Transverse Macrodispersivity Research Articles

Under What Conditions Does Transverse Macrodispersion Exist in Groundwater Flow?

AbstractIn recent years there has been vigorous debate whether asymptotic transverse macrodispersion exists in steady three‐dimensional (3D) groundwater flows in the purely advective limit. This question is tied to the topology of 3D flow paths (termed the Lagrangian kinematics), specifically whether streamlines can undergo braiding motions or can wander freely in the transverse direction. In this study we determine which Darcy flows do admit asymptotic transverse macrodispersion for purely advective transport on the basis of the conductivity structure. We prove that porous media with smooth, locally isotropic hydraulic conductivity exhibit zero transverse macrodispersion under pure advection due to constraints on the Lagrangian kinematics of these flows, whereas either non‐smooth or locally anisotropic conductivity fields can generate transverse macrodispersion. This has implications for upscaling locally isotropic porous media to the block scale as this can result in a locally anisotropic conductivity, leading to non‐zero macrodispersion at the block scale that is spurious in that it does not arise for the fully resolved Darcy scale flow. We also show that conventional numerical methods for computation of particle trajectories do not explicitly preserve the kinematic constraints associated with locally isotropic Darcy flow, and propose a novel psuedo‐symplectic method that preserves these constraints. These results provide insights into the mechanisms that govern transverse macrodispersion in groundwater flow, and unify seemingly contradictory results in the literature in a consistent framework. These insights call into question the ability of smooth, locally isotropic conductivity fields to represent flow and transport in real heterogeneous porous media.

Evidence Based Estimation of Macrodispersivity for Groundwater Transport Applications.

The scope of this work is to discuss the proper choice of macrodispersion coefficients in modeling contaminant transport through the advection dispersion equation (ADE). It is common to model solute concentrations in transport by groundwater with the aid of the ADE. Spreading is quantified by macrodispersivity coefficients, which are much larger than the laboratory observed pore-scale dispersivities. In the frame of stochastic theory, longitudinal macrodispersivity is related to the hydraulic conductivity spatial variability via its statistical moments (mean, variance, integral scales), which are generally determined by geostatistical analysis of field measurements. In many cases, especially for preliminary assessment of contaminant spreading, these data are not available and ad hoc values are adopted by practitioners. The present study aims at recommending dispersivity values based on a thorough analysis of tens of field experiments. Aquifers are classified as of weak, medium, and high heterogeneity and for each class a range of macrodispersivity values is recommended. Much less data are available for the transverse macrodispersivities, which are significantly smaller than the longitudinal one. Nevertheless, a few realistic values based on field data, are recommended for applications. Transport models using macrodispersivities can predict mean concentrations, different from the local ones. They can be used for estimation of robust measures, like plumes spatial moments, longitudinal mass distribution and breakthrough curves at control planes.

Statistical analysis and modeling of particle trajectories in 2-D fractured porous media

Solute transport observed in subsurface formations shows complex behavior, particularly in the presence of fractures. In this work, we focus on formations with fractures that are smaller compared to the domain of interest and which are distributed in a heterogeneous matrix at densities below the percolation threshold. We developed a simplified Lagrangian approach to characterize and predict advective transport in 2-D domains containing differently oriented fractures. To this end, we performed Monte Carlo simulation (MCS) studies using ensembles of random domain realizations and gathered/analyzed tracer particle displacement statistics. The series of displacement steps were defined by the locations of particles entering the matrix after traversing through one or several connected fractures. Then, we identify key correlation structures in the evolution of and between displacement step coordinates, namely, the step length, its orientation and the traverse time. Subsequently, a correlated random walk model was derived which is able to accurately reproduce longitudinal and transverse macrodispersion as recorded in the MCS. Finally, we explored the predictive capabilities of our stochastic model by simulating macrodispersion in stratified media composed of heterogeneous zones with different fracture orientations.

The Lagrangian kinematics of three-dimensional Darcy flow

Abstract

Effects of the hydraulic conductivity microstructure on macrodispersivity

AbstractHeterogeneity of the hydraulic properties is one of the main causes of the seemingly random distribution of solute concentration observed in contaminated aquifers, with macrodispersivity providing a global measure of spreading. Earlier studies on transport of solutes in heterogeneous formations, either theoretical or numerical, expressed dispersivity as a function of the geostatistical properties of the hydraulic conductivity K. In most cases, K follows a second‐order statistical characterization, which may not be adequate when heterogeneity is high. In this work, we adopt the Multi‐Indicator Model–Self Consistent Approach (MIMSCA) to compute the longitudinal and transverse macrodispersivity. This methodology enables to model the K field by using geological inclusions of different shapes and orientation (defined here as the microstructure), while replicating the heterogeneous macrostructure obtained by the second‐order statistics. The above scheme attempts to reproduce the effect on macrodispersion of different distribution and orientation of local facies, and for instance it may represent the orientation and spatial features of the layers that are often observed in aquifers. The relevant impact of the microstructure on effective conductivity, longitudinal and transverse macrodispersivities is analyzed and discussed, for both binary and lognormally distributed K fields.

The Boundaries Influence in Trending Hydraulic Conductivity Fields

We analyze the impact of the boundary conditions on flow and transport solutions for the case of a two-dimensional heterogeneous medium with a linear trend in the mean log-conductivity. The influence of the boundaries is analyzed for different values of the domain size and aspect ratio, and different directions of the trend respect to the mean flow direction. The analysis involving a steady, spatially nonstationary velocity field, is developed by using the stochastic finite element method. Results of this study show that the impact of the trend in finite domains may disagree with that obtained in unbounded domains and it is more complex. The trend alters the result obtained for the medium with stationary conductivity, leading to a larger or smaller value of the macrodispersion coefficients, depending on the size and aspect ratio of the domain, and this effect may vary with the travel distance from the solute source. Moreover in limited domains here analyzed also for particles starting at distance of several integral scales from the boundaries and traveling in the middle of the domain, the behavior of macrodispersion coefficients is affected by the boundary conditions and the influence of the trend on longitudinal and transverse macrodispersion coefficients may vary with the imposed constraints.

Capture and release zones of permeable reactive barriers under the influence of advective–dispersive transport in the aquifer

The problem of permeable reactive barrier (PRB) capture and release behavior is investigated by means of an approximate analytical approach exploring the invariance of steady-state solutions of the advection–dispersion equation to conformal mapping. PRB configurations considered are doubly-symmetric funnel-and-gate as well as less frequent drain-and-gate systems. The effect of aquifer heterogeneity on contaminant plume spreading is hereby incorporated through an effective transverse macro-dispersion coefficient, which has to be known. Results are normalized and graphically represented in terms of a relative capture efficiency M of contaminant mass or groundwater passing a control plane (transect) at a sufficient distance up-stream of a PRB as to comply with underlying assumptions. Factors of safety FS are given as the ratios of required capture width under advective–dispersive and purely advective transport for achieving equal capture efficiency M. It is found that M also applies to the release behavior down-stream of a PRB, i.e., it describes the spreading and dilution of PRB treated groundwater possibly containing incompletely remediated contamination and/or remediation reaction products. Hypothetical examples are given to demonstrate results.

Numerical assessment of 3D macrodispersion in heterogeneous porous media

Hydrodynamic dispersion is a key controlling factor of solute transport in heterogeneous porous media. It critically depends on dimensionality. The asymptotic macrodispersion, transverse to the mean velocity direction, vanishes only in 2D and not in 3D. Using the classical Gaussian correlated permeability fields with a lognormal distribution of variance σY 2 , the longitudinal and transverse dispersivities are determined numerically as a function of heterogeneity and dimensionality. We show that the transverse macrodispersion steeply increases with σY 2 underlying the essential role of flow lines braiding, a mechanism specific to 3D systems. The transverse macrodispersion remains however at least two orders of magnitude smaller than the longitudinal macrodispersion, which increases even more steeply with σY 2 . At moderate to high levels of heterogeneity, the transverse macrodispersion also converges much faster to its asymptotic regime than do the longitudinal macrodispersion. Braiding cannot be thus taken as the sole mechanism responsible for the high longitudinal macrodispersions. It could be either supplemented or superseded by stronger velocity correlations in 3D than in 2D. This assumption is supported by the much larger longitudinal macrodispersions obtained in 3D than in 2D, up to a factor of 7 for σY 2 = 7.56.

Asymptotic dispersion for two‐dimensional highly heterogeneous permeability fields under temporally fluctuating flow

Temporal fluctuations of water flux have been investigated as a mechanism that strongly enhances transverse dispersion in heterogeneous media. Unfortunately, most results have been obtained by linear stochastic theories on permeability fields of limited variability. Worse, results are inconsistent regarding the impact of fluctuations on longitudinal dispersion, which motivates our work to find the effect of temporal velocity fluctuations on macrodispersion. We perform numerical Monte Carlo simulations for highly variable permeability fields of up to 800 correlation lengths. We find that fluctuations longitudinal to the main flow direction hardly modify macrodispersion because they do not alter the flow lines. Fluctuations transverse to the main flow direction not only increase transverse dispersion, which is well known, but also reduce the longitudinal macrodispersion in a significant and consistent way, which contradicts previous findings. The reduction of the longitudinal dispersion is comparable to the increase of transverse dispersion. Most surprisingly, for high heterogeneity, temporal fluctuations cause total (longitudinal plus transverse) macrodispersion to drop with respect to the steady state one. Enhancement of the transverse macrodispersion comes from both the increase of the transverse velocity variability and Lagrangian correlation. Reduction of the longitudinal macrodispersion results from the reduction of the Lagrangian correlation of the longitudinal velocity. That is, temporal fluctuations reduce longitudinal spreading both by breaking the fastest velocity paths on the plume front and by letting solute bypass the low‐permeability zones that tend to block or trap the solute in steady state flow conditions.

Influence of temporal fluctuations and spatial heterogeneity on pollution transport in porous media

The combined influence of temporal fluctuations and spatial heterogeneity on non-reactive solute transport mechanisms in porous media can be understood by performing simulations of steady and unsteady flow and transport in heterogeneous media. The study focuses on issues such as the degree of heterogeneity, correlation length, separation of the combined effects of temporal and spatial variations, and ergodicity conditions under unsteady flow conditions. It is shown that the effect of temporal variations on solute transport is masked by the strong effect of spatial heterogeneity. There is no obvious difference in plume shape between steady and unsteady flow conditions; the first and the second spatial moments of the plume of the unsteady-state flow condition fluctuate around the steady-state flow condition with the same period of oscillations as the input signal at small storage coefficient (S ≤ 0.001). At a relatively high standard deviation in hydraulic conductivity and a small storage coefficient, the unsteady flow condition sharpens the temporal variations in macrodispersion coefficients. The magnitude of the longitudinal macrodispersion coefficient under unsteady flow condition is almost doubled at the maximum values. However, the transverse macrodispersion coefficient fluctuates around zero. The Kubo number and Peclet number ranges are 1.2–64 and 10–250, respectively.

Stochastic flux‐related analysis of transverse mixing in two‐dimensional heterogeneous porous media

Transverse mixing of solutes in steady state transport is of utmost importance for assessing mixing‐controlled reactions of compounds that are continuously introduced into the subsurface. Classical spatial moments analysis fails to describe mixing because the tortuous streamlines in heterogeneous formations cause plume meandering, squeezing, and stretching, which affect transverse spatial moments even if there is no mass transfer perpendicular to the direction of flow. For transverse solute mixing, however, the decisive process is the exchange of solute mass between adjacent stream tubes. We therefore reformulate the advection‐dispersion equation in streamline coordinates (i.e., in terms of the potential and the stream function values) and analyze how flux‐related second central moments of plumes increase with dropping hydraulic potential. We compare the ensemble behavior of these second central moments in random two‐dimensional heterogeneous flow fields with the moments in an equivalent homogeneous system, thus defining an equivalent effective transverse dispersion coefficient. Unlike transverse macrodispersion coefficients derived by traditional moment analysis, our mixing‐relevant, flux‐related coefficient does not increase with travel distance. We present closed‐form solutions for the mean enhancement of transverse mixing by heterogeneity in two‐dimensional isotropic media for linear laws of local‐scale transverse dispersion. The mixing enhancement factor increases with the log conductivity variance but remains fairly low. We also evaluate the variance of our cumulative measure of transverse mixing, showing that heterogeneity causes substantial uncertainty of mixing. The analytical expressions are compared to numerical Monte Carlo simulations for various values of log conductivity variance, indicating good agreement with the analytical results at low variability. In the numerical simulations, we also consider nonlinear models of local‐scale transverse dispersion.

Numerical Monte Carlo analysis of the influence of pore‐scale dispersion on macrodispersion in 2‐D heterogeneous porous media

We investigate the influences of pore‐scale dispersion and of larger‐scale permeability heterogeneities on the macrodispersion without the molecular diffusion. Permeability follows a lognormal exponentially correlated distribution characterized by its correlation length λ and its lognormal variance σ2. Macrodispersion is evaluated numerically by using parallel simulations on grids of characteristic size ranging from 200λ to 1600λ. We note αL and αT the pore‐scale longitudinal and transversal dispersivities. For αL/λ < 10−2 and αT/λ < 10−3, the influence of pore‐scale dispersion on the macrodispersion is smaller than 5% of the macrodispersion due only to permeability heterogeneities. Larger dispersivities (αL/λ ≥ 10−2 or αT/λ ≥ 10−3) induce larger effects than those obtained by the semianalytical expression of Salandin and Fiorotto (1998) for σ2 > 1. The effects of local dispersion on the longitudinal macrodispersion remain limited to 25% at most of the macrodispersion due only to permeability heterogeneities. For σ2 > 1, isotropic local dispersion induces a reduction of the longitudinal macrodispersion, whereas anisotropic local dispersion lets it increase. The longitudinal and transverse local dispersions induce opposite effects on the longitudinal macrodispersion, which are respectively an increase and a reduction. The transverse macrodispersion null without local dispersion or molecular diffusion becomes strictly positive with local dispersion. Because of the velocity field heterogeneities, it is amplified by a factor of 2 to 50 from the grid scale to the macro scale. The transverse dispersion is triggered by both longitudinal and transverse local dispersions. A reduction of a factor of 2 of the transverse local dispersion at fixed longitudinal local dispersion yields only a reduction of a factor of 4 at most of the transverse macrodispersion for σ2 ≥ 2.25.

Characteristics and controlling factors of dispersion in bounded heterogeneous porous media

In this study, we present first‐order (in terms of the log conductivity variance) analytical solutions to displacement covariances Xij and macrodispersion coefficients Dij for transport of conservative solutes in two‐dimensional, bounded heterogeneous porous media. These solutions are presented as infinity series and are explicit functions of the statistics of the log hydraulic conductivity. Using numerical examples, the convergence of these solutions in terms of the number of terms in truncated finite summations is first investigated, and the accuracy of these solutions is examined by comparing them with results from Monte Carlo simulations and the semianalytical solutions of Osnes (1998). Through several examples, the general features of time‐dependent Xij and Dij are discussed. Unlike in unbounded domains, the longitudinal macrodispersivity D11 for bounded domains does not approach an asymptotic value at large dimensionless time but instead increases quickly near the downstream constant head boundary. In addition, D11 for bounded domains is always larger than that in unbounded domains, and accordingly, the transverse macrodispersivity D22 in bounded domains is smaller than that in unbounded domains. These differences stem from the lateral no‐flow boundaries in our bounded domain models. We also investigated the effect of domain sizes on Xij and Dij. Our study shows that both Xij and Dij depend not only on the dimensionless size of the domain but also on its aspect ratio. The dependence of Xij and Dij on the particles' initial location has also been investigated at detail. Our results indicate that while this dependence is very small for X11 and D11, the release location has a significant impact on both X22 and D22. Our solutions to both displacement covariances and macrodispersivity are compared against those derived from tracer test data at the Borden site. The comparison shows that our solutions are quite close to observed data, indicating that they may be applicable to predict solute transport at the field scale.

Is transverse macrodispersivity in three‐dimensional groundwater transport equal to zero? A counterexample

In advective transport through weakly heterogeneous aquifers of random stationary and isotropic three‐dimensional permeability distribution, transverse macrodispersivity αT is found to be zero. This was determined in the past by solving the transport equation at first order in the log conductivity variance σY2. However, field findings indicate the presence of small but finite αT. The aim of the paper is to determine αT for highly heterogeneous formations using a model that contains inclusions of conductivity K, submerged in a matrix of conductivity K0, for large κ = K/K0. In the dilute medium approximation, valid for small volume fraction n, but arbitrary κ, and for spherical inclusions, it is found that αT = 0 because of the axisymmetry of flow past a sphere. A medium made up of rotational ellipsoids of arbitrary random orientation, macroscopically isotropic, and of the same κ and n is devised as a counterexample. It is found that because of the intertwining of streamlines αT > 0, being of order (κ − 1)4 for κ → 1. These findings are confirmed by accurate numerical simulations of flow through a large number of interacting inclusions; for κ = 10 and n = 0.2 (jamming limit), the large value αT/αL ≃ 0.15 is attained. The numerical simulations display the strong permanent deformation of stream tubes responsible for this phenomenon, coined as “advective mixing.” The two‐point covariance, used in practice in order to characterize the aquifer structure, is not able to detect the structures that produce advective mixing. Nevertheless, the presence of high‐conductivity lenses inclined with respect to the mean flow may explain the occurrence of finite αT in field applications.

Spatial moments analysis of kinetically sorbing solutes in aquifer with bimodal permeability distribution

We analyze the influence of the formation structure upon transport of nonreactive and kinetically sorbing solutes in heterogeneous porous formations. Formation structure is represented through a bimodal distribution of hydraulic conductivity with two scales of spatial variability, which is a simplified, yet realistic, model of spatial variability. Kinetic sorption is modeled using a linear reversible rate expression. We derive new closed‐form solutions of the second ergodic spatial moments of a kinetically sorbing solute. Although these solutions are obtained through a perturbation expansion in term of logconductivity variance, they are not limited to any particular model of spatial variability. We show that the formation structure, as described by the bimodal model, has a significant impact on plume moments of a non‐reactive tracer at moderate contrast between mean hydraulic conductivity of the two hydrofacies, when the two scales of variability have comparable weight. On the contrary, the impact diminishes when bimodality weakens at both low and high contrast. As expected, the large time asymptotic longitudinal macrodispersion coefficient is influenced by the structure only through the variance and integral scale of log‐conductivity, and thus the result for the bimodal model is identical to that of a unimodal Gaussian field sharing the same second‐order statistics. The impact of kinetic sorption depends upon the plume travel time since injection. At early times, sorption kinetics tends to accentuate the difference between the longitudinal macrodispersivity in bimodal and unimodal fields, while at later times the difference is minimal because overall macrodispersivity is dominated by the spreading caused by nonequilibrium sorption. In a bimodal formation transverse macrodispersivity peaks later and declines to zero slower than in a Gaussian unimodal field with the same second‐order statistics. These results demonstrate that even a very simplified facies model, such as the bimodal model relying on a statistical distribution of high‐ and low‐conductivity zones, has important impacts upon global parameters describing the migration of nonreactive and reactive tracers. Consequently, larger effects are expected for quantities such as early arrival times and BTC tailing which are more directly influenced by the spatial patterns of high hydraulic conductivity than global parameters, such as macrodispersivity. Furthermore, these solutions contribute to bridge the gap between stochastic modeling and applications, since bimodal fields can be obtained to honor soft data typically available in applications, for example through conditional simulations using stratigraphic information from well logs.

Large‐time behavior of macrodispersion in heterogeneous trending aquifers

This paper presents a stochastic analysis of the large‐time behavior of macrodispersion in a three‐dimensional heterogeneous aquifer with a linear trend in the mean log hydraulic conductivity. To solve the problem analytically, focus is placed on the particular case where the linear trend is aligned in the direction of mean hydraulic head gradient. A spectral approach based on Fourier‐Stieltjes representations for the perturbed quantities is used to develop closed‐form expressions that describe variability of flow velocity, the second‐order mean flow, and asymptotic macrodispersion. The impact of the mean log hydraulic conductivity gradient on these results is examined. It is found that the asymptotic longitudinal and transverse macrodispersion coefficients decrease with the increasing trend gradient of mean log hydraulic conductivity in the case of finite Peclet numbers. This feature is a consequence of the reduction in variability of flow velocity with the trend gradient.

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.

Travel time and trajectory moments of conservative solutes in three dimensional heterogeneous porous media under mean uniform flow

We present expressions satisfied by the first statistical moments (mean and variance–covariance) of travel time and trajectory of conservative solute particles advected in a three-dimensional heterogeneous aquifer under uniform in the mean flow conditions. Closure of the model is obtained by means of a consistent second-order expansion in σ Y (standard deviation of the log hydraulic conductivity) of (statistical) moments of quantities of interest. As such, the results obtained are nominally limited to mildly non-uniform fields, with σ Y < 1. Resulting mean and variance of particles travel time and trajectory are functions of first and second moments and cross-moments of trajectory and velocity components. Our solution is applicable to infinite domains and is free of distributional assumptions. As an important application of the methodology we obtain closed-form expressions for the unconditional mean and variance of travel time and particle trajectory for isotropic log-conductivity domain characterized by an exponential variogram. This allows us to recover the non linear behavior of mean travel time versus distance, in agreement with numerical results published in the literature, as well as a non-linear effect in the mean trajectory. The analysis of trajectory variance allows recovering some known results regarding transverse macro-dispersion, evidencing some limitations typical of perturbation theory.

Can we determine the transverse macrodispersivity by using the method of moments?

One of the main assumptions that renders the stochastic theories applicable to real aquifers is the ergodic hypothesis, i.e. the possibility to exchange ensemble and spatial averages of a variable of interest. The principal aim of this paper is to elucidate the conditions that allow for an exchange between ensemble and spatially averaged second moments of concentration field ( S ij ); the fulfillment of the ergodic condition underlies the applicability of the dispersion coefficients or other related quantities obtained by the stochastic theories to actual aquifers. The fulfillment of the ergodic hypothesis is assessed here by analyzing the diminishing of the variance of S ij as the initial size of the plume V 0 grows, i.e. the tendency of S ij toward its expected value 〈 S ij 〉. It is shown that it is not always possible to assume ergodicity for solute plumes in heterogeneous aquifers. For the typical plume configurations encountered in applications, transverse and vertical spreading are the most problematic in this respect. In particular, satisfying the ergodic hypothesis depends largely on the initial plume configuration and size, on one hand, and the direction of the moment of interest, on the other. Numerical simulations based on the analytic element method elucidate the results. The relevance of the results is mostly felt for the inference of macrodispersive parameters, which are often derived through S ij . The work indicates that S ij may be a distorted and inadequate measure of the plume spread. This should serve as a warning against application of results based on ensemble averages to real-life plumes, particularly when estimating macrodispersion coefficient from field experiment.

Flow and Solute Transport in Strongly Heterogeneous Porous Media

The stochastic analysis of flow and solute transport provides the required probabilistic information for the risk assessment of hazard and radioactive waste management. Only low-order approximations of the velocity covariance based on the perturbation theory exit in literature. The high-order stochastic theory is needed to describe the flow and solute transport in strongly heterogeneous porous media. A conjecture on the high-order transverse velocity covariance is proposed in this study. It is based on the analytical results of first- and second-order velocity covariances from the perturbation theory. The conjecture is of an algebraic form with a power of 3/4. It leads to a good fit with the results of Monte Carlo simulations found in literature. Theoretical transverse macrodispersion coefficients are investigated for the first-order advection transport associated with different forms of velocity covariances including first-order~in variance of log-hydraulic conductivity!, second-order, and the conjectured high-order forms. The conjecture shows a peak transverse macrodis- persion coefficient growing with the variance of log conductivity. The proposed high-order velocity covariance will generate plumes that grow faster than the first-order but slower than the second-order approximation. The validity of the conjecture requires further investigation.