Effective dispersion in a chemically heterogeneous medium under temporally fluctuating flow conditions

We investigate effective solute transport in a chemically heterogeneous medium subject to temporal fluctuations of the flow conditions. Focusing on spatial variations in the equilibrium adsorption properties, the corresponding fluctuating retardation factor is modeled as a stationary random space function. The temporal variability of the flow is represented by a stationary temporal random process. Solute spreading is quantified by effective dispersion coefficients, which are derived from the ensemble average of the second centered moments of the normalized solute distribution in a single disorder realization. Using first-order expansions in the variances of the respective random fields, we derive explicit compact expressions for the time behavior of the disorder induced contributions to the effective dispersion coefficients. Focusing on the contributions due to chemical heterogeneity and temporal fluctuations, we find enhanced transverse spreading characterized by a transverse effective dispersion coefficient that, in contrast to transport in steady flow fields, evolves to a disorder-induced macroscopic value (i.e., independent of local dispersion). At the same time, the asymptotic longitudinal dispersion coefficient can decrease. Under certain conditions the contribution to the longitudinal effective dispersion coefficient shows superdiffusive behavior, similar to that observed for transport in s stratified porous medium, before it decreases to its asymptotic value. The presented compact and easy to use expressions for the longitudinal and transverse effective dispersion coefficients can be used for the quantification of effective spreading and mixing in the context of the groundwater remediation based on hydraulic manipulation and for the effective modeling of reactive transport in heterogeneous media in general.     

Analytical solutions for the coefficient of variation of the volume-averaged solute concentration in heterogeneous aquifers

Under the assumption that local solute dispersion is negligible, a new general formula (in the form of a convolution integral) is found for the arbitrary k-point ensemble moment of the local concentration of a solute convected in arbitrary m spatial dimensions with general sure initial conditions. From this general formula new closed-form solutions in m=2 spatial dimensions are derived for 2-point ensemble moments of the local solute concentration for the impulse (Dirac delta) and Gaussian initial conditions. When integrated over an averaging window, these solutions lead to new closed-form expressions for the first two ensemble moments of thevolume-averaged solute concentration and to the corresponding concentration coefficients of variation (CV). Also, for the impulse (Dirac delta) solute concentration initial condition, the second ensemble moment of thesolute point concentration in two spatial dimensions and the corresponding CV are demonstrated to be unbound. For impulse initial conditions the CVs for volume-averaged concentrations axe compared with each other for a tracer from the Borden aquifer experiment. The point-concentration CV is unacceptably large in the whole domain, implying that the ensemble mean concentration is inappropriate for predicting the actual concentration values. The volume-averaged concentration CV decreases significantly with an increasing averaging volume. Since local dispersion is neglected, the new solutions should be interpreted as upper limits for the yet to be derived solutions that account for local dispersion; and so should the presented CVs for Borden tracers. The new analytical solutions may be used to test the accuracy of Monte Carlo simulations or other numerical algorithms that deal with the stochastic solute transport. They may also be used to determine the size of the averaging volume needed to make a quasi-sure statement about the solute mass contained in it.     

Natural Gradient Tracer Test to Evaluate Natural Attenuation of MTBE Under Anaerobic Conditions

A natural gradient tracer test using perdeuterated MTBE was conducted in an anaerobic aquifer to determine the relative importance of dispersion and degradation in reducing MTBE concentrations in ground water. Preliminary ground water chemistry and hydraulic conductivity data were used to place the tracer within an existing dissolved MTBE plume at Port Hueneme, California. Following one year of transport, the tracer plume was characterized in detail.
Longitudinal dispersion was identified as the dominant mechanism for lowering the perdeuterated MTBE concentrations. The method of moments was used to determine the longitudinal and lateral dispersion coefficients (0.85 m²/day and 0.08 m²/day, respectively). A mass-balance analysis, carried out after one year of transport, accounted for 110% of the injected mass and indicated that no significant mass loss occurred. The plume structure created by zones of higher and lower hydraulic conductivity at the site was complex, consisting of several localized areas of high tracer concentration in a lower concentration plume. This is important because the aquifer has generally been characterized as exhibiting fairly minor heterogeneity. In addition, the tracer plume followed a curved flowpath that deviated from the more macroscopic direction of ground water flow inferred from local ground water elevation measurements and the behavior of the existing plume. Understanding the mass balance, plume structure, curvature of the tracer plume, and consequently natural attenuation behavior required the detailed sampling approach employed in this study. These data imply that a detailed understanding of site hydrogeology and an extensive sampling network may be critical for the correct interpretation of monitored natural attenuation of MTBE.     

On the aquifer's integrated balance equations

A general methodology is presented for describing transport phenomena in porous media at a macroscopic level. Then, these macroscopic balance equations are integrated (or averaged) along the vertical for confined, leaky and phreatic aquifers.The results are employed to derive (averaged) aquifer equations for the flow of water and of a solute (hydrodynamic dispersion). It is shown that in all cases, the resulting equation is identical to that derived on the basis of an assumption of horizontal flow (the Dupuit assumption).Macrodispersion, occurring at the aquifer level, is discussed and appropriate coefficients are proposed.     

Monte Carlo evaluation of microbial-mediated contaminant reactions in heterogeneous aquifers

Monte Carlo simulations are conducted to evaluate microbial-mediated contaminant reactions in an aquifer comprised of spatially variable microbial biomass concentrations, aquifer hydraulic conductivities, and initial electron donor/acceptor concentrations. A finite element simulation model is used that incorporates advection, dispersion, and Monod kinetic expressions to describe biological processes. Comparisons between Monte Carlo simulations of heterogeneous systems and simulations using homogeneous formulation of the same two-dimensional transport problem are presented. For the assumed set of parameters, physical aquifer heterogeneity is found to have a minor effect on the mass of contaminant biodegraded/transformed when compared to a homogeneous system; however, it noticeably changes the dispersion, skewness, and peakness of contaminant concentration distributions. Similarly, for low microbial growth rate, given favorable microbial growth characteristics, biological heterogeneity has minor effect on the mass of contaminant biodegraded/transformed when compared to a homogeneous system. On the other hand, when higher effective growth rates are assumed, biological heterogeneity and spatial heterogeneities in essential electron donor/acceptors reduce the efficiency of biotic contaminant reactions; consequently, model simulations derived from heterogeneous biomass distributions predict remediation time scales that are longer than those simulated for homogeneous systems. When correlations between physical aquifer and biological heterogeneities are considered, the assumed correlation affects predicted mean and variance of contaminant concentration and biomass distributions. For example, an assumed negative correlation between hydraulic conductivity and the initial biomass distribution produces a plume where less efficient biotic contaminant reactions occur at the leading edge of the plume; this is consistent with less degradation/transformation occurring over regions of higher groundwater velocities. However, the presence and absence of these correlations do not appear to affect the efficiency of microbial-mediated contaminant attenuation.     

Upstream Dispersion in Solute Transport Models: A Simple Evaluation and Reduction Methodology

This article outlines analytical solutions to quantify the length scale associated with “upstream dispersion,” the artificial movement of solutes in the opposite direction to groundwater flow, in solute transport models. Upstream dispersion is an unwanted artifact in common applications of the advection-dispersion equation (ADE) in problems involving groundwater flow in the direction of increasing solute concentrations. Simple formulae for estimating the one-dimensional distance of upstream dispersion are provided. These show that under idealized conditions (i.e., steady-state flow and transport, and a homogeneous aquifer), upstream dispersion may be a function of only longitudinal dispersivity. The scale of upstream dispersion in a selection of previously presented situations is approximated to highlight the utility of the presented formulae and the relevance of this ADE anomaly in common transport problems. Additionally, the analytical solution is applied in a hypothetical scenario to guide the modification of dispersion parameters to minimize upstream dispersion.     

Solute transport scales in an unsaturated stony soil

Solute transport parameters are known to be scale-dependent due mainly to the increasing scale of heterogeneities with transport distance and with the lateral extent of the transport field examined. Based on a transect solute transport experiment, in this paper we studied this scale dependence by distinguishing three different scales with different homogeneity degrees of the porous medium: the observation scale, transport scale and transect scale. The main objective was to extend the approach proposed by van Wesenbeeck and Kachanoski to evaluating the role of textural heterogeneities on the transition from the observation scale to the transport scale. The approach is based on the scale dependence of transport moments estimated from solute concentrations distributions. In our study, these moments were calculated starting from time normalized resident concentrations measured by time domain reflectometry (TDR) probes at three depths in 37 soil sites 1 m apart along a transect during a steady state transport experiment. The Generalized Transfer Function (GTF) was used to describe the evolution of apparent solute spreading along the soil profile at each observation site by analyzing the propagation of the moments of the concentration distributions. Spectral analysis was used to quantify the relationship between the solid phase heterogeneities (namely, texture and stones) and the scale dependence of the solute transport parameters. Coupling the two approaches allowed us to identify two different transport scales (around 4-5 m and 20 m, respectively) mainly induced by the spatial pattern of soil textural properties. The analysis showed that the larger transport scale is mainly determined by the skeleton pattern of variability. Our analysis showed that the organization in hierarchical levels of soil variability may have major effects on the differences between solute transport behavior at transport scale and transect scale, as the transect scale parameters will include information from different scales of heterogeneities.     

Computational issues in the determination of solute discharge moments and implications for comparison to analytical solutions

Solute discharge moments (mean and variance) are computed using numerical modeling of flow and advective transport in two-dimensional heterogeneous aquifers and are compared to theoretical results. The solute discharge quantifies the temporal evolution of the total contaminant mass crossing a certain compliance boundary. In addition to analyzing the solute discharge moments within a classical absolute dispersion framework, we also analyze relative dispersion formulation, whereby plume meandering (deviation from mean flow path caused by velocity variations at scales larger than plume size) is removed. This study addresses some important issues related to the computation of solute discharge moments from random walk particle tracking experiments, and highlights some of the important differences between absolute and relative dispersion frameworks. Relative dispersion formulation produces maximum uncertainty that coincides with the peak mean discharge. Absolute dispersion, however, results in earlier arrival of the uncertainty peak as compared to the first moment peak. Simulations show that the standard deviation of solute discharge in a relative dispersion framework requires increasingly large temporal sampling windows to smooth out some of the large fluctuations in breakthrough curves associated with advective transport. Using smoothing techniques in particle tracking to distribute the particle mass over a volume rather than at a point significantly reduces the noise in the numerical simulations and removes the need to use large temporal windows. Same effect can be obtained by adding a local dispersion process to the particle tracking experiments used to model advective transport. The effect of the temporal sampling window bears some relevance and important consequences for evaluating risk-related parameters. The expected value of peak solute discharge and its standard deviation are very sensitive to this sampling window and so will be the risk distribution relying on such numerical models.     

Exposure concentration statistics in the subsurface transport

The concentration fluctuations resulting from hazardous releases in the subsurface are modeled through the concentration moments. The local solute exposure concentration, resulting from the heterogeneous velocity field and pore scale dispersion in the subsurface, is a random function characterized by its statistical moments. The approximate solution to the exact equation that describes the evolution of concentration standard moments in the aquifer transport is proposed in a recursive form. The expressions for concentration second, third and fourth central moments are derived and evaluated for various flow and transport conditions. The solutions are sought by starting from the exact upper bound solution with the zero pore scale dispersion and introducing the physically based approximation that allows the inclusion of the pore scale dispersion resulting in simple closed-form expressions for the concentration statistical moments. The concentration moments are also analyzed in the relative and absolute frame of reference indicating their combined importance in the practical cases of the subsurface contaminant plume migration. The influence of pore scale dispersion with different source sizes and orientations are analyzed and discussed with respect to common cases in the environmental risk assessment problems. The results are also compared with the concentration measurements of the conservative tracer collected in the field experiments at Cape Cod and Borden Site.     

Analysis of solute transport in flow fields influenced by preferential flowpaths at the decimeter scale

Several recent studies at the Macrodispersion Experiment (MADE) site in Columbus, Mississippi, have indicated that the relative preferential flowpaths and flow barriers resulting from decimeter-scale aquifer heterogeneities appear to have a dominant effect on plume-scale solute transport. Numerical experiments are thus conducted in this study to explore the key characteristics of solute transport in two-dimensional flow fields influenced by decimeter-scale preferential flowpaths. A hypothetical but geologically plausible network of 10 cm wide channels of high hydraulic conductivity is used to represent the relative preferential flowpaths embedded in an otherwise homogeneous aquifer. When the hydraulic conductivity in the channels is 100 times greater than that in the remaining portion of the aquifer, the calculated concentration distributions under three source configurations all exhibit highly asymmetrical, non-Gaussian patterns. These patterns, with peak concentrations close to the source and extensive spreading downgradient, resemble that observed at the MADE site tracer tests. When the contrast between the channel and nonchannel hydraulic conductivities is reduced to 30:1 from 100:1, the calculated mass distribution curve starts to approach a Gaussian one with the peak concentration near the central portion of the plume. Additional analysis based on a field-scale model demonstrates that the existence of decimeter-scale preferential flowpaths can have potentially far-reaching implications for ground water remediation. Failure to account for them in numerical simulation could lead to overestimation of the effectiveness of the remedial measure under consideration.     

Combining 3D Hydraulic Tomography with Tracer Tests for Improved Transport Characterization

Hydraulic tomography (HT) is a method for resolving the spatial distribution of hydraulic parameters to some extent, but many details important for solute transport usually remain unresolved. We present a methodology to improve solute transport predictions by combining data from HT with the breakthrough curve (BTC) of a single forced‐gradient tracer test. We estimated the three dimensional (3D) hydraulic‐conductivity field in an alluvial aquifer by inverting tomographic pumping tests performed at the Hydrogeological Research Site Lauswiesen close to Tübingen, Germany, using a regularized pilot‐point method. We compared the estimated parameter field to available profiles of hydraulic‐conductivity variations from direct‐push injection logging (DPIL), and validated the hydraulic‐conductivity field with hydraulic‐head measurements of tests not used in the inversion. After validation, spatially uniform parameters for dual‐domain transport were estimated by fitting tracer data collected during a forced‐gradient tracer test. The dual‐domain assumption was used to parameterize effects of the unresolved heterogeneity of the aquifer and deemed necessary to fit the shape of the BTC using reasonable parameter values. The estimated hydraulic‐conductivity field and transport parameters were subsequently used to successfully predict a second independent tracer test. Our work provides an efficient and practical approach to predict solute transport in heterogeneous aquifers without performing elaborate field tracer tests with a tomographic layout.     

Effects of rate-limited mass transfer on water sampling with partially penetrating wells

Nonequilibrium concentration type curves are numerically developed and sensitivity analyses are performed to examine the relationships between effluent concentrations in partially penetrating monitoring/extraction wells, the vertical plume shape, and the mass transfer characteristics of the aquifer. The governing two-dimensional, axisymmetric nonequilibrium solute transport equation is solved in three stages using an operator-splitting approach. In the first two stages, the advection and dispersion terms are solved with the Eulerian-Lagrangian method, based on the backward method of characteristics for advection and the standard implicit Galerkin finite element method for dispersion. In the third step, the first-order, immobile-mobile domain mass transfer term is computed analytically for both two-site and lognormally distributed, multirate models. Effluent concentration variations with time and contour plots of the pore water concentration distribution in the aquifer are compared for a wide range of field- and laboratory-measured mass transfer rates, various plume shapes, and relevant physical/chemical parameter values, including pumping rate, vertical anisotropy ratio, retardation factor, and porosity. The simulation results show that rate-limited mass transfer can have a significant impact on sample and aquifer pore water concentrations during three-dimensional transport to a partially penetrating well. An alternative dimensionless form of the nonequilibrium solute transport equation is derived to illustrate the key parameter groupings that quantify rate-limited sorption effects and show the relative importance of individual parameters. A hypothetical field application example demonstrates the fitting of dimensional type curves to discrete-interval sampling data in order to evaluate the mass transfer characteristics of an aquifer and shows how type curve superposition can be used to model complex plume shapes.     

Scale effect and calibration of contaminant transport models

The influence on solute transport of the small-scale spatial variation of aquifer hydraulic conductivity (K) was analyzed by comparing results from fine-grid (2 m by 2 m) simulations of a synthetic heterogeneous aquifer to those from coarse-grid (8 m by 4 m) simulations of an equivalent homogeneous aquifer. Realizations of the K field of the heterogeneous aquifer were generated, using the Monte Carlo approach, from a lognormal distribution with mean log K of 2 (K in m/d) and three levels of log K variance of 0.1, 0.5, and 1.0. Numerical simulation results show that the average standard deviation of point concentrations increased from 1.21 to 5.78 when the value of log K variance was increased from 0.1 to 1.0. The average discrepancy between modeled concentrations (obtained from a coarse-grid deterministic numerical simulation) and the actual mean point concentrations (obtained from fine-grid Monte Carlo numerical simulations) increased from 0.91 to 4.23 with the increase in log K variance. The results from this study illustrate the uncertainty in predictions from contaminant transport models due to their inability to simulate the effects of heterogeneities at scales smaller than the model grid.     

Characterization of mixing and spreading in a bounded stratified medium

Matheron and de Marsily [Matheron M, de Marsily G. Is the transport in porous media always diffusive? A counter-example. Water Resour Res 1980;16:901–17] studied transport in a perfectly stratified infinite medium as an idealized aquifer model. They observed superdiffusive solute spreading quantified by anomalous increase of the apparent longitudinal dispersion coefficient with the square root of time. Here, we investigate solute transport in a vertically bounded stratified random medium. Unlike for the infinite medium at asymptotically long times, disorder-induced mixing and spreading is uniquely quantified by a constant Taylor dispersion coefficient. Using a stochastic modeling approach we study the effective mixing and spreading dynamics at pre-asymptotic times in terms of effective average transport coefficients. The latter are defined on the basis of local moments, i.e., moments of the transport Green function. We investigate the impact of the position of the initial plume and the initial plume size on the (highly anomalous) pre-asymptotic effective spreading and mixing dynamics for single realizations and in average. Effectively, the system “remembers” its initial state, the effective transport coefficients show so-called memory effects, which disappear after the solute has sampled the full vertical extent of the medium. We study the impact of the intrinsic non-ergodicity of the confined medium on the validity of the stochastic modeling approach and study in this context the transition from the finite to the infinite medium.     

Modeling the effects of nonlinear equilibrium sorption on the transport of solute plumes in saturated heterogeneous porous media

Transport of sorbing solutes in 2D steady and heterogeneous flow fields is modeled using a particle tracking random walk technique. The solute is injected as an instantaneous pulse over a finite area. Cases of linear and Freundlich sorption isotherms are considered. Local pore velocity and mechanical dispersion are used to describe the solute transport mechanisms at the local scale. This paper addresses the impact of the degree of heterogeneity and correlation lengths of the log-hydraulic conductivity field as well as negative correlation between the log-hydraulic conductivity field and the log-sorption affinity field on the behavior of the plume of a sorbing chemical. Behavior of the plume is quantified in terms of longitudinal spatial moments: center-of-mass displacement, variance, 95% range, and skewness. The range appears to be a better measure of the spread in the plumes with Freundlich sorption because of plume asymmetry. It has been found that the range varied linearly with the travelled distance, regardless of the sorption isotherm. This linear relationship is important for extrapolation of results to predict behavior beyond simulated times and distances. It was observed that the flow domain heterogeneity slightly enhanced the spreading of nonlinearly sorbing solutes in comparison to that which occurred for the homogeneous flow domain, whereas the spreading enhancement in the case of linear sorption was much more pronounced. In the case of Freundlich sorption, this enhancement led to further deceleration of the solute plume movement as a result of increased retardation coefficients produced by smaller concentrations. It was also observed that, except for plumes with linear sorption, correlation between the hydraulic conductivity and the sorption affinity fields had minimal effect on the spatial moments of solute plumes with nonlinear sorption.     

Transport of variable-density solute plumes in beach aquifers in response to oceanic forcing

A comprehensive numerical study was undertaken to investigate transport of a variable-density, conservative solute plume in an unconfined coastal aquifer subject to high and low frequency oceanic forcing. The model combined variable-density saturated flow for groundwater and solute transport, and wave hydrodynamics from a 2D Navier–Stokes solver. A sinusoidal tidal signal was specified by implementing time-varying heads at the seaward boundary. The solute plume behavior was investigated under different oceanic forcing conditions: no forcing, waves, tide, and combined waves and tide. For each forcing condition, four different injected solute densities (freshwater, brackish water, seawater, brine) were used to investigate the effects of density on the transport of the injected plume beneath and across the beach face. The plume’s low-order spatial moments were computed, viz., mass, centroid, variance and aspect ratio. The results confirmed that both tide- and wave-forcing produce an upper saline plume beneath the beach face in addition to the classical saltwater wedge. For the no-forcing and tide-only cases (during rising tides), an additional small circulation cell below the beach face was observed. Oceanic forcing affects strongly the solute plume’s flow path, residence time and discharge rate across the beach face, as well as its spreading. For the same oceanic forcing, solute plumes with different densities follow different trajectories from the source to the discharge location (beach face). The residence time and plume spreading increased with plume density. It was concluded that simulations that neglect the effect of waves or tides cannot reproduce accurately solute plume dispersion and also, in the case of coasts with small waves or tides, the solute residence time in the aquifer.     

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. Received: December 28, 1998     

Laplace transform solutions for solute transport in fissured aquifers

The main processes affecting the migration of a solute in a fissured aquifer will be advection and dispersion in the fissures, diffusion into the porous matrix; and adsorption. This paper considers solute transport in an idealized fissured aquifer consisting of slabs of saturated rock-matrix separated by equally spaced, planar fissures. The solution of the transport equations is developed as far as Laplace transforms of the solute concentrations in the fissure and matrix water. Numerical inversion of the transforms is used to investigate characteristic behaviour of the model for a number of special cases.