A two-region nonequilibrium model for solute transport in solution conduits in karstic aquifers

A two-region nonequilibrium model was used to calibrate initial solute-transport parameter estimates generated from tracer-breakthrough curves (TBCs) developed from tracer tests conducted in uni-axial solution conduits in karstic aquifers. Two-region nonequilibrium models account for partitioning of solute into mobile- and immobile-fluid regions to produce a more representative model fit to the strong tails associated with TBCs than do equilibrium models. The nonequilibrium model resulted in an increase in average flow velocities and a decrease in longitudinal dispersion coefficients over comparable estimates using an equilibrium model. Increases in velocity and decreases in dispersion were obtained at the expense of including parameters that describe solute partitioning and mass transfer rate for the mobile- and immobile-fluid regions. In addition, nonidentifiable sorption and mass transfer parameters for the immobile-fluid regions could only be described in terms of upper and lower bounds using readily determined identifiable ratios representing solute partitioning and system constraints based on known physical properties. The identifiable ratios and system constraints serve to minimize model nonuniqueness and renders the nonidentification problem trivial.     

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.     

Interpreting organic solute transport data from a field experiment using physical nonequilibrium models

In a field experiment, two inorganic tracers and five organic solutes were injected into an unconfined sand aquifer. Breakthrough response curves were obtained at several points downgradient of the injection zone. These response curves are analyzed using a model which assumes equilibrium sorption and two models which postulate physical nonequilibrium. The physical nonequilibrium models hypothesize the existence of zones of immobile water, which act as diffusion sources and sinks for the solutes. The physical nonequilibrium models better simulate the sharp breakthrough and extended tailing exhibited by the experimental responses than does the model assuming equilibrium sorption. The reasonableness of parameters obtained from curve-fitting the data is assessed. The two physical nonequilibrium models are compared.     

Application of the method of temporal moments to interpret solute transport with sorption and degradation

In this note, we applied the temporal moment solutions of [Das and Kluitenberg, 1996. Soil Sci. Am. J. 60, 1724] for one-dimensional advective-dispersive solute transport with linear equilibrium sorption and first-order degradation for time pulse sources to analyse soil column experimental data. Unlike most other moment solutions, these solutions consider the interplay of degradation and sorption. This permits estimation of a first-order degradation rate constant using the zeroth moment of column breakthrough data, as well as estimation of the retardation factor or sorption distribution coefficient of a degrading solute using the first moment. The method of temporal moment (MOM) formulae was applied to analyse breakthrough data from a laboratory column study of atrazine, hexazinone and rhodamine WT transport in volcanic pumice sand, as well as experimental data from the literature. Transport and degradation parameters obtained using the MOM were compared to parameters obtained by fitting breakthrough data from an advective-dispersive transport model with equilibrium sorption and first-order degradation, using the nonlinear least-square curve-fitting program CXTFIT. The results derived from using the literature data were also compared with estimates reported in the literature using different equilibrium models. The good agreement suggests that the MOM could provide an additional useful means of parameter estimation for transport involving equilibrium sorption and first-order degradation. We found that the MOM fitted breakthrough curves with tailing better than curve fitting. However, the MOM analysis requires complete breakthrough curves and relatively frequent data collection to ensure the accuracy of the moments obtained from the breakthrough data.     

Applied tracer tests in fractured rock: Can we predict natural gradient solute transport more accurately than fracture and matrix parameters?

Applied tracer tests provide a means to estimate aquifer parameters in fractured rock. The traditional approach to analysing these tests has been using a single fracture model to find the parameter values that generate the best fit to the measured breakthrough curve. In many cases, the ultimate aim is to predict solute transport under the natural gradient. Usually, no confidence limits are placed on parameter values and the impact of parameter errors on predictions of solute transport is not discussed. The assumption inherent in this approach is that the parameters determined under forced conditions will enable prediction of solute transport under the natural gradient. This paper considers the parameter and prediction uncertainty that might arise from analysis of breakthrough curves obtained from forced gradient applied tracer tests. By adding noise to an exact solution for transport in a single fracture in a porous matrix we create multiple realisations of an initial breakthrough curve. A least squares fitting routine is used to obtain a fit to each realisation, yielding a range of parameter values rather than a single set of absolute values. The suite of parameters is then used to make predictions of solute transport under lower hydraulic gradients and the uncertainty of estimated parameters and subsequent predictions of solute transport is compared. The results of this study show that predictions of breakthrough curve characteristics (first inflection point time, peak arrival time and peak concentration) for groundwater flow speeds with orders of magnitude smaller than that at which a test is conducted can sometimes be determined even more accurately than the fracture and matrix parameters.     

Prediction of mass-transfer coefficient for solute transport in porous media

Several previously reported laboratory studies related to transport of solutes through packed columns were utilized to develop predictive relationships for mass-transfer rate coefficient. The data were classified into two groups: those obtained under rate-limited mass transfer between mobile and immobile water regions (physical nonequilibrium conditions), and those derived from rate-limited mass transfer between instantaneous and slow sorption sites (sorption nonequilibrium conditions). The mass-transfer coefficient in all these studies was obtained by fitting breakthrough data to a transport model employing a first-order rate limitations with a "constant" mass-transfer coefficient, independent of flow conditions. This study demonstrated that the mass-transfer coefficient in these models is dependent on system parameters including pore-water velocity, length-scale, retardation coefficient, and particle or aggregate size. Predictive relationships were developed, through regression analysis, relating mass-transfer coefficient to residence time. The developed relationships adequately estimated previously reported field mass-transfer values. Successful simulations of field desorption data reported by Bahr [J. Contam. Hydrol. 4 (1989) 205] further demonstrate the potential applicability of the developed relationships.     

Modelling the solute transport under nonequilibrium conditions on the basis of mass transfer equations

A solute transport model that describes nonequilibrium adsorption in soil/groundwater systems by mass transfer equations for film and intraparticle diffusion is presented. The model is useful in cases where breakthrough curve spreading cannot be explained by dispersion only. To evaluate its validity, the model was applied to several data sets from column experiments. The validity was also proved by a comparison with an analytical solution for the limiting case of predominating dispersion. Furthermore, a sensitivity analysis was performed to illustrate the influence of different process and sorption parameters (pore water velocity, intraparticle mass transfer coefficient, isotherm nonlinearity) on the shape of the calculated breakthrough curves. The application of the proposed model is discussed in comparison to the widely used dispersed flow/local equilibrium model, and a relationship between both models, which is based on a lumped parameter approach, is shown.     

Estimating spatially-variable first-order rate constants in groundwater reactive transport systems

Numerical reactive transport models are often used as tools to assess aquifers contaminated with reactive groundwater solutes as well as investigating mitigation scenarios. The ability to accurately simulate the fate and transport of solutes, however, is often impeded by a lack of information regarding the parameters that define chemical reactions. In this study, we employ a steady-state Ensemble Kalman Filter (EnKF), a data assimilation algorithm, to provide improved estimates of a spatially-variable first-order rate constant λ through assimilation of solute concentration measurement data into reactive transport simulation results. The methodology is applied in a steady-state, synthetic aquifer system in which a contaminant is leached to the saturated zone and undergoes first-order decay. Multiple sources of uncertainty are investigated, including hydraulic conductivity of the aquifer and the statistical parameters that define the spatial structure of the parameter field. For the latter scenario, an iterative method is employed to identify the statistical mean of λ of the reference system. Results from all simulations show that the filter scheme is successful in conditioning the λ ensemble to the reference λ field. Sensitivity analyses demonstrate that the estimation of the λ values is dependent on the number of concentration measurements assimilated, the locations from which the measurement data are collected, the error assigned to the measurement values, and the correlation length of the λ fields.     

Comparison of nonideal sorption formulations in modeling the transport of phthalate esters through packed soil columns

Sorption of dimethyl phthalate (DMP), diethyl phthalate (DEP) and dipropyl phthalate (DPP) to two soil materials that vary in organic matter content was investigated using miscible displacement experiments under saturated flow conditions. Generated breakthrough curves (BTCs) were inversely simulated using linear, equilibrium sorption (LE), nonlinear, equilibrium sorption (NL), linear, first-order nonequilibrium sorption (LFO), linear, radial diffusion (LRD), and nonlinear, first-order nonequilibrium sorption (NFO) models. The Akaike information criterion was utilized to determine the preferred model. The LE model could not adequately describe phthalate ester (PE) BTCs in higher organic matter soil or for more hydrophobic PEs. The LFO and LRD models adequately described the BTCs but a slight improvement in curve-fitting was gained in some cases when the NFO model was used. However, none of the models could properly describe the desorptive tail of DPP for the high organic matter soil. Transport of DPP through this soil was adequately predicted when degradation or sorption hysteresis was considered. Using the optimized parameter values along with values reported by others it was shown that the organic carbon distribution coefficient (K(oc)) of PEs correlates well with the octanol/water partition coefficient (K(ow)). Also, a strong relationship was found between the first-order sorption rate coefficient normalized to injection pulse size and compound residence time. A similar trend of timescale dependence was found for the rate parameter in the radial diffusion model. Results also revealed that the fraction of instantaneous sorption sites is dependent on K(ow) and appears to decrease with the increase in the sorption rate parameter.     

An interpretation of potential scale dependence of the effective matrix diffusion coefficient

Matrix diffusion is an important process for solute transport in fractured rock, and the matrix diffusion coefficient is a key parameter for describing this process. Previous studies have indicated that the effective matrix diffusion coefficient values, obtained from a large number of field tracer tests, are enhanced in comparison with local values and may increase with test scale. In this study, we have performed numerical experiments to investigate potential mechanisms behind possible scale-dependent behavior. The focus of the experiments is on solute transport in flow paths having geometries consistent with percolation theories and characterized by multiple local flow loops formed mainly by small-scale fractures. The water velocity distribution through a flow path was determined using discrete fracture network flow simulations, and solute transport was calculated using a previously derived impulse-response function and a particle-tracking scheme. Values for effective (or up-scaled) transport parameters were obtained by matching breakthrough curves from numerical experiments with an analytical solution for solute transport along a single fracture. Results indicate that a combination of local flow loops and the associated matrix diffusion process, together with scaling properties in flow path geometry, seems to be an important mechanism causing the observed scale dependence of the effective matrix diffusion coefficient (at a range of scales).     

Analytical solutions for reactive transport under an infiltration-redistribution cycle

Transport of reactive solute in unsaturated soils under an infiltration-redistribution cycle is investigated. The study is based on the model of vertical flow and transport in the unsaturated zone proposed by Indelman et al. [J. Contam. Hydrol. 32 (1998) 77], and generalizes it by accounting for linear nonequilibrium kinetics. An exact analytical solution is derived for an irreversible desorption reaction. The transport of solute obeying linear kinetics is modeled by assuming equilibrium during the redistribution stage. The model which accounts for nonequilibrium during the infiltration and assumes equilibrium at the redistribution stage is termed partial equilibrium infiltration-redistribution model (PEIRM). It allows to derive approximate closed form solutions for transport in one-dimensional homogeneous soils. These solutions are further applied to computing the field-scale concentration by adopting the Dagan and Bresler [Soil Sci. Soc. Am. J. 43 (1979) 461] column model. The effect of soil heterogeneity on the solute spread is investigated by modeling the hydraulic saturated conductivity as a random function of horizontal coordinates. The quality of the PEIRM is illustrated by calculating the critical values of the Damk?hler number which provide the achievable accuracy in estimating the solute mass in the mobile phase. The distinguishing feature of transport during the infiltration-redistribution cycle as compared to that of infiltration only is the finite depth of solute penetration. For irreversible desorption, the maximum solute penetration W/theta(r) is determined by the amount of applied water W and the residual water content theta(r). For sorption-desorption kinetics, the maximum depth of penetration z(r)(e, infinity ) also depends on the ratio between the rate of application and the column-saturated conductivity. It is shown that z(r)(e, infinity ) is bounded between the depths W/(theta(r)+K(d)) and W/theta(r) corresponding to the maximum solute penetration for equilibrium transport and for irreversible desorption, respectively. This feature of solute penetration explains the unusual phenomena of plume contraction after an initial period of spreading [Lessoff, S.C., Indelman, P., Dagan, G., 2002. Solute transport in infiltration-redistribution cycles in heterogeneous soils. In Raats, P.A.C., Smiles, D.,Warrick, A.W. (Eds), Environmental Mechanics: Water, Mass and Energy Transport in the Biosphere. American Geophysical Union, pp. 133-144]. Unlike transport under equilibrium conditions, when the solute is completely concentrated at the front, the solute under nonequilibrium conditions is spread out behind the front. Heterogeneity leads to additional spreading of the plume.     

Sorption of hydrophobic organic pollutants in saturated soil systems

Sorption equilibria and rates were characterized for a matrix of four aquifer sands and two slightly to moderately hydrophobic organic solutes (nitrobenzene and lindane), and the effects of sorption on the behavior of these solutes in saturated systems of the soils were determined. Experimental data were used to test and evaluate a variety of mathematical models for predicting contaminant fate and transport in groundwater systems.Observed equilibrium relationships between soil and solution phase solute concentrations were found to be described best by the nonlinear Freundlich isotherm model. It was further determined that the sorption process in the systems tested is rate controlled, requiring several days to approach equilibrium in completely mixed batch reactors. Subsequent modeling of solute transport in continuous flow soil column reactors was found to be most successful when rate-controlled models were used, the best results were obtained with a dual-resistance model incorporating the coupled mass transport steps of boundary-layer and intraparticle diffusion.     

Multi-process herbicide transport in structured soil columns: experiments and model analysis

Model predictions of pesticide transport in structured soils are complicated by multiple processes acting concurrently. In this study, the hydraulic, physical, and chemical nonequilibrium (HNE, PNE, and CNE, respectively) processes governing herbicide transport under variably saturated flow conditions were studied. Bromide (Br-), isoproturon (IPU, 3-(4-isoprpylphenyl)-1,1-dimethylurea) and terbuthylazine (TER, N2-tert-butyl-6-chloro-N4-ethyl-1,3,5-triazine-2,4-diamine) were applied to two soil columns. An aggregated Ap soil column and a macroporous, aggregated Ah soil column were irrigated at a rate of 1 cm h(-1) for 3 h. Two more irrigations at the same rate and duration followed in weekly intervals. Nonlinear (Freundlich) equilibrium and two-site kinetic sorption parameters were determined for IPU and TER using batch experiments. The observed water flow and Br- transport were inversely simulated using mobile-immobile (MIM), dual-permeability (DPM), and combined triple-porosity (DP-MIM) numerical models implemented in HYDRUS-1D, with improving correspondence between empirical data and model results. Using the estimated HNE and PNE parameters together with batch-test derived equilibrium sorption parameters, the preferential breakthrough of the weakly adsorbed IPU in the Ah soil could be reasonably well predicted with the DPM approach, whereas leaching of the strongly adsorbed TER was predicted less well. The transport of IPU and TER through the aggregated Ap soil could be described consistently only when HNE, PNE, and CNE were simultaneously accounted for using the DPM. Inverse parameter estimation suggested that two-site kinetic sorption in inter-aggregate flow paths was reduced as compared to within aggregates, and that large values for the first-order degradation rate were an artifact caused by irreversible sorption. Overall, our results should be helpful to enhance the understanding and modeling of multi-process pesticide transport through structured soils during variably saturated water flow.     

Entrapment and dissolution of DNAPLs in heterogeneous porous media

Two-dimensional multiphase flow and transport simulators were refined and used to numerically investigate the entrapment and dissolution behavior of tetrachloroethylene (PCE) in heterogeneous porous media containing spatial variations in wettability. Measured hydraulic properties, residual saturations, and dissolution parameters were employed in these simulations. Entrapment was quantified using experimentally verified hydraulic property and residual saturation models that account for hysteresis and wettability variations. The nonequilibrium dissolution of PCE was modeled using independent estimates of the film mass transfer coefficient and interfacial area for entrapped and continuous (PCE pools or films) saturations. Flow simulations demonstrate that the spatial distribution of PCE is highly dependent on subsurface wettability characteristics that create differences in PCE retention mechanisms and the presence of subsurface capillary barriers. For a given soil texture, the maximum and minimum PCE infiltration depth was obtained when the sand had intermediate (an organic-wet mass fraction of 25%) and strong (water- or organic-wet) wettability conditions, respectively. In heterogeneous systems, subsurface wettability variations were also found to enhance or diminish the performance of soil texture-induced capillary barriers. The dissolution behavior of PCE was found to depend on the soil wettability and the spatial PCE distribution. Shorter dissolution times tended to occur when PCE was distributed over large regions due to an increased access of flowing water to the PCE. In heterogeneous systems, capillary barriers that produced high PCE saturations tended to exhibit longer dissolution times.     

Modelling tritium and phosphorus transport by preferential flow in structured soil

Subsurface solute transport through structured soil is studied by model interpretation of experimental breakthrough curves from tritium and phosphorus tracer tests in three intact soil monoliths. Similar geochemical conditions, with nearly neutral pH, were maintained in all the experiments. Observed transport differences for the same tracer are thus mainly due to differences in the physical transport process between the different monoliths. The modelling is based on a probabilistic Lagrangian approach that decouples physical and chemical mass transfer and transformation processes from pure and stochastic advection. Thereby, it enables explicit quantification of the physical transport process through preferential flow paths, honouring all independently available experimental information. Modelling of the tritium breakthrough curves yields a probability density function of non-reactive solute travel time that is coupled with a reaction model for linear, non-equilibrium sorption–desorption to describe the phosphorus transport. The tritium model results indicate that significant preferential flow occurs in all the experimental soil monoliths, ranging from 60–100% of the total water flow moving through only 25–40% of the total water content. In agreement with the fact that geochemical conditions were similar in all experiments, phosphorus model results yield consistent first-order kinetic parameter values for the sorption–desorption process in two of the three soil monoliths; phosphorus transport through the third monolith cannot be modelled because the apparent mean transport rate of phosphorus is anomalously rapid relative to the non-adsorptive tritium transport. The occurrence of preferential flow alters the whole shape of the phosphorus breakthrough curve, not least the peak mass flux and concentration values, and increases the transported phosphorus mass by 2–3 times relative to the estimated mass transport without preferential flow in the two modelled monoliths.     

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.     

Modeling solute transport in one-dimensional homogeneous and heterogeneous soil columns with continuous time random walk

In this paper, we used the continuous time random walk (CTRW) framework to characterize the transport process in 1250-cm long one-dimensional homogenous and heterogeneous soil columns at the experiments conducted by Huang et al. [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]. The transport process was also simulated by using the advection-dispersion equation (ADE) and the spatial fractional advection-dispersion equation (FADE) for comparison. In the homogeneous soil column, the non-Fickian behavior is found at the distances less than 1000cm with beta values larger than 1.60, but less than 2, and Fickian form transport is obtained at distances larger than 1000cm with beta values larger than 2. In the heterogeneous soil column, we found the most anomalous behavior at distances from 200cm to 700cm with beta values ranging from 0.894 to 0.958, and non-Fickian transport process is observed at distances larger than 800cm with beta values in the range between 1 and 1.3. More significant non-Fickian behavior is found for transport in the heterogeneous soil column than that in the homogeneous soil column. The CTRW fits to the breakthrough curves (BTCs) have lower values of root mean square error (RMSE) and higher values of determination coefficient (r(2)), with respect to the fits of ADE and FADE. The CTRW model also is better captures the full evolution of BTCs, and especially their tails.     

Spatial and temporal distribution of solute leaching in heterogeneous soils: analysis and application to multisampler lysimeter data

Accurate assessment of the fate of salts, nutrients, and pollutants in natural, heterogeneous soils requires a proper quantification of both spatial and temporal solute spreading during solute movement. The number of experiments with multisampler devices that measure solute leaching as a function of space and time is increasing. The breakthrough curve (BTC) can characterize the temporal aspect of solute leaching, and recently the spatial solute distribution curve (SSDC) was introduced to describe the spatial solute distribution. We combined and extended both concepts to develop a tool for the comprehensive analysis of the full spatio-temporal behavior of solute leaching. The sampling locations are ranked in order of descending amount of total leaching (defined as the cumulative leaching from an individual compartment at the end of the experiment), thus collapsing both spatial axes of the sampling plane into one. The leaching process can then be described by a curved surface that is a function of the single spatial coordinate and time. This leaching surface is scaled to integrate to unity, and termed S can efficiently represent data from multisampler solute transport experiments or simulation results from multidimensional solute transport models. The mathematical relationships between the scaled leaching surface S, the BTC, and the SSDC are established. Any desired characteristic of the leaching process can be derived from S. The analysis was applied to a chloride leaching experiment on a lysimeter with 300 drainage compartments of 25 cm2 each. The sandy soil monolith in the lysimeter exhibited fingered flow in the water-repellent top layer. The observed S demonstrated the absence of a sharp separation between fingers and dry areas, owing to diverging flow in the wettable soil below the fingers. Times-to-peak, maximum solute fluxes, and total leaching varied more in high-leaching than in low-leaching compartments. This suggests a stochastic-convective transport process in the high-flow streamtubes, while convection dispersion is predominant in the low-flow areas. S can be viewed as a bivariate probability density function. Its marginal distributions are the BTC of all sampling locations combined, and the SSDC of cumulative solute leaching at the end of the experiment. The observed S cannot be represented by assuming complete independence between its marginal distributions, indicating that S contains information about the leaching process that cannot be derived from the combination of the BTC and the SSDC.     

Capability and limitations of first-order and diffusion approaches to describe long-term sorption of chlortoluron in soil

This paper compares the capability of a first-order and a spherical diffusion model to describe and predict long-term sorption and desorption processes of chlortoluron in two soils. Chlortoluron sorption was investigated at different time scales utilizing one rate experiment (120 days) and two sorption/desorption experiments. Experimental periods for sorption and desorption were set to 1 day (five desorption steps) and 30 days (three desorption steps), respectively. Upon fitting, the two models satisfactorily described the whole set of data. The spherical diffusion model performed better than the first-order model. We then tested the predictive capability of the models by predicting 30-day sorption/desorption data using kinetic parameters fitted on 1-day sorption/desorption data only. While the spherical diffusion model was able to predict the 30-day data set, the first-order model failed completely. Fitting both models to subsets of the data corresponding to different experimental time scales revealed that the rate parameter as well as the Freundlich coefficient of the first-order model are strongly time-dependent--a property that is not shared by parameters of the spherical diffusion model. The apparent stability of the spherical diffusion model with regard to time dependency of its parameters indicates that sorptive uptake may be diffusion-controlled. This also explains the models greater predictive power across different time scales compared to the first-order model. Finally, we investigate the suitability of solute class specific log-linear relationships between the first-order rate parameter and the Freundlich coefficient presented by earlier researchers in the light of the time dependency observed for the parameters of the first-order model.