Analysis of solute transport with a hyperbolic scale-dependent dispersion model

An empirical hyperbolic scale-dependent dispersion model, which predicts a linear growth of dispersivity close to the origin and the attainment of an asymptotic dispersivity at large distances, is presented for deterministic modelling of field-scale solute transport and the analysis of solute transport experiments. A simple relationship is derived between local dispersivity, which is used in numerical simulations of solute transport, and effective dispersivity, which is estimated from the analysis of tracer breakthrough curves. The scale-dependent dispersion model is used to interpret a field tracer experiment by nonlinear least-squares inversion of a numerical solution for unsaturated transport. Simultaneous inversion of concentration-time data from several sampling locations indicates a linear growth of the dispersion process over the scale of the experiment. These findings are consistent with the results of an earlier analysis based on the use of a constant dispersion coefficient model at each of the sampling depths.     

Density-dependent dispersion in heterogeneous porous media Part I: A numerical study

In this paper, we describe carefully conducted numerical experiments, in which a dense salt solution vertically displaces fresh water in a stable manner. The two-dimensional porous media are weakly heterogeneous at a small scale. The purpose of these simulations, conducted for a range of density differences, is to obtain accurate concentration profiles that can be used to validate nonlinear models for high-concentration-gradient dispersion. In this part we focus on convergence of the computations, in numerical and statistical sense, to ensure that the uncertainty in the results is small enough.Concentration variances are computed, which give estimates of the uncertainty in local concentration values. These local variations decrease with increasing density contrast. For tracer transport, obtained longitudinal dispersivities are in accordance with analytical findings. In the case of high-density contrasts, stabilizing gravity forces counteract the growth of dispersive fingers, decreasing the effective width of the transition zone. For small log-permeability variances, the decrease of the apparent dispersivity that is found is in agreement with laboratory results for homogeneous columns.     

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.     

Experimental analysis of pore-scale flow and transport in porous media

A novel, non-intrusive fluorescence imaging technique has been used to quantitatively measure the pore geometry, fluid velocity, and solute concentration within a saturated, three-dimensional porous medium. Discrete numerical averages of these quantities have been made over a representative volume of the medium and used to estimate macroscopic quantities that appear in conventional continuum models of flow and transport. The approach is meant to illustrate how microscopic information can be measured, averaged, and used to characterize medium-scale processes that are typically approximated constitutively. The experimental system consisted of a clear, cylindrical column packed with clear spherical beads and a refractive index-matched fluid seeded with fluorescent tracer particles and solute dye. By illuminating the fluid within the column with a scanning planar laser beam, details of flow and concentration within the pore spaces can be quantitatively observed, allowing for three-dimensional, dimensional, time dependent information to be obtained at good resolution. In time dependent information to be obtained at good resolution. In the current experiment, volumetrically averaged velocities and void-to-volume ratios are first compared with bulk measurements of fluid flux and medium porosity. Microscopic measurements of concentration are then used to construct cross-sectionally averaged profiles, mean breakthrough curves, and direct measurements of the dispersive flux, velocity variance, and concentration variance. In turn, the dispersive flux measurements are compared with mean concentration gradients to provide a basis for confirming the Fickian dispersion model and estimating dispersion coefficients for the medium. Coefficients determined in this manner are compared with others based upon traditional length-scale arguments, mean breakthrough analyses, and curve fits with numerical simulations.     

A 3D ERT study of solute transport in a large experimental tank

A high resolution, cross-borehole, 3D electrical resistivity tomography (ERT) study of solute transport was conducted in a large experimental tank. ERT voxels comprising the time sequence of electrical images were converted into a 3D array of ERT estimated fluid conductivity breakthrough curves and compared with direct measurements of fluid conductivity breakthrough made in wells. The 3D ERT images of solute transport behaviour were also compared with predictions based on a 3D finite-element, coupled flow and transport model, accounting for gravity induced flow caused by concentration differences.The tank (dimensions 185×245×186 cm) was filled with medium sand, with a gravel channel and a fine sand layer installed. This heterogeneous system was designed to complicate solute transport behaviour relative to a homogeneous sand tank, and to thus provide a challenging but insightful analysis of the ability of 3D ERT to resolve transport phenomena. Four ERT arrays and 20 piezometers were installed during filling. A NaCl tracer (conductivity 1.34 S/m) was injected and intensively monitored with 3D ERT and direct sampling of fluid chemistry in piezometers.We converted the bulk conductivity estimate for 250 voxels in the ERT imaged volume into ERT estimated voxel fluid conductivity by assuming that matrix conduction in the tank is negligible. In general, the ERT voxel response is in reasonable agreement with the shape of fluid conductivity breakthrough observed in six wells in which direct measurements of fluid conductivity were made. However, discrepancies occur, particularly at early times, which we attribute to differences between the scale of the image voxels and the fluid conductivity measurement, measurement errors mapped into the electrical inversion and artificial image roughness resulting from the inversion.ERT images revealed the 3D tracer distribution at 15 times after tracer injection. The general pattern and timing of solute breakthrough observed with ERT agreed with that predicted from the flow/transport modelling. However, the ERT images indicate a vertical component of tracer transport and preferential flow paths in the medium sand. We attribute this to transient vertical gradients established during tracer injection, and heterogeneity caused by sorting of the sand resulting from the filling procedure. In this study, ERT provided a unique dataset of 250 voxel breakthrough curves in 1.04 m³. The use of 3D ERT to generate an array of densely sampled estimated fluid conductivity breakthrough curves is a potentially powerful tool for quantifying solute transport processes.     

A dispersion scale effect in model calibrations and field tracer experiments

Field tracer experiments and model calibrations indicate that the magnitude of dispersivity increases as a function of the scale at which observations are made. Calculations presented in this study suggest that some part of this scaling may be explained as an artifact of the models used. Specifically, a scaling-up of dispersivity will occur whenever an (n − 1)-dimensional model is calibrated or otherwise employed to describe an n-dimensional system. The calibrated coefficients for such models will depend not only on size of the contaminant plume or tracer experiment at the time of calibration, but will exhibit a size dependency beyond the calibration period. The magnitude of scaling appears to be sufficient to encompass the range of differences between laboratory measurements of dispersivity and model calibrations.     

Simultaneous analyses and applications of multiple fluorobenzoate and halide tracers in hydrologic studies

An analytical method that employs ion chromatography has been developed to exploit the use of fluorobenzoic acids (FBAs) and halides more fully as hydrologic tracers. In a single run, this reliable, sensitive, and robust method can simultaneously separate and quantify halides (fluoride, chloride, bromide, and iodide) and up to seven FBAs from other common groundwater constituents (e.g. nitrate and sulphate). The usefulness of this analytical method is demonstrated in both field and laboratory tracer experiments. The field study examines the hydrologic response of fractures and the matrix to different flow rates and the contribution of matrix diffusion in chemical transport. Laboratory tracer experiments with eight geologic media from across the USA—mostly from Department of Energy facilities where groundwater contamination is prevalent and where subsurface characterization employing tracers has been ongoing or is in need—reveal several insights about tracer transport behaviour: (1) bromide and FBAs are not always transported conservatively; (2) the delayed transport of these anionic tracers is likely related to geologic media characteristics, such as organic matter, pH, iron oxide content, and clay mineralogy; (3) use of iodine as a hydrologic tracer should take into account the different sorption behaviours of iodide and iodate and the possible conversion of iodine's initial chemical form; (4) the transport behaviour of potential FBA and halide tracers under relevant geochemical conditions should be evaluated before beginning ambitious, large‐scale field tracer experiments. Copyright © 2005 John Wiley & Sons, Ltd.     

Tracer Tests and Image Analysis of Biological Clogging in a Two-Dimensional Sandbox Experiment

A two-dimensional flow experiment on biological clogging was carried out by biostimulating a sandbox packed with sand inoculated with bacteria. Biostimulation consisted of continuously injecting nutrients (acetate and nitrate). Clogging was visualized by frequently carrying out colored tracer experiments using Brilliant Blue. The tracer experiments were recorded with a digital camera and converted to concentration maps using an image-analysis method that revealed in detail the complex spreading pattern surrounding clogged areas. Clogging resulted in a finger-like spreading of the tracer around the main clogged area. Fingers were asymmetric and their dominant direction changed over time. Although the flow field was complex around the main clogged area, the effect on the bulk hydraulic conductivity at the scale of the sandbox was very small.     

The impact of boundary on the fractional advection–dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives

The inherent heterogeneity of geological media often results in anomalous dispersion for solute transport through them, and how to model it has been an interest over the past few decades. One promising approach that has been increasingly used to simulate the anomalous transport in surface and subsurface water is the fractional advection–dispersion equation (FADE), derived as a special case of the more general continuous time random walk or the stochastic continuum model. In FADE, the dispersion is not local and the solutes have appreciable probability to move long distances, and thus reach the boundary faster than predicted by the classical advection–dispersion equation (ADE). How to deal with different boundaries associated with FADE and their consequent impact is an issue that has not been thoroughly explored. In this paper we address this by taking one-dimensional solute movement in soil columns as an example. We show that the commonly used FADE with its fractional derivatives defined by the Riemann–Liouville definition is problematic and could result in unphysical results for solute transport in bounded domains; a modified method with the fractional dispersive flux defined by the Caputo derivatives is presented to overcome this problem. A finite volume approach is given to numerically solve the modified FADE and its associated boundaries. With the numerical model, we analyse the inlet-boundary treatment in displacement experiments in soil columns, and find that, as in ADE, treating the inlet as a prescribed concentration boundary gives rise to mass-balance errors and such errors could be more significant in FADE because of its non-local dispersion. We also discuss a less-documented but important issue in hydrology: how to treat the upstream boundary in analysing the lateral movement of tracer in an aquifer when the tracer is injected as a pulse. It is shown that the use of an infinite domain, as commonly assumed in literature, leads to unphysical backward dispersion, which has a significant impact on data interpretation. To avoid this, the upstream boundary should be flux-prescribed and located at the upstream edge of the injecting point. We apply the model to simulate the movement of Cl⁻ in a tracer experiment conducted in a saturated hillslope, and analyse in details the significance of upstream-boundary treatments in parameter estimation.     

Sensitivity of nitrate concentration-discharge patterns to soil nitrate distribution and drainage properties in the vertical dimension

We propose a conceptual model that examines the ‘variable source area’ (VSA) and ‘nitrate flushing’ hypothesis in the vertical direction, and use this approach to explain nitrate concentration–discharge relationships in a semi-arid watershed. We use an eco-hydrology simulation model (RHESSys) to show that small changes in the vertical distribution of nitrate mass and their interaction with soil hydraulic conductivity can result in abrupt changes in the nitrate concentration–discharge relationship. We show that the estimated concentration–discharge relationship is sensitive to the parameters governing soil vertical nitrate distribution and soil hydraulic conductivity, at both patch scale and watershed scale, where lateral redistribution of water and nitrate is also accounted for. Given heterogeneity in nitrogen inputs, uptake processes, soil drainage and storage processes, substantial variation in parameters that describe rate of changes in vertical distribution of soil nitrate and hydraulic properties is likely both within and between watersheds. Thus, we argue that vertical ‘variable source area’ processes may be as important as lateral VSA in determining concentration discharge relationships.     

Perspective on theories of non-Fickian transport in heterogeneous media

Subsurface fluid flow and solute transport take place in a multiscale heterogeneous environment. Neither these phenomena nor their host environment can be observed or described with certainty at all scales and locations of relevance. The resulting ambiguity has led to alternative conceptualizations of flow and transport and multiple ways of addressing their scale and space–time dependencies. We focus our attention on four approaches that give rise to nonlocal representations of advective and dispersive transport of nonreactive tracers in randomly heterogeneous porous or fractured continua. We compare these approaches theoretically on the basis of their underlying premises and the mathematical forms of the corresponding nonlocal advective–dispersive terms. One of the four approaches describes transport at some reference support scale by a classical (Fickian) advection–dispersion equation (ADE) in which velocity is a spatially (and possibly temporally) correlated random field. The randomness of the velocity, which is given by Darcy’s law, stems from random fluctuations in hydraulic conductivity (and advective porosity though this is often disregarded). Averaging the stochastic ADE over an ensemble of velocity fields results in a space–time-nonlocal representation of mean advective–dispersive flux, an approach we designate as stnADE. A closely related space–time-nonlocal representation of ensemble mean transport is obtained upon averaging the motion of solute particles through a random velocity field within a Lagrangian framework, an approach we designate stnL. The concept of continuous time random walk (CTRW) yields a representation of advective–dispersive flux that is nonlocal in time but local in space. Closely related to the latter are forms of ADE entailing fractional derivatives (fADE) which leads to representations of advective–dispersive flux that are nonlocal in space but local in time; nonlocality in time arises in the context of multirate mass transfer models, which we exclude from consideration in this paper. We describe briefly each of these four nonlocal approaches and offer a perspective on their differences, commonalities, and relative merits as analytical and predictive tools.     

Imaging Pathways in Fractured Rock Using Three‐Dimensional Electrical Resistivity Tomography

Major challenges exist in delineating bedrock fracture zones because these cause abrupt changes in geological and hydrogeological properties over small distances. Borehole observations cannot sufficiently capture heterogeneity in these systems. Geophysical techniques offer the potential to image properties and processes in between boreholes. We used three‐dimensional cross borehole electrical resistivity tomography (ERT) in a 9 m (diameter) × 15 m well field to capture high‐resolution flow and transport processes in a fractured mudstone contaminated by chlorinated solvents, primarily trichloroethylene. Conductive (sodium bromide) and resistive (deionized water) injections were monitored in seven boreholes. Electrode arrays with isolation packers and fluid sampling ports were designed to enable acquisition of ERT measurements during pulsed tracer injections. Fracture zone locations and hydraulic pathways inferred from hydraulic head drawdown data were compared with electrical conductivity distributions from ERT measurements. Static ERT imaging has limited resolution to decipher individual fractures; however, these images showed alternating conductive and resistive zones, consistent with alternating laminated and massive mudstone units at the site. Tracer evolution and migration was clearly revealed in time‐lapse ERT images and supported by in situ borehole vertical apparent conductivity profiles collected during the pulsed tracer test. While water samples provided important local information at the extraction borehole, ERT delineated tracer migration over spatial scales capturing the primary hydrogeological heterogeneity controlling flow and transport. The fate of these tracer injections at this scale could not have been quantified using borehole logging and/or borehole sampling methods alone.     

Incorporating Super‐Diffusion due to Sub‐Grid Heterogeneity to Capture Non‐Fickian Transport

Numerical transport models based on the advection‐dispersion equation (ADE) are built on the assumption that sub‐grid cell transport is Fickian such that dispersive spreading around the average velocity is symmetric and without significant tailing on the front edge of a solute plume. However, anomalous diffusion in the form of super‐diffusion due to preferential pathways in an aquifer has been observed in field data, challenging the assumption of Fickian dispersion at the local scale. This study develops a fully Lagrangian method to simulate sub‐grid super‐diffusion in a multidimensional regional‐scale transport model by using a recent mathematical model allowing super‐diffusion along the flow direction given by the regional model. Here, the time randomizing procedure known as subordination is applied to flow field output from MODFLOW simulations. Numerical tests check the applicability of the novel method in mapping regional‐scale super‐diffusive transport conditioned on local properties of multidimensional heterogeneous media.     

Solute movement in drained fen peat: a field tracer study in a Somerset (UK) wetland

Little is known about solute transport in peats, despite the obvious importance of solute transport on eco‐hydrological processes in both managed and natural peatlands. To address this lack of knowledge, we investigated solute transport processes in an agricultural fen peat using a conservative KBr tracer. The main aim of the study was to elucidate solute transport behaviour in general in this peat, with a more specific aim of investigating whether preferential or bypassing flow occurred. The tracer moved through the peat more rapidly than expected, and the pattern of movement showed clear evidence of plot‐scale bypassing flow. The data also provide evidence that bypassing flow occurs in pores at smaller scales. The implications of this study for management of wetland pastures in the Somerset Moors in south‐west England are discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

Nitrate removal and young stream water fractions at the catchment scale

Despite extensive research on nitrate export and removal, nutrient contamination remains a major threat to water bodies worldwide. At the local scale, nitrate removal is governed by biogeochemical conditions that vary in space and time, making integration to entire landscapes critical. Water transit times have often been used to describe solute transport, but the relation between water age and nitrate removal at the catchment scale is still poorly understood. We test the hypothesis that nitrate removal peaks when the fraction of young water in discharge is at its minimum, because nitrate removal occurs mostly under dry conditions where deeper, older groundwater dominates streamflow. We tested this hypothesis by exploring a detailed water quality record from the Kervidy–Naizin catchment (FR) and comparing the dynamics of nitrate to those of a conservative solute (chloride). We find that estimates of nitrate removal are consistent with previous estimates at the site and they show a good (inverse) correlation with the fraction of streamflow that is younger than 2.5 months. However, this young water fraction cannot be used to predict nitrate removal in the winter–spring period, when no removal is observed regardless of streamflow age. While this leads us to reject our hypothesis during the winter period, it also suggests that water age distributions and their correlation with nitrate removal can possibly reveal distinct sources of stream water at different hydrologic regimes and relevant biogeochemical reactions.     

Investigation of runoff generation in a pristine,poorly gauged catchment in the Chilean Andes II: Qualitative and quantitative use of tracers at three spatial scales

Understanding runoff generation processes is important for flood prediction, water management, erosion control, water quality, contaminant transport and the evaluation of impacts of land use change. However, little process research has been carried out in southern Chile. In particular the young volcanic ash soils, which are typical for this area, are not well understood in their hydrologic behaviour. To establish a ‘reference study’ which can then be used for comparison with other (disturbed) sites, this study focuses on the investigation of runoff generation processes in an undisturbed, forested catchment in the Chilean Andes. The paper reports on an investigation of these processes with different tracer methods at different spatial scales. Hydrograph separation with environmental isotopes and geochemical constituents was used on the catchment scale. Thermal energy was used as a tracer to investigate groundwater–surface water interactions at the local stream reach scale and dye tracers were used to study infiltration and percolation characteristics at the plot scale. It was found that pre‐event water dominates the storm hydrograph. In the lower reaches, however, water usually exfiltrates from the stream into the adjacent aquifer. The dye tracer experiments showed that while preferential vertical flow dominates under forest, water infiltrates as a straight horizontal front in the bare volcanic ashes (no vegetation) on the catchment rim. Subsurface flow patterns in the forest differ significantly from summer to winter. All three approaches used in this study suggest an important shift in dominant processes from dry to wet season. Copyright © 2008 John Wiley & Sons, Ltd.     

Simulation of less‐mobile porosity dynamics in contrasting sediment water interface porous media

Considering heterogeneity in porous media pore size and connectivity is essential to predicting reactive solute transport across interfaces. However, exchange with less‐mobile porosity is rarely considered in surface water/groundwater recharge studies. Previous research indicates that a combination of pore‐fluid sampling and geoelectrical measurements can be used to quantify less‐mobile porosity exchange dynamics using the time‐varying relation between fluid and bulk electrical conductivity. For this study, we use macro‐scale (10 s of cm) advection–dispersion solute transport models linked with electrical conduction in COMSOL Multiphysics to explore less‐mobile porosity dynamics in two different types of observed sediment water interface porous media. Modeled sediment textures contrast from strongly layered streambed deposits to poorly sorted lakebed sands and cobbles. During simulated ionic tracer perturbations, a lag between fluid and bulk electrical conductivity, and the resultant hysteresis, is observed for all simulations indicating differential loading of pore spaces with tracer. Less‐mobile exchange parameters are determined graphically from these tracer time series data without the need for inverse numerical model simulation. In both sediment types, effective less‐mobile porosity exchange parameters are variable in response to changes in flow direction and fluid flux. These observed flow‐dependent effects directly impact local less‐mobile residence times and associated contact time for biogeochemical reaction. The simulations indicate that for the sediment textures explored here, less‐mobile porosity exchange is dominated by variable rates of advection through the domain, rather than diffusion of solute, for typical low‐to‐moderate rate (approximately 3–40 cm/day) hyporheic fluid fluxes. Overall, our model‐based results show that less‐mobile porosity may be expected in a range of natural hyporheic sediments and that changes in flowpath orientation and magnitude will impact less‐mobile exchange parameters. These temporal dynamics can be assessed with the geoelectrical experimental tracer method applied at laboratory and field scales.     

Investigation of Small‐Scale Preferential Flow with a Forced‐Gradient Tracer Test

A new tracer experiment (referred to as MADE‐5) was conducted at the well‐known Macrodispersion Experiment (MADE) site to investigate the influence of small‐scale mass‐transfer and dispersion processes on well‐to‐well transport. The test was performed under dipole forced‐gradient flow conditions and concentrations were monitored in an extraction well and in two multilevel sampler (MLS) wells located at 6, 1.5, and 3.75 m from the source, respectively. The shape of the breakthrough curve (BTC) measured at the extraction well is strongly asymmetric showing a rapidly arriving peak and an extensive late‐time tail. The BTCs measured at seven different depths in the two MLSs are radically different from one another in terms of shape, arrival times, and magnitude of the concentration peaks. All of these characteristics indicate the presence of a complex network of preferential flow pathways controlling solute transport at the test site. Field‐experimental data were also used to evaluate two transport models: a stochastic advection‐dispersion model (ADM) based on conditional multivariate Gaussian realizations of the hydraulic conductivity field and a dual‐domain single‐rate (DDSR) mass‐transfer model based on a deterministic reconstruction of the aquifer heterogeneity. Unlike the stochastic ADM realizations, the DDSR accurately predicted the magnitude of the concentration peak and its arrival time (within a 1.5% error). For the multilevel BTCs between the injection and extraction wells, neither model reproduced the observed values, indicating that a high‐resolution characterization of the aquifer heterogeneity at the subdecimeter scale would be needed to fully capture 3D transport details.     

Prediction of regional-scale groundwater recharge and nitrate storage in the vadose zone: A comparison between a global model and a regional model

Extensive nitrogen loads at the soil surface exceed plant uptake and soil biochemical capacity, and therefore lead to nitrogen accumulation in the deep vadose zone. Studies have shown that stored nitrogen in the vadose zone can eventually reach the water table and affect the quality of groundwater resources. Recently, global scale models have been implemented to quantify nitrate storage and nitrate travel time in the vadose zone. These global models are simplistic and relatively easy to implement and therefore facilitate analysis of the considered transport processes at a regional scale with no further requirements. However, the suitability of applying these models at a regional scale has not been tested. Here, we evaluate, for the first time, the performance and utility of global scale models at the regional scale. Applied to the Loess Plateau of China, we compare estimates of groundwater recharge and nitrate storage derived from global scale models with results from a regional scale approach utilizing the Richards and advection-dispersion equations. The estimated nitrate storage was compared to nitrate observations collected in the deep vadose zone (>50 m) at five sites across the Loess Plateau. Although both models predict similar spatial patterns of nitrate storage, the recharge fluxes were three times smaller and the nitrate storage was two times higher compared with the regional model. The results suggest that global scale models are a potentially useful screening tool, but require refinement for local scale applications.