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Soil Science Society of America Journal 67:449-457 (2003)
© 2003 Soil Science Society of America

DIVISION S-1—SOIL PHYSICS

Effect of Water Flux on Solute Velocity and Dispersion

M. K. Shukla*,a, T. R. Ellsworthb, R. J. Hudsonb and D. R. Nielsenc

a School of Natural Resources, The Ohio State Univ., 422A Kottman Hall, 2021 Coffey Road, Columbus, OH 43210
b Dep. of Natural Resources and Environmental Sci., Univ. of Illinois, Urbana, IL 61801
c Dep. of Land and Water Resources, Univ. of California, Davis, CA 95616

* Corresponding author (shukla.9{at}osu.edu)


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Equilibrium Convective...
 Dispersion Processes
 METHODS AND MATERIALS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
A systematic analysis of the effect of displacement length, soil texture, and steady state water flux (q) on solute transport was performed on saturated repacked loam and sandy loam soil columns to identify the simplest possible description that was consistent with the observations across all of the breakthrough curves (BTCs) for each data set. All 39 BTCs obtained from loam and 21 from sandy loam soils, with column lengths of 10, 20, and 30 cm, and a two-order magnitude variation in measured pore water velocity (vm), were fitted simultaneously and global estimates of parameters [dispersivity ({lambda}) and molecular diffusion coefficient (D0)] were obtained. We showed that a two-parameter global dispersion relationship (D = {lambda}vm + D0, where D is the apparent diffusion coefficient) accurately represents the spreading process for all 39 loam soil BTCs (r2 > 0.99), whereas a single global parameter (D = {lambda}vm) was all that was required for the sandy soil (r2 > 0.98). The globally fitted {lambda} was independent of displacement length and vm. D remained independent of vm in the lower velocity ranges; however, a linear relationship between vm and fitted D was obtained for vm > 0.1 cm h-1. The results of this study illustrated the importance of molecular diffusion. In addition, we identified a nonlinear relationship between vm and average solute velocity (vs), which suggested that the anion exclusion volume decreased with increasing vm.

Abbreviations: AEM, anion exclusion model • BTC, breakthrough curves • CDE, convective dispersion transport equation • RMSE, rootmean square error


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Equilibrium Convective...
 Dispersion Processes
 METHODS AND MATERIALS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
DISPLACEMENT studies are important tools in soil physics for understanding transport of adsorbed and nonadsorbed solutes through soil. Such experiments provide valuable insight about the porous media, the behavior of chemicals, and associated processes such as diffusion, dispersion, anion exclusion, sorption, or exchange during transport (Biggar and Nielsen, 1967). The mathematical conceptualization of these processes at an individual pore scale is available (e.g., Navier-Stokes equation). However, it is not possible to characterize completely the variability or heterogeneity of the pore structure. Even if it were, in most practical applications the transport domain is much too great for computer simulation on the basis of a solution using the Navier-Stokes equation. Therefore, most models provide the description of solute transport at a macroscopic scale. The convective dispersion transport equation (CDE; Lapidus and Amundson, 1952) remains the foundation on which most analyses of solute transport in porous media have been based. In this model, the partitioning between solute concentration in sorbed and solution phases in equilibrium determines the retardation factor (R). According to Valocchi (1984), for linearly sorbed solutes, an average effective value of R for a whole range of solute concentrations and water velocities can be used. In the CDE model, solute spreading about the mean solute velocity is represented using a lumped effective diffusion coefficient (Deff). Conceptually, Deff is assumed to reflect both the phenomenon of ionic or molecular diffusion and mechanical dispersion, although these effects may not be additive (Nielsen et al., 1986). The parameter Deff has frequently been reported to be proportional to measured pore water velocity (vm) provided longitudinal diffusion has negligible influence on the dispersion coefficient and water content does not vary appreciably among different experiments (van Genuchten and Wierenga, 1977; Nkedi-Kizza et al., 1984; Shukla and Kammerer, 1998). The proportionality constant between dispersion and water velocity is generally referred to as the dispersivity ({lambda}, Jury et al., 1991). For transient flow, it has been shown that the temporal average vm can be used instead of a more complicated transient description (Wierenga, 1977). However, for spatial scales within which pronounced local variations in vm exist, the CDE may not be a valid description (Gelhar et al., 1979).

A large number of displacement experiments have been performed both in the laboratory and in the field (Nielsen and Biggar, 1961; Rao et al., 1980; Li et al., 1994). These experiments have examined the validity of the CDE model to quantify solute transport in porous media. However, limitations of much of this work are that a narrow range of velocity and experimental conditions usually prevailed. The present study is designed to extend these efforts and provide an in-depth analysis of the validity of the CDE model. We performed a systematic examination of the CDE model for two soils types under steady, saturated flow, with a wide range in water flux (q), spatial scale (i.e., column length), and two separate solute boundary conditions. We employed packed soil columns and assumed that these different soil columns behave similarly with respect to solute transport. The first objective of our study was to examine diffusion and dispersion as a function of pore water velocity and displacement length, as well as pore size distribution. The second objective was to identify the simplest possible description that was consistent with the observations across all of the BTCs for each data set. The remaining objectives included the evaluation of different transport mechanisms such as solute exclusion and physical nonequilibrium.


   Equilibrium Convective Dispersion Equation
TOP
 ABSTRACT
 INTRODUCTION
 Equilibrium Convective...
Dispersion Processes
 METHODS AND MATERIALS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
For the prescribed experimental conditions, which corresponded to detection of flux concentrations (Parker and van Genuchten, 1984a), we employed the one-dimensional CDE of Lapidus and Amundson (1952). The column was assumed to be semiinfinite with a constant flux boundary condition at the inlet as described in Eq. [1] and [2].


[1]

[2]

In Eq. [1] and [2], vm is the average measured pore water velocity (L T-1), x is the distance from the inflow boundary in the direction of flow (L), C is the solution concentration [mass (M) L-1], D is the apparent (or effective) diffusion coefficient (L2 T-1), t is the time (T), and R is the retardation factor. For linear sorption, R = 1 + ({rho}bKD/{theta}), where {theta} is the volumetric soil water content (L3 L-3), {rho}b is the soil bulk density (M L-3), KD is the distribution coefficient (S C-1), S is the sorbed concentration (M M-1), t0 is duration of pulse (T), and C0 is the solute concentration of applied solution at the inlet boundary. The solutions of Eq. [1] for a variety of boundary conditions for step and pulse type solute applications are given in van Genuchten and Alves (1982).


   Dispersion Processes
TOP
 ABSTRACT
 INTRODUCTION
 Equilibrium Convective...
 Dispersion Processes
METHODS AND MATERIALS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Taylor (1953), De Josselin De Jong (1958), Bear and Bachmat (1967), and Fried and Combarnous (1971) developed the classical dispersion theory assuming the random capillary bundle concept and suggested a dispersion equation similar to Fick's law, which takes into account both dispersive and diffusive fluxes. Several mechanisms causing macroscopic mixing are generally accounted for in the dispersion coefficient. Some of these are: mixing due to tortuosity, inaccessibility of pore water, recirculation due to flow restrictions, macroscopic and hydrodynamic dispersion, and turbulence in flow paths (Greenkorn, 1983). In addition, molecular diffusion, the presence of dead end pores, sorption, exclusion, and physical nonequilibrium affect the degree of asymmetry in BTCs in different proportions (Nielsen et al., 1986). According to Krupp and Elrick (1968), {lambda} is dependent on the water content of the porous media. Wilson and Gelhar (1974) reported a 10-fold increase in longitudinal {lambda} when water content decreased from saturation. According to Jury et al. (1991), a typical value of {lambda} is 0.5 to 2 cm for laboratory soil columns. The effective dispersion coefficient generally varies with mean microscopic flow velocity. On the basis of the magnitude of the Peclet number (Pe, defined as vmL/D, where L is a characteristic length), Pfannkuch (1962) developed an empirical relationship between the dispersion coefficient and Pe for five flow regimes and experimentally demonstrated the applicability of the relationship. The five dispersion regimes identified by Pfannkuch (1962) were (i) pure molecular diffusion, (ii) molecular diffusion and kinematic dispersion, (iii) predominant kinematic dispersion, and (iv) and (v) were pure kinematic dispersion regimes. In Regimes 2 to 5, an increase in average vm increased mixing and reduced the impact of D0 in the direction of flow. Perkins and Johnston (1963), using mixing cell approximations, also showed that in the region 0.01 < Pe < 50, dispersion was directly proportional to vm. Further increases in Pe resulted in a nonlinear relation to velocity (Pe {propto} vnm, with n > 1). Pfannkuch (1962) and Torelli and Scheidegger (1972) reported an n value of 1.2, whereas Taylor (1953) gave an n value of 2.

Despite these early theoretical works, in practice, the CDE model parameters are often calibrated to observations, without making a distinction between diffusion and dispersion mechanisms (Butters et al., 1989; Porro et al., 1993; Ellsworth et al., 1996; Lee et al., 2000).


   METHODS AND MATERIALS
TOP
 ABSTRACT
 INTRODUCTION
 Equilibrium Convective...
 Dispersion Processes
 METHODS AND MATERIALS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
We selected a loam (Entisol) and a sandy loam (Mollisol) soil for this study. Approximately 40 kg of both soils were collected from the field. The soils were air-dried and sieved through a 2-mm sieve and stored in the laboratory. Several representative samples from sieved portions were analyzed for their physical and chemical properties (Table 1). Air-dried soil was packed into 10-cm ID acrylic plastic cylinders having lengths of 10, 20, and 30 cm. Care was taken to follow exactly the same procedure for packing all of the soil columns. The particle density ({rho}p) and {rho}b were quantified for each soil column (Tables 1, 2). The porosity of each column was calculated from the {rho}p and {rho}b relationship. Each soil column was slowly saturated via capillary rise with 0.1 M CaBr2 from the column bottom using a point source. Later, the head at the inlet was slowly raised till the column started flowing. Steady state flow was verified by monitoring inflow and effluent volumes with respect to time. The effluent solutions were collected at fixed time intervals. To span a wide range of flow conditions, displacement experiments using 0.1 M MgCl2 at a series of measured pore water velocities were performed on each column starting with the lowest vm. The step input experiments were terminated when Br- was no longer detected in the effluent (Fig. 1a, b, d) . For a pulse input, the original saturating solution followed 300 mL of displacing solution (Fig. 1c,e,f and 2a,b,c) . All the experiments were conducted at a controlled temperature of 20 ± 2°C. The concentrations of Cl- and Br- were determined using a silver electrode and titrating the solution with silver nitrate solution. The weight of the soil column was recorded both at the beginning and at the end of an experiment to get an accurate value of soil water content for each soil column. All together, sixty displacement experiments, 39 within loam soil and 21 within sandy loam, were conducted using 14 different soil columns (Table 2). The columns were kept flowing throughout the experimental period and were never allowed to dry. A more elaborate description on the experimental methods, soil, and chemical analysis can be found in Shukla et al. (2000) and Shukla et al. (2002).


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  		Table 1. Physical and chemical properties of loam (L) and sandy loam (SL) soil, where CEC is cation exchange capacity, {rho}p is particle density of soil, and EC is electrical conductivity.

 

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  		Table 2. Details of the experiments through soil columns along with locally fitted retardation coefficient (R) and pulse duration (T0).

 


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  		Fig. 1. Measured and connective dispersion transport equation (CDE) fitted breakthrough curves from loam soil columns for G{lambda}D0Vl model for step chloride application in (a, b) 10- and (d) 20-cm columns and pulse application in (c) 10-, (e) 20-, and (f) 30-cm columns. Note that time of effluent arrival is plotted on a log scale.

 


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  		Fig. 2. Measured and connective dispersion transport equation (CDE) fitted breakthrough curves from sandy loam soil columns for G{lambda}D0Vl model for pulse chloride application in (a) 10-, (b) 20-, and (c) 30-cm columns. Time of effluent arrival is plotted on a log scale.

 
Parameter Estimation
The parameters of the classical CDE were fitted to the measured BTCs using the nonlinear curve-fitting program CXTFIT (Parker and van Genuchten, 1984b). The CXTFIT code provides optimal (in a least-squares sense) transport model parameters for each experiment at a given vm. In our preliminary evaluation of the BTCs for each soil, we first fitted D and R values for each experiment individually using CXTFIT (Parker and van Genuchten, 1984b). However, this resulted in R values that varied between 0.8 and 1.2, which seemed unrealistic. In summary, this approach did not prove useful to understand transport mechanisms across the entire range of vms. In hindsight, we believe this was a consequence of confounding mechanisms operative in the columns, an anion exclusion volume that varied systematically with vm, the influence of molecular diffusion as well as slight transport variations between columns. Thus, we developed an alternate strategy to evaluate the influence of vm and column length on transport. We did this by simultaneously fitting global (i.e., for all 39 BTCs for loam and all 21 BTCs for sandy loam) parameters to each of the BTCs. This was achieved by modifying the original CXTFIT code for a simultaneous optimization of the CDE parameters for all 39 experiments for the loam soil and a separate fit to all 21 experiments with the sandy loam columns. According to the CDE model, the effective {lambda} should be a constant at different flow rates and displacement lengths (Khan and Jury, 1990). Instead of fitting the dispersion coefficient directly, we modified the CXTFIT code to fit {lambda} with and/or without a molecular diffusion coefficient (D0), using the relationship D = {lambda}vm + D0. We evaluated several alternative parameterizations of the CDE to describe either all 39 BTCs for the loam soil or all 21 BTCs for the sandy loam soil, to identify the simplest and most consistent description across all of the BTCs for each data set. This objective was sought under the hypothesis that the several columns would behave similarly with respect to solute transport.

We also examined the physical nonequilibrium CDE as well in the model parameterization effort and concluded that the data did not support this more complicated description (this decision was made on the basis of the travel time variance analysis, discussed subsequently). We do not discuss the nonequilibrium CDE further herein. In all of these model calibration and evaluation efforts, we used the measured pulse duration and applied concentrations. The several parameterizations, or models, we evaluated for each soil type differ with respect to the number of parameters employed to fit all 39 BTCs (or 21 BTCs) for loam (or sandy loam) soils. The first model had the least parameters. In this approach, we fitted a global {lambda} value while keeping R = 1. We denote this model as G{lambda}. The second model differed only with the inclusion of a global estimated diffusion coefficient (D0). We denote this model as G{lambda}D0. We included a global D0 while fitting {lambda} because several of our vm were relatively low and diffusion and mechanical dispersion should be operative simultaneously (Pfannkuch, 1962). In the third model, we fit both global {lambda} and R values, using the vm. We denote this model as G{lambda}R (i.e., global {lambda} and R values). The fourth model was G{lambda}RD0. We noted from the preliminary fitting of D and R values separately to each BTC, a relatively systematic variation between vm and the average solute velocity (vs = vm/R). Since R is theoretically constant with variations in vm, we concluded that this systematic variation was due to an effective anion exclusion volume that varied with vm. We developed a two-parameter global description of this systematic variation (with parameters A and B as defined subsequently in Eq. [4]). Thus, we also evaluated the performance of this effective solute velocity model with the global {lambda} model and with and/or without diffusion. We denote these two parameterizations as G{lambda}AB and G{lambda}D0AB, being the three- and four-parameter description of all 39 BTCs for loam soil and all 21 BTCs for sandy loam soil, respectively. We also desired to evaluate the global dispersion process without the confounding influence of the systematic variations in effective solute velocity. To do this, we used a locally fitted velocity (Vl, where l denotes local) for each BTC and a global dispersion model with and/or without diffusion. We denote these parameterizations as G{lambda}Vl and G{lambda}D0Vl. These parameterizations were evaluated by comparing the sum of squares, root mean square error (RMSE), and coefficient of determination between observed and fitted relative chloride concentration.

Time Moment Analysis
As a preliminary evaluation of the data, we performed time moment analysis. Time moment analysis provides a model independent tool for characterizing the solute BTCs. The first temporal moment provides the mean breakthrough time, the second central temporal moment (i.e., the variance) describes the solute spreading, and the third (skewness) describes the degree of asymmetry of the BTCs (Valocchi, 1985). The equation for calculating the nth travel time moment is (Jury and Roth, 1990):

[3]

The mean travel time to depth L is T1/T0 and the travel time variance is T2/T0 - (T1/T0)2. These numerical estimates can be compared with the CDE theoretical travel time moments to provide estimates of the CDE model parameters, in contrast to least-squares fitting of the analytical solution to Eq. [1] and [2].

The solute application corresponded with either a step or a finite pulse input. For a finite pulse, the expected or theoretical mean travel time to depth L is (RL/vm) + (t0/2) and the theoretical travel time variance is + . For the step input experiments, we first fit a smooth cubic spline to each BTC, and then computed the derivative with respect to time from this smooth spline fit. The numerical temporal moments were then estimated using this derivative function (the corresponding CDE theoretical moments are for a dirac source, equivalent to setting t0 = 0 in the above theoretical moment expressions; see Parker and van Genuchten, 1984a). To allow comparison between travel time moments estimated from both finite pulse and derivative-based step input applications, adjustments were made to the finite pulse applications (for example, the quantity, t20/12 was subtracted from both the numerical and theoretical travel time variance estimates for a given pulse input experiment).


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 Equilibrium Convective...
 Dispersion Processes
 METHODS AND MATERIALS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Our hypothesis was that all loam (or sandy loam) soil columns were macroscopically homogeneous with respect to solute transport. The {rho}b and water contents of the columns showed small deviations (Table 2). The coefficient of variation of {rho}b for all loam soil columns was 0.018 (average 1.447 g cm-3) and for all sandy loam soil columns was 0.018 (average 1.469 g cm-3). The average and coefficients of variation of water contents from all loam and sandy loam soil columns were 0.434, 0.03, and 0.424, 0.014, respectively. The vm was measured by determining the total water flux, based on volume of effluent collected during a given time interval, converting to a flux density, and then to an average pore water velocity by dividing the flux density by the measured volumetric water content (Table 2). The mass recoveries from both loam and sandy loam soil columns were very high (always >97% of the applied pulse of solute, Shukla et al., 2002).

Travel Time Analysis
The center of mass of an inert solute pulse under steady flow at a given average vm is model independent and moves at the same rate as the average vm. However, different process models often result in quite different rates of spreading or dispersion but these do not effect the mean travel time (Valocchi, 1985; Jury et al., 1991). The best-fit linear model (intercept = 0 and r2 > 0.99) for observed and theoretical travel times for loam soil experiments suggested an R value of {approx}1.04. However, in sandy loam experiments, the best-fit linear model gave a global R close to one (R = 0.99, r2 > 0.99). The second moment or variance after adjusting for differences in pulse duration showed a very strong linear relationship between the observed and CDE theoretical variance with r2 > 0.99 in loam experiments and >0.96 in sandy loam experiments. Therefore, we concluded that the equilibrium CDE was appropriate for describing the solute spreading process.

Parameter Estimates for Individual BTCs
To estimate the transport parameters of the CDE model, we first used the original nonlinear curve-fitting program of Parker and van Genuchten (1984b). We fitted both D and R simultaneously for each experiment. In general, for a step input of chloride, R values increased with increasing measured vm in 10- and 20-cm columns. However, R remained largely uncorrelated with vm when a pulse input of chloride was applied. This may have been due to the narrow range of velocities used in the pulse-input experiments (Table 2). Some of the fitted R values were significantly <1 in loam soil columns, indicating that chloride was subject to anion exclusion. Since the loam soil had 21% clay, exclusion of chloride was a possibility. In this case, 1–R could be viewed as a relative anion exclusion volume (van Genuchten, 1981). As can be seen in Table 2, as vm increased, the apparent anion exclusion volume decreased. This apparent decrease in exclusion volume with increasing vm is consistent with findings by Melamed et al. (1994). However, Melamed et al. (1994) observed this phenomenon for an oxic soil in which the pH-dependent charge would likely be significantly greater than for the loam soil in our study.

The fitted local R values in 10- and 20-cm loam soil columns became >1 for vm {approx} 0.1 cm h-1 and increased further with increasing vm. This suggested sorption of chloride. By definition, R is not a function of vm (as also noted in the review by Valocchi, 1984), therefore, the increase in fitted R values with vm did not seem realistic. Smaller random fluctuations in fitted R values could be attributed to small differences in packing, humus or organic matter content, water contents, and bulk densities of columns. However, as vm increased, the R values increased from about 0.8 to 1.2 in our columns. The time moment analysis also showed that R (observed/theoretical travel time), as given in Fig. 3 , varied systematically with vm. We hypothesize, without pore scale evidence, that increasing water velocity decreased the anion excluded region, perhaps via altering the relative geometry of the electokinetic plane of shear (see Fig. 3.3 in Sposito, 1984). We also fitted anion exclusion model (AEM) of van Genuchten (1981) to measured BTCs and found that dimensionless partition coefficient (ß) increased with increasing vm. Since AEM assumes that anion exclusion is restricted to immobile water phase only, the decreasing immobile water with velocity indicated decrease in anion exclusion volume. Such mechanisms are consistent with our results, although there may have been some other cause, as we did not directly measure either exclusion volumes or anion sorption in these experiments.



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  		Fig. 3. The relationship between measured pore water velocity (vm), ratio of observed and theoretical travel times, and ratio of measured and effective solute water velocity (v) for (a) loam and (b) sandy loam soil columns.

 
Global Parameter Estimates
Theoretically, R would be a constant for all soil columns and should not vary significantly between columns, because the pH of both soils was constant and close to 7 (Table 1), and the variations in bulk densities and water contents were also small (Table 2). The time moment analysis suggested global R values near 1. Therefore, we fitted D values for each experiment separately with the CDE with R = 1. These resulted in poor fits to most of the measured BTCs. Hence, we decided to use the modified CXTFIT to fit global parameters for all 39 loam BTCs. We first assumed an R = 1 and evaluated the one-parameter global dispersion model (G{lambda}). The G{lambda} model produced r2 and RMSE of 0.83 and 0.143 for experiments in loam soil columns (0.75 and 0.164 for sandy loam). The addition of one more global parameter (D0, model G{lambda}D0) improved the predictions with a lower RMSE and a higher r2 (Table 3). However, neither global model fit the BTCs very well for the loam soil with r2 values of 0.83 and 0.88, respectively (Table 3). Similar fitting of experiments in sandy loam soil columns also revealed that these parameterizations of the CDE model did not fit the measured BTCs very well, with r2 values of only {approx}0.75 for both one- and two-parameter global dispersion models (Table 3). We note that interpretation of r2 values for nonlinear models such as the CDE is not as simple as for a linear model. Thus, we provide figures and RMSE values to further evaluate model performance.


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  		Table 3. Global measures of model performance for the various global parameter estimates (number of data points in loam are 1159 and 602 in sandy loam).

 
The global fitting of {lambda} and R in (model G{lambda}R) for loam (and sandy loam) experiments produced r2 and RMSE of 0.86 and 0.143 (and 0.88 and 0.113), respectively. The global fitting of {lambda}, D0, and R in (model G{lambda}D0R) for loam (and sandy loam) gave r2 and RMSE of 0.88 and 0.118 (and 0.89 and 0.08), respectively. We note that the globally fitted R values were close to 1 for loam and 1.07 for sandy loam (Table 4). We also note from Eq. [2] that if we assume in fitting that R = 1, the estimated dispersion parameters will simply be proportional to the dispersion parameters estimated using a true R value. Thus, we decided to fit the subsequent dispersion models assuming R = 1 (i.e., G{lambda}AB, G{lambda}D0AB, G{lambda}Vl and G{lambda}D0Vl).


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  		Table 4. Global convective dispersion transport equation parameter estimates for each model and 95% linear confidence intervals for loam and sandy loam soil columns.

 
The experiments for very slow water velocities for both the step- and pulse-type boundary condition showed early arrival of chloride in effluent solution and a long flat BTC. While model G{lambda}D0 (Table 3), which explicitly accounted for diffusion, simulated the experiments significantly better in loam soil columns, it provided little improvement in fit for the sandy loam soil columns. Most of the vms in sandy loam columns were relatively high, which probably explains the lack of importance of molecular diffusion (Table 4). The sandy loam soil columns also had a greater average pore size associated with the higher sand content; therefore, convective mass transfer would likely be the dominating mechanism in the transport process. Note the significant reduction in global {lambda} values using model G{lambda}D0 for both soil types. Generally, as {lambda} decreased, the solute concentration predictions improved, as evidenced graphically and by a decrease in RMSE values.

The strong correlation (r2 > 0.98, in both loam and sandy loam soils) between the observed and theoretical variances in solute travel times also suggested that the convective dispersion process was consistent with the data, and that the CDE should fit all of the BTCs reasonably well. In the current study, the likely sources of variability between BTCs would be due to measurements of bulk density, water content, and average pore water velocity.

To examine the solute velocity variations with respect to vm, we first examined the variations due to column experimental conditions. We concluded that the observed average vs appeared randomly distributed about a given vm, with the exception of one column. This column appeared to have a slight systematic bias between observed vs and vm, with the observed vs consistently less than the vm. We found that this apparent systematic bias did not have a significant influence on the global fitting efforts. This was accomplished by employing locally fitted v values for these columns and then using the measured vm values; in doing so, the r2 improved from 0.927 to 0.931. Hence, we ignored this apparent bias, and treated the BTCs in this column the same as for all other columns in the fitting efforts. We assumed that at a given water velocity, the local deviations between vm and average vs were a consequence of rather slight variations due to packing, organic matter and clay content distributions, and/or measurement errors.

We then examined the data for a systematic relationship between observed average vs and vm across the entire range of vm values. To obtain an effective solute velocity for each BTC, we used the dispersion coefficients (D) estimated from the best-fit global estimates of {lambda} and D0 from model G{lambda}D0 for each experiment. Using these D values and an R = 1, we optimized the pore water velocity for each experiment separately using the original CXTFIT. A plot of the measured vs. fitted v suggested a slight but clear nonlinear relationship (Fig. 4) . As discussed earlier, we hypothesized that this was due to the existence of an anion exclusion volume that varied systematically with vm.



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  		Fig. 4. The relationship between measured pore water velocity and fitted pore water velocity for loam and sandy loam soil columns.

 
The systematic discrepancies between the vm for each BTC and the vs was assumed to be due to an effective solute transport volume that was less than the steady state volumetric water content. We thus related the measured water velocity, q/{theta} = vm, to an effective solute velocity (vef = 1/R x q/{theta}ef) by assuming a {theta}ef that is always < {theta} and with {theta}ef an increasing function of vm (or q). To do this, as an empirical approach, we investigated several simple two- and three-parameter functions and finally chose the following:

[4]
where v0 is 1 cm h-1 and A and B are fitting parameters. Parameter A is thus conceptually equivalent to 1/R. The parameter B is related to the effective transport volume (and thus indirectly to the excluded volume). This resulted in models G{lambda}AB and G{lambda}D0AB. The globally optimized value for A in loam soil was 0.83, which is equivalent to an R value of 1.2. The corresponding value of B was 0.21, which gave an effective transport volume of {theta}ef = {theta} [1 - 0.21 x exp (-vm/v0)]. For sandy loam columns, the fitted value of A was 0.92, which gave an R of 1.09. The corresponding value of B was 0.03, which provided an effective transport volume of {theta}ef = {theta} [1 - 0.03 x exp (-vm/v0)].

The simultaneous fitting of all the experiments with model G{lambda}AB and G{lambda}D0AB reduced the RMSE significantly for both soil types (Table 3). Model G{lambda}D0AB provided a 23% reduction in RMSE in loam experiments and 55% reduction in sandy loam experiments relative to model G{lambda}D0 (Table 3). The r2 improved to about 0.92 in loam soil columns and to 0.89 in sandy loam soil columns (Table 3). The global {lambda} and D0 values changed slightly for both soils (Table 4). Note that incorporating the nonlinearity in effective vs. measured velocities had a very significant influence on model performance, as a global R value performed poorly in comparison. The significant reduction in sum of squares (SS) by using a local effective water velocity (Table 3) thus suggests that the effective transport volume varied systematically with vm, and that this was especially evident in loam soil (Fig. 3).

To examine the global dispersion process without confounding results from velocity variations, we used locally fitted pore water velocities (vl) in model G{lambda}Vl and G{lambda}D0Vl (Table 3) and obtained global {lambda} and D0 using the modified CXTFIT program (Table 4). Model G{lambda}D0Vl produced a very significant reduction in RMSE (>81%) and a corresponding increase in r2 ({approx}0.98) in both loam and sandy loam soils (Table 3). Thus, a two-parameter global dispersivity and diffusion coefficient model G{lambda}D0Vl accurately represented the spreading process in all loam soil columns (Fig. 1). In sandy loam soil columns, the spreading was well described by just a one-parameter dispersivity model G{lambda}Vl (Fig. 2).

In our study, fitted {lambda} values were small and were independent of water velocity and displacement scale but were dependent on soil type or aggregate scale. Khan and Jury (1990) also reported small and spatially independent {lambda} in repacked soil columns. However, Wierenga and van Genuchten (1989) reported a five-fold increase in {lambda} in the repacked columns when the displacement length was increased from 30 cm to 6 m. The best fit globally optimized D0 in loam soil column was 0.0203 cm2 h-1, which was close to the reported D0 values in the literature, 0.02 cm2 h-1 by Jury et al. (1991), 0.01 cm2 h-1 by Sposito (1989), and 0.022 cm2 h-1 by Shukla et al. (2002). However, for sandy loam soil, D0 was marginally important (Table 4).

The linear relationship between fitted and measured pore water velocities (r2 > 0.99) gave an estimate of R = 1.15 in loam experiments and R = 1.08 in sandy loam experiments. The fitted function in Eq. [4] provided an estimate of R = 1.2 in loam soil and R = 1.09 in sandy loam soil. The average value of R was also obtained from the product of measured water velocity per unit displacement length and observed travel times and was 1.13 for loam experiments and 1.08 for sandy loam columns. Thus, we estimated the value of R as the average of these three estimation methods. This gave R values of 1.16 and 1.08 for loam and sandy loam soil respectively. These values were used to scale the best-fit dispersion coefficients from model G{lambda}D0Vl to get an estimate of the dispersion coefficients. For sandy loam soil experiments, a perfect linear relationship between scaled measured water velocities and scaled dispersion coefficients was obtained (Fig. 5) . However, for loam soil, Fig. 5 clearly shows that D was not directly proportional to vm in the water velocity region 0.02 < vm < 0.09 cm h-1. From vm >= 0.09 cm h-1 a perfect linear relationship between vm and D was also obtained for loam soil (Fig. 5).



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  		Fig. 5. Dispersion coefficient (D) and measured pore water velocity (vm) relationships in loam and sandy loam soil columns. CDE = connective dispersion transport equation.

 

   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 Equilibrium Convective...
 Dispersion Processes
 METHODS AND MATERIALS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
The results from this work clearly demonstrate the validity of the CDE dispersion process model for saturated, homogeneous soil across a wide range of pore water velocities, two soil types, and several column lengths. This work is important in that it simultaneously provides, in contrast to previous work, a systematic examination of the CDE process model across both a wide range in pore water velocities and column lengths. A two-parameter global dispersion relationship accurately represented the spreading process for all 39 loam soil BTCs, whereas a single global parameter was sufficient for the sandy soil. The dispersivity remained independent of spatial and velocity scale for both soils. These results also illustrated the importance of molecular diffusion in loam column experiments. The globally fitted value of D0 was close to the reported values in literature. A linear relationship between vm and D was obtained for vm >= 0.1 cm h-1 in both loam and sandy loam soil columns. The results from this work also demonstrated a nonlinear relationship between the average measured pore water velocity and effective solute velocity. This suggested that even in a relatively homogeneous system, the effective solute transport volume is a function of pore water velocity.

Received for publication October 9, 2001.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Equilibrium Convective...
 Dispersion Processes
 METHODS AND MATERIALS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 





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