyzed using the spatial moments of numerically generated solute plumes. Among different realizations of the heterogeneous soil, the discrepancies between second‐order moments and macrodispersion coefficients were large. Macrodispersivities of unsaturated soils increased with decreasing water content. Also, heterogeneous adsorption of solute enhanced the solute spreading. When the adsorption coefficient was negatively correlated with the saturated hydraulic conductivity, solute spreading was greater than when adsorption was uncorrelated or positively correlated with the conductivity.

On the Velocity Covariance and Transport Modeling in Heterogeneous Anisotropic Porous Formations: 1. Saturated Flow

Velocity covariances for saturated heterogeneous formations with anisotropic structures were derived for mean flow parallel and perpendicular to the formation bedding and were employed for modeling transport in these formations by the use of a Lagrangian formulation. Relatively simple analytical expressions, each requiring only a single numerical integration, were obtained for the longitudinal and transverse components of the velocity covariance, the displacement covariance, and the macrodispersion coefficient tensors, all as functions of the anisotropy of the formation. Closed‐form analytical expressions were derived for the longitudinal and transverse components of the velocity variance and for the asymptotic values of the transverse components of the displacement covariance. The applicability of this approach for modeling flow and transport in the vadose zone is tested in the companion to this paper by the use of the stochastic theory of Yeh et al. (1985a, b) for unsaturated flow.

On the Velocity Covariance and Transport Modeling in Heterogeneous Anisotropic Porous Formations: 2. Unsaturated Flow

Velocity covariances, and the resultant macrodispersion coefficient tensor, derived by Russo (this issue) for saturated flow conditions, are applied for unsaturated flow conditions, employing the assumption that for a given mean capillary pressure head, water saturation is a deterministic constant and log conductivity is a multivariate normal, stationary random space function. The applicability of the approach for modeling flow and transport in the vadose zone was evaluated by the use of the stochastic theory of Yeh et al. (1985a, b) for steady, unsaturated flow. Results of the analyses suggest that the approach may be applicable to vadose zone flow and transport, as long as the scale of heterogeneity in the direction of the mean flow is smaller than approximately one tenth of the characteristic length of unsaturated flow. For porous formation of given statistics, the magnitude of macrodispersion in unsaturated flow is larger than that in saturated flow, and increases as water saturation decreases. For a given water saturation, transport in unsaturated flow may approach asymptotic Fickian behavior more slowly than in saturated flow, when the two formation properties log Ks and α are positively cross‐correlated and when the correlation scale of α is relatively large as compared with the correlation scale of log Ks.

The significance of heterogeneity of evolving scales to transport in porous formations

Flow takes place in a heterogeneous formation of spatially variable conductivity, which is modeled as a stationary space random function. To model the variability at the regional scale, the formation is viewed as one of a two‐dimensional, horizontal structure. A constant head gradient is applied on the formation boundary such that the flow is uniform in the mean. A plume of inert solute is injected at t = 0 in a volume V0. Under ergodic conditions the plume centroid moves with the constant, mean flow velocity U, and a longitudinal macrodispersion coefficient dL may be defined as half of the time rate of change of the plume second spatial moment with respect to the centroid. For a log‐conductivity covariance CY of finite integral scale I, at first order in the variance σY2 and for a travel distance L = Ut ≫ I, dL → σY2UI and transport is coined as Fickian. Ergodicity of the moments is ensured if l ≫ I, where l is the initial plume scale. Some field observations have suggested that heterogeneity may be of evolving scales and that the macrodispersion coefficient may grow with L without reaching a constant limit (anomalous diffusion). To model such a behavior, previous studies have assumed that CY is stationary but of unbounded integral scale with CY ∼ arβ (−1 < β < 0) for large lag r. Under ergodic conditions, it was found that asymptotically dL ∼ aUL1+β, i.e., non‐Fickian behavior and anomalous dispersion. The present study claims that an ergodic behavior is not possible for a given finite plume of initial size l, since the basic requirement that l ≫ I cannot be satisfied for CY of unbounded scale. For instance, the centroid does not move any more with U but is random (Figure 1), owing to the large‐scale heterogeneity. In such a situation the actual effective dispersion coefficient DL is defined as half the rate of change of the mean second spatial moment with respect to the plume centroid in each realization. This is the accessible entity in a given experiment. We show that in contrast with dL, the behavior of DL is controlled by l and it has the Fickian limit DL ∼ aUl1+β (Figure 3). We also discuss the case in which Y is of stationary increments and is characterized by its variogram γy. Then U and dL can be defined only if γY is truncated (equivalently, an “infrared cutoff” is carried out in the spectrum of Y). However, for a bounded U it is shown that DL depends only on γY. Furthermore, for γY = arβ, DL ∼ aUl2Lβ−1; i.e., dispersion is Fickian for 0 < β < 1, whereas for 1 < β < 2, transport is non‐Fickian. Since β < 2, DL cannot grow faster than L = Ut. This is in contrast with a recently proposed model (Neuman, 1990) in which the dispersion coefficient is independent of the plume size and it grows approximately like L1.5.

Investigation of flow and transport in certain cases of nonstationary conductivity fields

This paper describes an investigation of flow and transport phenomena in heterogeneous, nonstationary formations, where the log conductivity is the superposition of a linear trend and a stationary fluctuation. The problem of flow and transport in a steady, two‐dimensional flow without recharge is investigated analytically for an infinite flow domain. General results are obtained for any space dimensionality, but applications were limited to the case of two‐dimensional flow in the horizontal plane. The log conductivity field is assumed to have a linear trend. We limited the scope of this study to the cases where the trend is either parallel or normal to the head gradient and to small variance in the log conductivity. This paper presents results for the moments of the velocity and the hydraulic head, the effective conductivity, macrodispersion coefficients, solute mass breakthrough curves, and travel time distributions, all in terms of the parameters of the log conductivity field and the boundary conditions. The effects of the trend in the conductivity on these results are analyzed and compared with those obtained when the apparent trend is neglected. The results may be used for predicting transport phenomena extending over distances of many integral scales, as well as for interpretation of field data where the presence of a trend is suspected.

Transport in three‐dimensionally heterogeneous aquifers: 1. Dynamics of concentration fluctuations

The concentration variance, i.e., mean squared concentration fluctuations, undergoes mean advection, a local dispersive flux, and a macrodispersive flux due to a correlation between squared concentration perturbations and velocity perturbations. The products of the macrodispersion coefficient and the squared gradient of the mean concentration field determine the rate of production of concentration variance. The rate of dissipation of concentration variance is determined by the product of the local dispersion coefficient and the mean squared gradient of the concentration perturbation field. Variance dissipation is represented as a first‐order decay with the decay coefficient equal to twice the sum of the local dispersion coefficient divided by the squared concentration microscale. The concentration microscale, estimated for an advection‐dominated log hydraulic conductivity microscale, is an increasing function of the log conductivity microscale. Thus the larger the log conductivity microscale is, the slower is the rate of dissipation of concentration fluctuations by local dispersion and vice versa. The wave number squared dependence of fluctuation dissipation requires intensive sampling to realistically model the log conductivity spectrum and its microscale, which determines the rate of dissipation of concentration fluctuations by the action of local dispersion. There is no mechanism of destroying concentration fluctuations without the action of local dispersion.

The influence of the form of the log‐conductivity covariance on non‐Fickian dispersion in random permeability fields

AbstractThe paper deals with non‐Fickian dispersion of inert solutes in random permeability fields. Attention is focused critically on the statistical characterization of the porous medium which affects pollutant dispersion in groundwater. After a brief account of recent results of stochastic theories of transport in porous media and of the fundamental indications of large‐scale field experiments, it is inferred from numerical studies that the particular choice of an analytical form of covariance of log‐conductivity has a poor influence on the overall dispersion process. In fact, different covariance structures with the same macroscale (a measure of the distance between two points beyond which the permeability ceases to be correlated) yield very similar dispersion processes. The result has a noteworthy bearing on field studies of pollutant dispersion in groundwater because it underlines the reliability of exponential correlation structures yielding analytical expression for time‐varying macrodispersion coefficients.

Macrodispersion in Stratified Porous Media

A methodology has been presented by which field data on spatial variations in the hydraulic conductivity and local dispersivity may be utilized to determine the macrodispersion coefficient tensor of the formation. This technique has been applied to an idealized stratified formation, and the relative importance of several nondimensional parameters that influence the macrodispersivity have been studied. It has been shown that for flows parallel to the stratification the macrodispersivity may be several orders of magnitude greater than the local dispersivity. This effect is significantly reduced as the flow direction is rotated. In utilizing the macroscopic advection‐dispersion equation, it has been shown that the macrodispersion coefficient may be dependent on grid scale, source location, and flow direction, although scale and source dependence will generally disappear as the grid scale becomes very large.