Two-dimensional concentration distribution for mixing-controlled bioreactive transport in steady state

Under steady-state conditions, the degradation of contaminant plumes introduced continuously into an aquifer is controlled by transverse dispersion when the other reacting compound is provided from ambient groundwater. Given that the reaction is instantaneous and longitudinal dispersion can be neglected, the length of the plume is inversely proportional to the transverse dispersion coefficient. In typical scenarios of natural attenuation, however, the considered reaction is biotic and kinetic. The standard model of bioreactive transport relies on double-Monod kinetics and pseudo first-order biomass decay. Under these conditions, a fraction of the injected mass flux remains beyond the length of the plume determined for the instantaneous reaction. We present an analytical framework to derive the steady-state concentration distributions of the dissolved compounds and the biomass from the concentration distribution of a conservative compound, assuming double-Monod kinetics and two different models describing biomass decay. The first biomass-decay model assumes a constant first-order decay coefficient, while the second assumes that the decay coefficient depends upon the electron-acceptor concentration. We apply the method to the case of a line-injection in two-dimensional uniform flow. In general, the bioreactive concentration distributions are similar to the distributions computed for an instantaneous reaction. The similarity is greater when the biomass decay coefficient is assumed to depend on the electron-acceptor concentration rather than being constant.     

Exact transverse macro dispersion coefficients for transport in heterogeneous porous media

We study transport through heterogeneous media. We derive the exact large scale transport equation. The macro dispersion coefficients are determined by additional partial differential equations. In the case of infinite Peclet numbers, we present explicit results for the transverse macro dispersion coefficients. In two spatial dimensions, we demonstrate that the transverse macro dispersion coefficient is zero. The result is not limited on lowest order perturbation theory approximations but is an exact result. However, the situation in three spatial dimensions is very different: The transverse macro dispersion coefficients are finite – a result which is confirmed by numerical simulations we performed.     

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.     

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.     

Stochastic modeling of reactive transport in wetlands

This study describes the development of a general model for reaction in and performance of spatially heterogeneous bioreactors such as treatment wetlands. The modeled domain possesses local-scale velocities, reaction rates and transverse dispersion coefficients that are functions of an underlying heterogeneity variate representing one or more controlling biophysical attributes, for example, reactive surface area (submerged plant) density. Reaction rate coefficients are treated as related to local velocities in an inverse square fashion via their mutual dependence upon the variate. The study focuses on the solution for the steady-state case with constant inlet concentration. Results compare well with exact solutions developed for laterally-bounded systems in which the heterogeneity is represented explicitly. Employing the bicontinuum analogue of a second-order model, an expression for an effective longitudinal dispersion coefficient as a function of travel distance is developed using the method of moments. The result provides insights into the behavior of concentration as transverse mixing drives the system asymptotically toward Fickian longitudinal dispersion. The model may represent an improvement over other approaches for characterizing treatment wetland performance because it accounts for evolving shear flow dispersion, and because parameters are few in number, physically based, and invariant with mean velocity.     

Effective dispersion in conditioned transmissivity fields

We analyze the impact of conditioning to measurements of hydraulic transmissivity on the transport of a conservative solute. The effects of conditioning on solute transport are widely discussed in the literature, but most of the published works focuses on the reduction of the uncertainty in the prediction of the plume dispersion. In this study both ensemble and effective plume moments are considered for an instantaneous release of a solute through a linear source normal to the mean flow direction, by taking into account different sizes of the source. The analysis, involving a steady and spatially inhomogeneous velocity field, is developed by using the stochastic finite element method. Results show that conditioning reduces the ensemble moment in comparison with the unconditioned case, whereas the effective dispersion may increase because of the contribution of the spatial moments related to the lack of stationarity in the flow field. As the number of conditioning points increases, this effect increases and it is significant in both the longitudinal and transverse directions. Furthermore, we conclude that the moment derived from data collected in the field can be assessed by the conditioned second-order spatial moment only with a dense grid of measured data, and it is manifest for larger initial lengths of the plume. Nevertheless, it seems very likely that the actual dispersion of the plume may be underestimated in practical applications.     

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.     

The effect of non divergence-free velocity fields on field scale ground water solute transport

Solute plume subjected to field scale hydraulic conductivity heterogeneity shows a large dispersion/macrodispersion, which is the manifestation of existing fields scale heterogeneity on the solute plume. On the other hand, due to the scarcity of hydraulic conductivity measurements at field scale, hydraulic conductivity heterogeneity can only be defined statistically, which makes the hydraulic conductivity a random variable/function. Random hydraulic conductivity as a parameter in flow equation makes the pore flow velocity also random and the ground water solute transport equation is a stochastic differential equation now. In this study, the ensemble average of stochastic ground water solute transport equation is taken by the cumulant expansion method in order to upscale the laboratory scale transport equation to field scale by assuming pore flow velocity is a non stationary, non divergence-free and unsteady random function of space and time. Besides the stochastic explanation of macrodispersion and the velocity correction term obtained by Kavvas and Karakas (J Hydrol 179:321–351, 1996) before a new velocity correction term, which is a function of mean pore flow velocity divergence, is obtained in this study due to strict second order cumulant expansion (without omitting any term after the expansion) performed. The significance of the new velocity correction term is investigated on a one dimensional transport problem driven by a density dependent flow field.     

On the methods for determining the transverse dispersion coefficient in river mixing

In this study, the strengths and weaknesses of existing methods for determining the dispersion coefficient in the two-dimensional river mixing model were assessed based on hydraulic and tracer data sets acquired from experiments conducted on either laboratory channels or natural rivers. From the results of this study, it can be concluded that, when the longitudinal dispersion coefficient as well as the transverse dispersion coefficients must be determined in the transient concentration situation, the two-dimensional routing procedures, 2D RP and 2D STRP, can be employed to calculate dispersion coefficients among the observation methods. For the steady concentration situation, the STRP can be applied to calculate the transverse dispersion coefficient. When the tracer data are not available, either theoretical or empirical equations by the estimation method can be used to calculate the dispersion coefficient using the geometric and hydraulic data sets. Application of the theoretical and empirical equations to the laboratory channel showed that equations by Baek and Seo [[3], 2011] predicted reasonable values while equations by Fischer [23] and Boxwall and Guymer (2003) overestimated by factors of ten to one hundred. Among existing empirical equations, those by Jeon et al. [28] and Baek and Seo [6] gave the agreeable values of the transverse dispersion coefficient for most cases of natural rivers. Further, the theoretical equation by Baek and Seo [5] has the potential to be broadly applied to both laboratory and natural channels.     

Influence of transient flow on contaminant biodegradation

The rate of biodegradation in contaminated aquifers depends to a large extent on dispersive mixing processes that are now generally accepted to result from spatial variations in the velocity field. It has been shown, however, that transient flow fields can also contribute to dispersive mixing. The influence of transient flow on biodegrading contaminants is particularly important since it can enhance mixing with electron acceptors, further promoting the reactive process. Using numerical simulations, the effect of transient flow on the behavior of a biodegradable contaminant is evaluated here both with respect to the development of apparently large horizontal transverse dispersion and also with respect to enhanced mixing between the substrate (electron donor) and electron acceptor. The numerical model BIO3D, which solves for advective-dispersive transport coupled with Monod-type biodegradation of substrates in the presence of an electron acceptor, was used for the simulations. The model was applied in a two-dimensional plan view mode considering a single substrate. Transient flow fields were found to yield larger apparent transverse dispersion because the longitudinal dispersivity also acts transverse to the mean flow direction. In the reactive case, the transient flow field increases substrate-oxygen mixing, which in turn enhances the overall rate of biodegradation. The results suggest that in the case of moderate changes of flow directions, a steady-state flow field can be justified, thereby avoiding the higher computational costs of a fully transient simulation. The use of a higher transverse horizontal dispersivity in a steady flow field can, under these conditions, adequately forecast plume development.     

Transport of nonsorbing solutes in a streambed with periodic bedforms

Previous studies of hyporheic zone focused largely on the net mass transfer of solutes between stream and streambed. Solute transport within the bed has attracted less attention. In this study, we combined flume experiments and numerical simulations to examine solute transport processes in a streambed with periodic bedforms. Solute originating from the stream was subjected to advective transport driven by pore water circulation due to current–bedform interactions as well as hydrodynamic dispersion in the porous bed. The experimental and numerical results showed that advection played a dominant role at the early stage of solute transport, which took place in the hyporheic zone. Downward solute transfer to the deep ambient flow zone was controlled by transverse dispersion at the later stage when the elapsed time exceeded the advective transport characteristic time t_c (= L/u_c with L being the bedform length and u_c the characteristic pore water velocity). The advection-based pumping exchange model was found to predict reasonably well solute transfer between the overlying water and streambed at the early stage but its performance deteriorated at the later stage. With dispersion neglected, the pumping exchange model underestimated the long-term rate and total mass of solute transfer from the overlying water to the bed. Therefore both advective and dispersive transport components are essential for quantification of hyporheic exchange processes.     

Estimation of the dispersion coefficient in rivers with riparian vegetation

The study of the transport processes in the riparian environment is currently a subject of great interest. In this context, the evaluation of longitudinal dispersion is a fundamental issue. Although it plays an important role in the transport of chemicals, nutrients, and wood debris, few works have studied the effects of riparian vegetation on the dispersion coefficient in transversally non-uniform streams. Here, a quasi two-dimensional model is proposed to evaluate the transverse profile of the depth-averaged velocity in the presence of any vegetation distribution along the riparian transect. Once velocity profile is obtained, the dispersion coefficient is evaluated and the remarkable role of vegetation is shown. Both the velocity profile and the dispersion coefficient have been validated using our new experimental results and literature data. The work highlights the importance of taking into account the presence of vegetation along river banks. The estimation of the longitudinal dispersion in vegetated rivers can be affected by as much as 70–100% compared to the case without vegetation.     

A field study of the coupled effects of aquifer stratification,fluid density,and groundwater fluctuations on dispersivity assessments

A number of experimental studies have tackled the issue of solute transport parameter assessments either in the laboratory or in the field. But yet, the behavior of a plume in the field under density driven forces, is not well known due to possible development of instabilities. Some field tracer tests on the fate of plumes denser than native groundwater such as those encountered under waste disposal facilities, have pointed out the processes of sinking and splitting at the early stage of migration. The process of dispersion was widely investigated, but the range of dispersivity values obtained from either experimental tests, or numerical and theoretical calculations is still very large, even for the same type of aquifers. These discrepancies were considered to be essentially caused by soil heterogeneities and scale effects. In the meantime, studies on the influence of sinking and fingering have remained more scarce. The objective of the work is to analyze how transport parameters such as dispersivities can be affected by unstable conditions, which lead to plume sinking and fingering. A series of tracer tests were carried out to study under natural conditions, the transport of a dense chloride solution injected in a shallow two-layered aquifer. Two types of experiments were performed: in the first type, source injection was such that the plume could travel downward from one layer to the other of higher pore velocity, and in the second one, the migration took place only in the faster layer. The results suggest some new insights in the processes occurring at the early stages of a dense plume migration moving in a stratified aquifer under groundwater fluctuations, which can be summarized through the following points: (i) Above a stability criterion threshold, a fingering process and a multi modal plume transport take place, but local dispersivities can be cautiously derived, using breakthrough curves matching. (ii) When water table is subject to some cycling or rising, the plume can be significantly distorted in the transverse direction, leading to unusual values of the ratio between longitudinal and transverse dispersivities. (iii) Under stable conditions, for example in the case of straightforward injection in the faster aquifer layer, longitudinal dispersivity is greater than the transverse component as usually encountered, and the obtained transport parameters are closed to macro dispersivity values, which reach their asymptotic limit at very short distances. (iv) The classical scale effect about the varying dispersivity at short distances could be a process mainly due to the distance required for a plume stabilization.     

A Critical Analysis of Transverse Dispersivity Field Data

Transverse dispersion, or tracer spreading orthogonal to the mean flow direction, which is relevant e.g, for quantifying bio-degradation of contaminant plumes or mixing of reactive solutes, has been studied in the literature less than the longitudinal one. Inferring transverse dispersion coefficients from field experiments is a difficult and error-prone task, requiring a spatial resolution of solute plumes which is not easily achievable in applications. In absence of field data, it is a questionable common practice to set transverse dispersivities as a fraction of the longitudinal one, with the ratio 1/10 being the most prevalent. We collected estimates of field-scale transverse dispersivities from existing publications and explored possible scale relationships as guidance criteria for applications. Our investigation showed that a large number of estimates available in the literature are of low reliability and should be discarded from further analysis. The remaining reliable estimates are formation-specific, span three orders of magnitude and do not show any clear scale-dependence on the plume traveled distance. The ratios with the longitudinal dispersivity are also site specific and vary widely. The reliability of transverse dispersivities depends significantly on the type of field experiment and method of data analysis. In applications where transverse dispersion plays a significant role, inference of transverse dispersivities should be part of site characterization with the transverse dispersivity estimated as an independent parameter rather than related heuristically to longitudinal dispersivity.     

Comparison of theory and experiments for dispersion in homogeneous porous media

Modeling dispersion in homogeneous porous media with the convection–dispersion equation commonly requires computing effective transport coefficients. In this work, we investigate longitudinal and transverse dispersion coefficients arising from the method of volume averaging, for a variety of periodic, homogeneous porous media over a range of particle Péclet (Pe_p) numbers. Our objective is to validate the upscaled transverse dispersion coefficients and concentration profiles by comparison to experimental data reported in the literature, and to compare the upscaling approach to the more common approach of inverse modeling, which relies on fitting the dispersion coefficients to measured data. This work is unique in that the exact microscale geometry is available; thus, no simplifying assumptions regarding the geometry are required to predict the effective dispersion coefficients directly from theory. Transport of both an inert tracer and non-chemotactic bacteria is investigated for an experimental system that was designed to promote transverse dispersion. We highlight the occurrence of transverse dispersion coefficients that (1) depart from power-law behavior at relatively low Pe_p values and (2) are greater than their longitudinal counterparts for a specific range of Pe_p values. The upscaling theory provides values for the transverse dispersion coefficient that are within the 98% confidence interval of the values obtained from inverse modeling. The mean absolute error between experimental and upscaled concentration profiles was very similar to that between the experiments and inverse modeling. In all cases the mean absolute error did not exceed 12%. Overall, this work suggests that volume averaging can potentially be used as an alternative to inverse modeling for dispersion in homogeneous porous media.     

Coordinate mapping of analytical contaminant transport solutions to non-uniform flow fields

Existing analytical solutions to 2D and 3D contaminant transport problems are limited by the mathematically convenient assumption of uniform flow. An approximate method is developed herein for coordinate mapping of 2D (vertically-averaged) transport solutions to non-uniform steady-state irrotational and divergence-free flow fields in single-layer aquifers. The method enables existing analytical transport solutions to be applied to aquifer systems with wells, non-uniform saturated thickness, surface water features, and (to a limited degree) heterogeneous hydraulic conductivity and recharge. This mass-conservative coordinate mapping approach is inexact in its approximation of the dispersion process but is still sufficiently accurate for many simple flow systems. The degree of model error is directly proportional to the variation of velocity magnitude within the domain. These mapped analytical solutions are compared to numerical simulation results and the coordinate mapping errors are investigated. The methods described herein may be used in the traditional capacity of analytical transport models, i.e., screening and preliminary site assessment, without sacrificing accuracy by assuming locally uniform flow conditions or applying an ad-hoc coordinate transformation. The solutions benefit from the traditional advantages of analytical methods, particularly the removal of artifacts due to spatial and temporal discretization: no time-stepping or numerical discretization is required.     

One-dimensional solute dispersion along unsteady flow through a heterogeneous medium,dispersion being proportional to the square of velocity

Abstract

One-dimensional solute transport, originating from a continuous uniform point source, is studied along unsteady longitudinal flow through a heterogeneous medium of semi-infinite extent. Velocity is considered as directly proportional to the linear spatially-dependent function that defines the heterogeneity. It is also assumed temporally dependent. It is expressed in both the independent variables in degenerate form. The dispersion parameter is considered to be proportional to square of the velocity. Certain new independent variables are introduced through separate transformations to reduce the variable coefficients of the advection–diffusion equation to constant coefficients. The Laplace Transformation Technique (LTT) is used to obtain the desired solution. The effects of heterogeneity and unsteadiness on the solute transport are investigated.

Editor D. Koutsoyiannis; Associate editor F.F. Hattermann

Citation Kumar, A., Jaiswal, D.K., and Kumar, N., 2012. One-dimensional solute dispersion along unsteady flow through a heterogeneous medium, dispersion being proportional to the square of velocity. Hydrological Sciences Journal, 57 (6), 1223–1230.     

Box and Gaussian plume models of the exhaust composition evolution of subsonic transport aircraft in- and out of the flight corridor

A box and a Gaussian plume model including gas-phase photochemistry and with plume dispersion parameters estimated from the few available plume observations are proposed and used for evaluation of photochemical transformations of exhausts from a single subsonic transport aircraft. The effects of concentration inhomogeneities in the plume cross section on the photochemical sources and sinks in the plume are analyzed for various groups of compounds. The influence of these inhomogeneities on the rate and on the mass of ambient air entrainment into the plume are studied also by comparing the box and the Gaussian plume model simulations during the first hours of their life. Due to the enterance of HO_X and NO_X from ambient air into the plume with rates varying from the wind shear and turbulence conditions, the rate of emitted NOX oxidation in the plume is dependent on these and also on the background concentration levels of HO_X and NO_X.