Evidence of one-dimensional scale-dependent fractional advection-dispersion

A semi-analytical inverse method and the corresponding program FADEMain for parameter estimation of the fractional advection-dispersion equation (FADE) were developed in this paper. We have analyzed Huang et al.'s [Huang, K., Toride, N., van Genuchten, M.Th., 1995. Experimental investigation of solute transport in large homogeneous and heterogeneous saturated soil columns. Trans. Porous Media 18, 283-302.] laboratory experimental data of conservative solute transport in 12.5-m long homogeneous and heterogeneous soil columns to test the non-Fickian dispersion theory of FADE. The dispersion coefficient was calculated by fitting the analytical solution of FADE to the measured data at different transport scales. We found that the dispersion coefficient increased exponentially with transport scale for the homogeneous column, whereas it increased with transport scale in a power law function for the heterogeneous column. The scale effect of the dispersion coefficient in the heterogeneous soil was much more significant comparing to that in the homogeneous soil. The increasing rate of dispersion coefficient versus transport distance was smaller for FADE than that for the advection-dispersion equation (ADE). Finite difference numerical approximations of the scale-dependent FADE were established to interpret the experimental results. The numerical solutions were found to be adequate for predicting scale-dependent transport in the homogeneous column, while the prediction for the heterogeneous column was less satisfactory.     

Measurement and analysis of non-Fickian dispersion in heterogeneous porous media

Contaminant breakthrough behavior in a variety of heterogeneous porous media was measured in laboratory experiments, and evaluated in terms of both the classical advection-dispersion equation (ADE) and the continuous time random walk (CTRW) framework. Heterogeneity can give rise to non-Fickian transport patterns, which are distinguished by "anomalous" early arrival and late time tails in breakthrough curves. Experiments were conducted in two mid-scale laboratory flow cells packed with clean, sieved sand of specified grain sizes. Three sets of experiments were performed, using a "homogeneous" packing, a randomly heterogeneous packing using sand of two grain sizes, and an exponentially correlated structure using sand of three grain sizes. Concentrations of sodium chloride tracer were monitored at the inflow reservoir and measured at the outflow reservoir. Breakthrough curves were then analyzed by comparison to fitted solutions from the ADE and CTRW formulations. In all three systems, including the "homogeneous" one, subtle yet measurable differences between Fickian and non-Fickian transport were observed. Quantitative analysis demonstrated that the CTRW theory characterized the full shape of the breakthrough curves far more effectively than the ADE.     

Non-Fickian dispersion in a single fracture

Solute transport in fractured rocks is of major interest in many applications, from the petroleum industry to ground water management. This work focuses on the dispersion process in a transparent replica of a real single fracture. The fracture exhibits strong changes in heterogeneity, with the first half very heterogeneous and the second half fairly homogeneous. Three models have been used to interpret the tracer experiments: the classical advection-dispersion equation (ADE), the continuous time random walk (CTRW), and the stratified model. The main goals were to test these models and to study possible correlations between fitting parameters and heterogeneities. As expected, the solution derived from the ADE equation appears to be unable to model long-time tailing behavior. On the other hand, the results confirm the CTRW robustness and the coefficient beta seems well correlated to heterogeneities. Finally, the stratified model is also able to describe non-Fickian dispersion. The parameters defined by this model are correlated to the heterogeneities of the fracture.     

Continuous time random walk model better describes the tailing of atrazine transport in soil

Contaminant transport in soils is complicated and involves some physical and chemical nonequilibrium processes. In this research, the soil column displacement experiments of Cl⁻ and atrazine under different flow velocities were carried out. The data sets of Cl⁻ transport in sandy loam fitted to the convection dispersion equation (CDE) and the two-region model (TRM) indicated that the effects of physical nonequilibrium process produced by immobile water on the breakthrough curves (BTCs) of Cl⁻ and atrazine transport through the repacking soil columns were negligible. The two-site model (TSM) and the continuous time random walk (CTRW) were also used to fit atrazine transport behavior at the flow rate of 19.86 cm h⁻¹. The CTRW convincingly captured the full evolution of atrazine BTC in the soil column, especially for the part of long tailing. However, the TSM failed to characterize the tailing of atrazine BTC in the soil column. The calculated fraction of equilibrium sorption sites, F, ranging from 0.78 to 0.80 for all flow rates suggested the contribution of nonequilibrium sorption sites to the asymmetry of atrazine BTCs. Furthermore, the data sets for the flow rates of 6.68 cm h⁻¹ and 32.81 cm h⁻¹ were predicted by the TSM and the CTRW. As to the flow rate of 6.68 cm h⁻¹, the CTRW predicted the entire BTC of atrazine transport better than the TSM did. For the flow rate of 32.81 cm h⁻¹, the CTRW characterized the late part of the tail better, while the TSM failed to predict the tailings of atrazine BTC.     

Analysis of field observations of tracer transport in a fractured till

We analyze a set of observations from a recently published, field-scale tracer test in a fractured till. These observations demonstrate a dominant, underlying non-Fickian behavior, which cannot be quantified using traditional modeling approaches. We use a continuous time random walk (CTRW) approach which thoroughly accounts for the measurements, and which is based on a physical picture of contaminant motion that is consistent with the geometric and hydraulic characterization of the fractured formation. We also incorporate convolution techniques in the CTRW theory, to consider transport between different regions containing distinct heterogeneity patterns. These results enhance the possibility that limitations in predicting non-Fickian modes of contaminant migration can be overcome.     

Second-order modeling of arsenite transport in soils

Rate limited processes including kinetic adsorption-desorption can greatly impact the fate and behavior of toxic arsenic compounds in heterogeneous soils. In this study, miscible displacement column experiments were carried out to investigate the extent of reactivity during transport of arsenite in soils. Arsenite breakthrough curves (BTCs) of Olivier and Windsor soils exhibited strong retardation with diffusive effluent fronts followed by slow release or tailing during leaching. Such behavior is indicative of the dominance of kinetic retention reactions for arsenite transport in the soil columns. Sharp decrease or increase in arsenite concentration in response to flow interruptions (stop-flow) further verified that non-equilibrium conditions are dominant. After some 40-60 pore volumes of continued leaching, 30-70% of the applied arsenite was retained by the soil in the columns. Furthermore, continued arsenite slow release for months was evident by the high levels of residual arsenite concentrations observed during leaching. In contrast, arsenite transport in a reference sand material exhibited no retention where complete mass recovery in the effluent solution was attained. A second-order model (SOM) which accounts for equilibrium, reversible, and irreversible retention mechanisms was utilized to describe arsenite transport results from the soil columns. Based on inverse and predictive modeling results, the SOM model successfully depicted arsenite BTCs from several soil columns. Based on inverse and predictive modeling results, a second-order model which accounts for kinetic reversible and irreversible reactions is recommended for describing arsenite transport in soils.     

Eulerian derivation of the fractional advection-dispersion equation

A fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the second-order derivative is replaced with a fractional-order derivative. In contrast to the classical ADE, the fractional ADE has solutions that resemble the highly skewed and heavy-tailed breakthrough curves observed in field and laboratory studies. These solutions, known as alpha-stable distributions, are the result of a generalized central limit theorem which describes the behavior of sums of finite or infinite-variance random variables. We use this limit theorem in a model which sums the length of particle jumps during their random walk through a heterogeneous porous medium. If the length of solute particle jumps is not constrained to a representative elementary volume (REV), dispersive flux is proportional to a fractional derivative. The nature of fractional derivatives is readily visualized and their parameters are based on physical properties that are measurable. When a fractional Fick's law replaces the classical Fick's law in an Eulerian evaluation of solute transport in a porous medium, the result is a fractional ADE. Fractional ADEs are ergodic equations since they occur when a generalized central limit theorem is employed.     

Analysis of the long-term behavior of solute transport with nonlinear equilibrium sorption using breakthrough curves and temporal moments

We analyzed the long-term behavior of breakthrough curves (BTCs) and temporal moments of a solute subjected to Freundlich equilibrium sorption (s = kc(n)). For one-dimensional transport in a homogeneous porous medium, we derived a power-law relation between travel time, tau, and solute displacement, chi, with the exponent being equal to the Freundlich n exponent. The mean solute velocity, derived from the first time moment, was found to change as tau(n-1). For n values larger than 0.66, the second time moment could be related to c chi(2/n), where c is a constant. An approach based on the use of a critical concentration was developed to estimate the presence of the asymptotic regime in the tail of the BTC. This approach was tested successfully using numerical case studies. One-dimensional numerical simulations with varying values of k, n and initial mass were run to verify the closed form analytical expressions for the large time behavior of temporal moments and the tailing part of breakthrough curves. Good agreement between the slope of the tailing part of log-log transformed BTCs and the predicted slope using asymptotic theory was found. Asymptotic theory in general underestimated the magnitude of the concentration in the tail. The quality of the estimated concentrations in the tail improved for small values of the dispersivity. Experimental BTCs of uranin and benazolin were analyzed in combination with sorption/desorption batch experiments using asymptotic theory. A good agreement between the value of n parameter derived from desorption experiment with benazolin and the value of the n parameter derived from the tail of the BTC was found.     

Evaluating non-equilibrium solute transport in small soil columns

Displacement studies on leaching of bromide and two pesticides (atrazine and isoproturon) were conducted under unsaturated steady state flow conditions in 24 small undisturbed soil columns (5.7 cm in diameter and 10 cm long) each collected from two sites differing in soil structure and organic carbon content in North Germany. There were large and irregular variabilities in the characteristics of both soils, as well as in the shapes of breakthrough curves (BTCs) of different columns, including some with early breakthrough and increased tailing, qualitatively indicating the presence of preferential flow. It was estimated that one preferential flow column (PFC) at site A, and four at site B, contributed, respectively to 11% and 58% of the accumulated leached fraction and to more than 80% of the maximum observed standard deviation (SD) in the field-scale concentration and mass flux of pesticides at two sites. The bromide BTCs of two sites were analyzed with the equilibrium convection-dispersion equation (CDE) and a non-equilibrium two-region/mobile-immobile model. Transport parameters of these models for individual BTCs were determined using a curve fitting program, CXTFIT, and by the time moment method. For the CDE based equilibrium model, the mean values of retardation factor, R, considered separately for all columns, PFCs or non-preferential flow columns (NPFCs) were comparable for the two methods; significant differences were observed in the values of dispersion coefficients of two sites using the two estimation methods. It was inferred from the estimated parameters of non-equilibrium model that 5-12% of water at site A, and 12% at site B, was immobile during displacement in NPFCs. The corresponding values for PFCs of two sites were much larger, ranging from 25% to 51% by CXTFIT and from 24% to 72% by the moment method, suggesting the role of certain mechanisms other than immobile water in higher degrees of non-equilibrium in these columns. Peclet numbers in PFCs of both sites were consistently smaller than five, indicating the inadequacy of the non-equilibrium model to incorporate the effect of all forms of non-equilibrium in PFCs. Overall, the BTCs of individual NPFCs, PFCs and of field average concentration at the two sites were better reproduced with parameters obtained from CXTFIT than by the moment method. The moment method failed to capture the peak concentrations in PFCs, but tended to describe the desorption and tail branches of BTCs better than the curve fitting approach.     

Evaluating equilibrium and non-equilibrium transport of bromide and isoproturon in disturbed and undisturbed soil columns

In this study, displacement experiments of isoproturon were conducted in disturbed and undisturbed columns of a silty clay loam soil under similar rainfall intensities. Solute transport occurred under saturated conditions in the undisturbed soil and under unsaturated conditions in the sieved soil because of a greater bulk density of the compacted undisturbed soil compared to the sieved soil. The objective of this work was to determine transport characteristics of isoproturon relative to bromide tracer. Triplicate column experiments were performed with sieved (structure partially destroyed to simulate conventional tillage) and undisturbed (structure preserved) soils. Bromide experimental breakthrough curves were analyzed using convective-dispersive and dual-permeability (DP) models (HYDRUS-1D). Isoproturon breakthrough curves (BTCs) were analyzed using the DP model that considered either chemical equilibrium or non-equilibrium transport. The DP model described the bromide elution curves of the sieved soil columns well, whereas it overestimated the tailing of the bromide BTCs of the undisturbed soil columns. A higher degree of physical non-equilibrium was found in the undisturbed soil, where 56% of total water was contained in the slow-flow matrix, compared to 26% in the sieved soil. Isoproturon BTCs were best described in both sieved and undisturbed soil columns using the DP model combined with the chemical non-equilibrium. Higher degradation rates were obtained in the transport experiments than in batch studies, for both soils. This was likely caused by hysteresis in sorption of isoproturon. However, it cannot be ruled out that higher degradation rates were due, at least in part, to the adopted first-order model. Results showed that for similar rainfall intensity, physical and chemical non-equilibrium were greater in the saturated undisturbed soil than in the unsaturated sieved soil. Results also suggested faster transport of isoproturon in the undisturbed soil due to higher preferential flow and lower fraction of equilibrium sorption sites.     

Application of continuous time random walk theory to nonequilibrium transport in soil

Continuous time random walk (CTRW) formulations have been demonstrated to provide a general and effective approach that quantifies the behavior of solute transport in heterogeneous media in field, laboratory, and numerical experiments. In this paper we first apply the CTRW approach to describe the sorbing solute transport in soils under chemical (or) and physical nonequilibrium conditions by curve-fitting. Results show that the theoretical solutions are in a good agreement with the experimental measurements. In case that CTRW parameters cannot be determined directly or easily, an alternative method is then proposed for estimating such parameters independently of the breakthrough curve data to be simulated. We conduct numerical experiments with artificial data sets generated by the HYDRUS-1D model for a wide range of pore water velocities (υ) and retardation factors (R) to investigate the relationship between CTRW parameters for a sorbing solute and these two quantities (υ, R) that can be directly measured in independent experiments. A series of best-fitting regression equations are then developed from the artificial data sets, which can be easily used as an estimation or prediction model to assess the transport of sorbing solutes under steady flow conditions through soil. Several literature data sets of pesticides are used to validate these relationships. The results show reasonable performance in most cases, thus indicating that our method could provide an alternative way to effectively predict sorbing solute transport in soils. While the regression relationships presented are obtained under certain flow and sorption conditions, the methodology of our study is general and may be extended to predict solute transport in soils under different flow and sorption conditions.     

Stability of titania nanoparticles in soil suspensions and transport in saturated homogeneous soil columns

The stability of TiO₂ nanoparticles in soil suspensions and their transport behavior through saturated homogeneous soil columns were studied. The results showed that TiO₂ could remain suspended in soil suspensions even after settling for 10 days. The suspended TiO₂ contents in soil suspensions after 24 h were positively correlated with the dissolved organic carbon and clay content of the soils, but were negatively correlated with ionic strength, pH and zeta potential. In soils containing soil particles of relatively large diameters and lower solution ionic strengths, a significant portion of the TiO₂ (18.8-83.0%) readily passed through the soils columns, while TiO₂ was significantly retained by soils with higher clay contents and salinity. TiO₂ aggregate sizes in the column outflow significantly increased after passing through the soil columns. The estimated transport distances of TiO₂ in some soils ranged from 41.3 to 370 cm, indicating potential environmental risk of TiO₂ nanoparticles to deep soil layers.     

Dual-continuum analysis of a cadmium tracer field experiment

Cadmium penetration into a sandy-loam soil during the field ponded infiltration at Kralovska luka site in Southern Slovakia was observed in a controlled experiment. Adsorption of cadmium was examined using the radioactive tracer techniques in laboratory batch tests. Quite deep (115m)Cd penetration during the field experiment (65 cm beneath the soil surface) gives evidence of the particle-facilitated transport of cadmium through preferential pathways. More than 40% of the applied cadmium moved deeper than 10 cm. The field experiment was analyzed using both conventional single-continuum approach (based on Richards' equation for water flow and advection-dispersion equation for Cd transport) and dual-continuum approach. The latter approach allowed us to simulate more adequately the observed movement of cadmium.     

Analyses of locally measured bromide breakthrough curves from a natural gradient tracer experiment at Krauthausen

Solute travel time distributions were derived from breakthrough curves (BTCs) of bromide concentrations, which were measured during a large-scale tracer experiment in a quaternary fluviatile aquifer at Krauthausen. Travel time distributions to a specific point in the aquifer were derived from locally measured BTCs, using averaged absolute concentrations ?abs(x1,t), normalized concentrations ?norm(x1,t), and velocity-weighted normalized concentrations ?vw(x1,t). The travel time distributions were characterized in terms of equivalent convective-dispersive transport parameters: the equivalent solute velocity and equivalent dispersivity. Parameters were derived from BTCs using moment analyses and least-squares fits of the 1-D convection-dispersion equation (CDE). Both local and averaged BTCs showed pronounced tailing which was not well described by the 1-D CDE and which indicates the presence of macroscopic regions with low velocities in the aquifer. Therefore, dispersivities derived from CDE fits were significantly smaller than those derived from time moments. The BTCs of ?abs(x1,t) were dominated by only a few local BTCs with high concentrations and were less representative for the travel time distribution than BTCs of averaged normalized concentrations. Dispersivities derived from ?norm(x1,t) and ?vw(x1,t) were very similar. Finally, estimates of dispersivities and vertical correlation length of lnK, gamma 3, from BTCs were in agreement with a first-order estimate of the dispersivity and gamma 3 based on grain size data and flow meter measurements.     

One-dimensional simulation of solute transfer in saturated-unsaturated porous media using the discontinuous finite elements method

A one-dimensional transport model for simulating water flow and solute transport in homogeneous-heterogeneous, saturated-unsaturated porous media is presented. The model is composed of a combination of accurate numerical algorithms for solving the nonlinear Richard's and advection-dispersion equations (ADE). The mixed form of Richard's equation is solved using a standard finite element method (FEM) with primary variable switching. The transport equation is solved using operator splitting, with the discontinuous finite element method (DFE) for discretization of the advective term. A slope limiting procedure for DFE avoids numerical instabilities but creates very limited numerical dispersion for high Peclet numbers. An implicit finite differences scheme (FD) is used for the dispersive term. The unsaturated flow and transport model (Wamos-T) is applied to a variety of rigorous problems including transient flow, heterogeneous medium and abrupt variations of velocity in magnitude and direction due to time-varying boundary conditions. It produces accurate and mass-conservative solutions for a very large range of grid Peclet numbers. The Wamos-T model is a good and robust alternative for the simulation of mass transport in unsaturated domain.     

Determination of bromacil transport as a function of water and carbon content in soils

This study was conducted to determine the significance of bromacil transport as a function of water and carbon content in soils and to explore the implications of neglecting sorption when making assessments of travel time of bromacil through the vadose zone. Equilibrium batch sorption tests were performed for loamy sand and sandy soil added with four different levels of powdered activated carbon (PAC) content (0, 0.01, 0.05, and 0.1%). Column experiments were also conducted at various water and carbon contents under steady-state flow conditions. The first set of column experiments was conducted in loamy sand containing 1.5% organic carbon under three different water contents (0.23, 0.32, and 0.41) to measure breakthrough curves (BTCs) of bromide and bromacil injected as a square pulse. In the second set of column experiments, BTCs of bromide and bromacil injected as a front were measured in saturated sandy columns at the four different PAC levels given above. Column breakthrough data were analyzed with both equilibrium and nonequilibrium (two-site) convection-dispersion equation (CDE) models to determine transport and sorption parameters under various water and carbon contents. Analysis with batch data indicated that neglect of the partition-related term in the calculation of solute velocity may lead to erroneous estimation of travel time of bromacil, i.e. an overestimation of the solute velocity by a factor of R. The column experiments showed that arrival time of the bromacil peak was larger than that of the bromide peak in soils, indicating that transport of bromacil was retarded relative to bromide in the observed conditions. Extent of bromacil retardation (R) increased with decreasing water content and increasing PAC content, supporting the importance of retardation in the estimation of travel time of bromacil even at small amounts of organic carbon for soils with lower water content.     

A parametric transfer function methodology for analyzing reactive transport in nonuniform flow

We analyze reactive transport during in-situ bioremediation in a nonuniform flow field, involving multiple extraction and injection wells, by the method of transfer functions. Gamma distributions are used as parametric models of the transfer functions. Apparent parameters of classical transport models may be estimated from those of the gamma distributions by matching temporal moments. We demonstrate the method by application to measured data taken at a field experiment on bioremediation conducted in a multiple-well system in Oak Ridge, TN. Breakthrough curves (BTCs) of a conservative tracer (bromide) and a reactive compound (ethanol) are measured at multi-level sampling (MLS) wells and in extraction wells. The BTCs of both compounds are jointly analyzed to estimate the first-order degradation rate of ethanol. To quantify the tracer loss, we compare the approaches of using a scaling factor and a first-order decay term. Results show that by including a scaling factor both gamma distributions and inverse-Gaussian distributions (transfer functions according to the advection-dispersion equation) are suitable to approximate the transfer functions and estimate the reactive rate coefficients for both MLS and extraction wells. However, using a first-order decay term for tracer loss fails to describe the BTCs at the extraction well, which is affected by the nonuniform distribution of travel paths.     

Convergent-flow tracer tests in heterogeneous media: combined experimental-numerical analysis for determination of equivalent transport parameters

In modeling transport within naturally heterogeneous aquifers, it is usually assumed that the transport equations valid at local scales can also be applied at larger scales. At larger scales, the heterogeneous domain is represented by an equivalent homogeneous medium. Convergent-flow tracer tests constitute one of the most frequently used field tests to estimate effective input parameters of equivalent homogeneous aquifers. Traditionally, statistical approaches applied to groundwater flow and solute transport have provided tools to estimate these equivalent parameters. These approaches are based on a number of simplifications including the assumption that the point transmissivity values follow a multilog-normal random function. Several investigators have found that this assumption may not be valid in many field cases. In order to study the applicability of the equivalent homogeneous formulation in a nontraditional stochastic field, a number of experimental and numerical studies were conducted. The results are used to determine the apparent values of porosity and dispersivity that would be obtained if convergent-flow tracer tests were conducted in a deterministically generated heterogeneous transmissivity field displaying anisotropy in the correlation structure. It is shown that in this particular heterogeneous media, apparent porosity strongly depends on connectivity rather than on transmissivity. This dependence on connectivity questions the theoretical results obtained in continuum equivalent fields to estimate effective porosity.     

Transport of silver nanoparticles (AgNPs) in soil

The effect of soil properties on the transport of silver nanoparticles (AgNPs) was studied in a set of laboratory column experiments, using different combinations of size fractions of a Mediterranean sandy clay soil. The AgNPs with average size of ∼30 nm yielded a stable suspension in water with zeta potential of −39 mV. Early breakthrough of AgNPs in soil was observed in column transport experiments. AgNPs were found to have high mobility in soil with outlet relative concentrations ranging from 30% to 70%, depending on experimental conditions. AgNP mobility through the column decreased when the fraction of smaller soil aggregates was larger. The early breakthrough pattern was not observed for AgNPs in pure quartz columns nor for bromide tracer in soil columns, suggesting that early breakthrough is related to the nature of AgNP transport in natural soils. Micro-CT and image analysis used to investigate structural features of the soil, suggest that soil aggregate size strongly affects AgNP transport in natural soil. The retention of AgNPs in the soil column was reduced when humic acid was added to the leaching solution, while a lower flow rate (Darcy velocity of 0.17 cm/min versus 0.66 cm/min) resulted in higher retention of AgNPs in the soil. When soil residual chloride was exchanged by nitrate prior to column experiments, significantly improved mobility of AgNPs was observed in the soil column. These findings point to the importance of AgNP-soil chemical interactions as a retention mechanism, and demonstrate the need to employ natural soils rather than glass beads or quartz in representative experimental investigations.