Soil Science Society of America Journal 63:768-777 (1999)
© 1999 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
Predicting Solute Transport in Soils
Second-Order Two-Site Models
H.M. Selima,
Liwang Maa and
Hongxia Zhua
a Agronomy Dep., Louisiana State Univ., Baton Rouge, LA, 70803 USA
mselim{at}agctr.lsu.edu
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ABSTRACT
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Failure of numerous efforts to describe the movement of reactive solutes in soils is often due to inaccurate identification of solute–soil interactions or to the lack of independently derived model parameters. The main objective of this work was to test the applicability of chemical vs. physical nonequilibrium approaches in describing the transport of solutes in soils. The models evaluated are based on second-order two-site approaches (SOTS) with and without consideration of physical nonequilibrium in soils. The capability of these approaches for predicting the transport of metolachlor [2-Chloro-N-(2-ethyl-6-methylphenyl)-N-(2-methoxy-1-methylethyl)acetamide] in Sharkey clay (very-fine, smectitic, thermic Chromic Epiaquerts) soil columns of different aggregate sizes (<2, 2–4, and 4–6 mm) was examined. Moreover, two sets of model parameters were independently derived from the kinetic retention experiments. The first set was based on kinetic adsorption isotherms, and the second set utilized both adsorption and desorption kinetic retention (batch) results. Judging from data on the total root mean square error, parameters based on adsorption and desorption batch results provided breakthrough curve (BTC) predictions that are improved over those of parameters from adsorption kinetics only. The coupled physical and chemical nonequilibrium model (SOTS plus mobile–immobile [MIM]: SOTS–MIM) considerably improved BTC predictions for the 2- to 4- and 4- to 6-mm soil aggregate sizes. We conclude that the modified SOTS and SOTS–MIM methods provided an improved description of metolachlor transport in the Sharkey soil. Based on total root mean square errors, the modified SOTS–MIM with parameters derived from adsorption and desorption kinetic retention experiments provided best overall BTC predictions.
Abbreviations: BTC, breakthrough curve • CDE, convective–dispersive equation • MIM, mobile–immobile model • SOTS, second-order two-site
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INTRODUCTION
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BATCH AND MISCIBLE DISPLACEMENT EXPERIMENTS are commonly used to investigate solute retention and transport in porous media. Such studies have considerably contributed to our understanding of soil–solute–water interactions. Mathematical models have been developed to infer the mechanisms governing solute reactivity in soils by simulating solute concentration in the water phase. The earliest models used a linear adsorption isotherm based on batch experiments. Later, nonlinear adsorption isotherms such as the Freundlich and Langmuir equations were found to better describe reactive solutes in soils. For the past two decades, kinetic reaction models have been widely used to describe time-dependent solute adsorption and desorption (Ma and Selim, 1994a, 1994b, and 1998; Xue et al., 1997). Multiple reaction sites were also introduced to combine both equilibrium and kinetic reaction sites (Selim et al., 1976; Brusseau et al., 1989a and 1992). Selim and Amacher (1988) developed a second-order two-site (SOTS) model to account for the adsorption capacity (maximum adsorption sites) of soils. In their SOTS model, the adsorption capacity (Smax) was divided between two types of sites (S1 and S2), such that
and
. Here f is dimensionless and represents the fraction of S1 sites to the total sites. Ma and Selim (1994a) proposed a modified SOTS approach in which the adsorption maximum was not arbitrarily partitioned between S1 and S2. Specifically, the parameter f was no longer required. They found that such a modified approach resulted in improved prediction of atrazine [6-chloro-N-ethyl-N'-(1-methylethyl)-1,3,5-triazine-2,4-diamine] mobility in a Sharkey clay soil.
To account for physical nonequilibrium in solute transport, a mobile–immobile model (MIM) was introduced in the literature by Coats and Smith (1964) and later discussed by van Genuchten and Wierenga (1976). In this approach, soil water was partitioned into mobile and immobile phases with flow contribution from the mobile water only. Mass transfer between the mobile and immobile waters was assumed to be first order according to concentration gradients. van Genuchten and Wierenga (1976) further classified the fraction of soil in direct contact with the mobile water as the dynamic soil region and that in direct contact with the immobile water as the stagnant soil region. Selim and Ma (1995) modified the approach by not assigning a fixed number of adsorption sites to mobile or immobile water. Specifically, they coupled the modified SOTS concept with the MIM model, thus resulting in a coupled physical–chemical nonequilibrium approach. Though the new approach showed improved prediction of transport in soil columns for atrazine, the SOTS models have not been extensively evaluated.
The primary objectives of this study were (i) to evaluate two SOTS transport models with or without consideration of physical nonequilibrium in soils; (ii) to test the applicability of physical vs. chemical nonequilibrium approaches in describing the transport of metolachlor in Sharkey clay soil columns; and (iii) to evaluate the predication capability of adsorption vs. adsorption–desorption batch-derived model parameters for column transport results.
To accomplish these objectives, miscible displacement experiments were conducted to investigate the transport and reactivity of metolachlor in packed soil columns of different aggregate sizes (<2, 2–4, and 4–6 mm). Flow interruption was implemented to maximize the influence of nonequilibrium on metolachlor transport behavior. Breakthrough results of metolachlor were predicted using independently derived model parameters from retention kinetic (batch) adsorption–desorption experiments.
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Theory
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Second-Order Two-Site Models
Chemical degradation of metolachlor in soils is very minor, and microbial degradation of its side chain is the primary detoxication pathway of metolachlor. Microbial degradation products are chemically similar to metolachlor with slightly higher solubility in water (Chesters et al., 1989). Since we used ring-labeled metolachlor (as described below), our measurements did not differentiate between metolachlor and its degradation products. Therefore, degradation of metolachlor was not explicitly considered in our model formulation, and the simulation results may be best viewed as the cotransport of ring-labeled metolachlor and its degraded analogies. It was also assumed that an adsorption maxima (Smax) for metolachlor can be attained. Although an acceptable experimental measurement for Smax is not available, we obtained an independent estimate of Smax based on Langmuir formulation and our retention data at large times (see Results and Discussion).
Original Second-Order Two-Site Model
Based on our batch results, there was an initial fast adsorption followed by a slow adsorption of metolachlor. Therefore, an equilibrium–kinetic two-site model was invoked where an adsorption maximum was assumed. Specifically, two mathematical formulations were derived based on the SOTS chemical nonequilibrium concept, namely, the original second-order approach and a modified SOTS. The original SOTS was first proposed by Selim and Amacher (1988) and has not been used for pesticide adsorption. The adsorption maximum was divided between the two types of adsorption sites, such as
 | (1) |
where Smax is the total adsorption capacity (or adsorption maximum) (µg g-1 of soil); (Se)max and (Sk)max are the adsorption maxima for the equilibrium and kinetic sites, respectively (µg g-1 of soil); f is dimensionless, denoting the fraction of (Se)max sites to the total sites, Smax. Assuming 
e and k as the vacant or available sites (g g-1 of soil) for adsorption on the equilibrium and kinetic sites (Se and Sk), respectively, we have
 | (2) |
with total available sites of
. The associated retention mechanisms considered were first order with respect to solute concentration in the water phase and to vacant sites in the soil phase (Selim and Amacher 1988):
 | (3) |
 | (4) |
where 
is the soil water content (cm3 cm-3), Ke is a distribution constant [(g/mL)-1] for the equilibrium sites, and k1 and k2 are the forward and backward rate coefficients for the kinetic sites. The units of k1 and k2 are (g/mL)-1 h-1 and h-1, respectively. Solute concentration in water phase is C (g mL-1).
At equilibrium (t 
),
can be expressed as
 | (5) |
where
, and
and
are the corresponding affinity coefficients for the equilibrium and kinetic sites, respectively. Incorporating Eq. [3] and [4] into the convective–dispersive equation (CDE) (SOTS–CDE), we have
 | (6) |
where D is the hydrodynamic dispersion coefficient (cm2 h-1), v is the pore water velocity (cm h-1) where
and q is Darcy's water flux density (cm h-1), and 
is the soil bulk density (g cm-3).
is the associated retardation factor that results from the equilibrium reaction (dimensionless).
Modified Second-Order Two-Site Model
An approach was proposed by Ma and Selim (1994a) whereby the total adsorption sites were not partitioned between Se and Sk, but rather by assuming that the vacant sites are available to both Se and Sk. Therefore, f is not required, and the amount of solute adsorbed on each type of site is determined by the rate coefficients only. This approach has been used extensively for atrazine retention and transport (Ma and Selim, 1994a, 1994b, and 1998; Selim and Ma, 1995). Denoting as the number of sites available for solute adsorption and using the same symbols as the original SOTS described in the previous section, the associated retention mechanisms were
 | (7) |
 | (8) |
Here is related to the sorption capacity (Smax) by
 | (9) |
At equilibrium (t ), the total amount sorbed becomes
 | (10) |
Here
is the affinity coefficient of the combined equilibrium and kinetic retention mechanisms. The CDE (SOTS–CDE) thus can be written as
 | (11) |
is the retardation factor from the equilibrium reaction in the modified SOTS model.
Mobile–Immobile Physical Nonequilibrium Model
The MIM approach is a commonly used physical nonequilibrium model, in which the soil water is divided into mobile and immobile phases, based on flow velocities in the soil. Each water phase was in direct contact with a certain fraction of the soil matrix, namely, the dynamic and stagnant soil regions. Retention reaction of solutes present in each water phase was assumed to be identical for both soil regions (van Genuchten and Wierenga, 1976; Selim and Amacher, 1988). Solute transfer between the two water phases was described by an empirical first-order diffusion with 
as the mass transfer coefficient. Selim and Ma (1995) derived a mobile–immobile approach in which solute retention was assumed to follow second-order reactions. Based on their formulation, dividing the soil matrix into stagnant and dynamic regions was no longer necessary. Elimination of this arbitrary division provided improved solute predictions compared with the original MIM formulation (Selim and Ma, 1995).
The MIM model used in this study is expressed as
 | (12) |
and solute transfer between the mobile and immobile water phases was described by empirical first-order diffusion as
 | (13) |
where is the mass transfer coefficient (h-1), m and im are the respective mobile and immobile water content (cm3 cm-3), and Cm and Cim are solute concentrations of the mobile and immobile water phases (g mL-1), respectively. In addition, Dm is the dispersion coefficient (cm2 h-1), vm (cm h-1) is the pore water velocity of the mobile water phase, and Sm and Sim are the amounts of solute adsorbed from the mobile and immobile water phases (g g-1 soil), respectively. The estimate of used here was based on the method of Rao et al. (1980a, 1980b, and 1982), who derived a time-dependent for cubic soil aggregates based on physical diffusion, as
 | (14) |
where De is the effective diffusion coefficient in soil [
, where D0 is the molecular diffusion coefficient (cm2 h-1), and L and Le are the actual column length and effective solute path length (cm)]. The term l is the width of the cubic aggregates (cm) and F is the fraction of mobile water content
. Gamma (
) is a time-dependent variable and was estimated from Rao et al. (1980a and 1980b).
The MIM was coupled with the two SOTS formulations derived above (SOTS–MIM), namely, the original and modified second-order two-site chemical nonequilibrium models. Here we assume that the same reaction mechanisms prevail in the retention of solutes from the mobile and immobile water phases (Ma and Selim, 1998). A finite difference (Crank-Nicholson) method was used to provide numerical solutions of the physical and chemical nonequilibrium model presented. In addition, essential model parameters were independently derived from (other) batch experiments and were used to predict metolachlor BTCs for different soil aggregate sizes and different conditions of flow interruption.
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Materials and methods
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Adsorption–Desorption
Sharkey top soil was chosen in this study because of its highly stable aggregates (61% clay, 36% silt, 3% sand, 1.7% organic matter, and a pH of 6.48). Batch adsorption experiments with the Sharkey soil (<2 mm) were conducted to investigate the extent of kinetic retention and to provide essential model parameters for the SOTS approaches. The batch procedure followed here is described in Ma and Selim (1994a), in which a wide range of metolachlor concentrations (5, 10, 20, 40, 60, 80, and 100 g mL-1) in 0.005 M CaCl2 solution and reaction times (2, 8, 24, 48, 96, 192, 288, and 504 h) were used. The soil:water ratio was 1:2 (g soil:mL solution). In addition, triplicate samples were used. Input solutions were spiked with 14C-UL ring–labeled metolachlor. Desorption commenced immediately after the last adsorption time step (504 h). Each desorption step was conducted by replacing the supernatant with metolachlor-free 0.005 M CaCl2 solution and shaking for 24 h. Six desorptions were carried out with a total desorption time of 6 d. Metolachlor in solution was analyzed using liquid scintillation counting.
Column Experiments
Three aggregate size ranges (<2, 2–4, and 4–6 mm) were selected for use in our miscible displacement, metolachlor transport (column) experiments. Input metolachlor concentration (C0) of 20 g mL-1 in 0.005 M CaCl2 background solution was used. Carbon 14–UL ring–labeled metolachlor was utilized as a tracer in this study. Acrylic columns, 15 cm in length and 6.4 cm i.d., were used after we ascertained that this material was not active in metolachlor adsorption. Uniformly packed soil columns with bulk densities of 1.2 to 1.4 g cm-3 were purged with CO2 to improve water saturation. Column saturation was carried out upward with 0.005 M CaCl2 metolachlor-free solution for 1 wk prior to metolachlor pulse applications. Input velocity was controlled by a piston pump, and the effluent was collected by a fraction collector and analyzed by liquid scintillation counting. Each BTC from a soil column was obtained by introducing a pulse of metolachlor solution (about six pore volumes), followed by several pore volumes of 0.005 M CaCl2 metolachlor-free solution. Prior to metolachlor pulse application, a pulse of 3H2O in 0.005 M CaCl2 was applied as a tracer to each column. The tritium BTCs were similar to those published previously (Ma and Selim, 1994b) and were described using the classical CDE in which two parameters were fitted; the dispersion coefficient (D) and an equivalent or tortuous path length (Le) where Le 
L (soil column length). The use of Le was thought to be more meaningful than fitting the pore water velocity v or the pulse duration tp (Ma and Selim, 1994b). Estimates of D and Le that provided best fit of tritium BTCs are given in Table 1
.
Two metolachlor breakthrough results were obtained sequentially for each soil column. The two consecutive pulses were similar, except that during the second pulse, two flow interruptions or conditions of stop flow (4 d each) were imposed. The first flow interruption occurred after about four pore volumes of metolachlor pulse input, and the second was imposed after the metolachlor pulse was leached with two pore volumes of 0.005 M CaCl2 solution (Table 1). Flow interruption was accounted for in our models by assuming
and
(molecular diffusion coefficient) during flow interruption. Since we were unable to find a D0 value for metolachlor, D0 of 5.472 x 10-4 cm2 h-1 for atrazine was used as an approximation for metolachlor because both have a ring structure and are similar in molecular weights. At the end of each experiment, the soil columns were drained under 20-cm water suction, and the amount of water drained was used as an estimate for the mobile water content (m). This method is described in detail by Nkedi-Kizza et al. (1982) and Selim et al. (1987).
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Results and discussion
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Kinetics
A prerequisite for the use of the SOTS model is that an adsorption capacity (Smax) for a reactive solute must be determined for an individual soil. We utilized the adsorption results to obtain estimates for Smax and 
. Specifically, a linearlized Langmuir equation was fitted to metolachlor adsorption isotherm at reaction time of 192 h and yielded estimates for
and
. The corresponding standard errors were 32.96 and 0.0018. Although Chesters et al. (1989) claimed that the (equilibrium) Langmuir equation was inferior to the Freundlich equation, similar descriptions of isotherm results were obtained with both the Langmuir and the Freundlich equations for this Sharkey soil. Since the objective of the paper was to evaluate SOTS models, the kinetic Langmuir formulation was used, as described above.
Another necessary SOTS model parameter is the fraction of equilibrium sites (f). To obtain f, we fitted the linearlized Langmuir equation to metolachlor adsorption isotherm at reaction time of 2 h. Thus, the contribution from kinetic retention was assumed negligible at a reaction time of 2 h, and instantaneous equilibrium conditions were considered dominant. This assumption was based on the observed, slow kinetic retention during adsorption (for 504 h), as well as during desorption. We obtained an estimate of
and
with standard errors of 34.29 and 0.0024, respectively. Thus, an f value of 0.60 was obtained
and subsequently used to describe metolachlor sorption based on the SOTS model.
Metolachlor concentration (C) decreased with reaction time during adsorption and desorption (Fig. 1)
. Metolachlor adsorption was clearly strongly kinetic, as illustrated by the sharp decrease in concentration during early reaction times (<100 h). For convenience, we also present metolachlor concentration results during desorption in the same plot (Fig. 1). We recognize that such representation may not be strictly valid because of the displacement of supernatant at each desorption step; however, such a presentation provided a direct comparison among different adsorption–desorption isotherms (Xue and Selim, 1995). Based on the areas under each curve of the batch experiments, the total amount of metolachlor released from the soil after six desorption steps ranged from only 22 to 32% of that initially applied. Such low recoveries are indicative of the strong retention of metolachlor in the Sharkey soil. Strong retention is further illustrated by the selected adsorption and desorption isotherms that are presented in the traditional manner in Fig. 2
. These isotherms clearly show extensive hysteresis in this soil. This hysteretic behavior that results from a discrepancy between adsorption and desorption isotherms was not surprising in view of the kinetic retention behavior of metolachlor in Sharkey soil (Xue and Selim, 1995). The isotherms shown indicate the amount of irreversible and nonabsorbable phase that increased with time of reaction. Metolachlor may be retained by heterogeneous sites having a wide range of binding energies where the amount of nondesorbable almost always increases with reaction time (Chester et al., 1989).
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Fig. 1 Metolachlor concentrations in soil solution vs. reaction time. Symbols are for different initial concentrations (Ci) of 5, 10, 20, 40, 60, 80, and 100 µg mL-1 (from bottom to top). Solid curves are simulations using both adsorption and desorption batch data sets, and dashed curves are simulations using adsorption data sets only
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Fig. 2 Metolachlor adsorption (solid symbols) at reaction times of 504 h and desorption isotherms (open circles). Solid and dashed curves are simulations based on the modified second-order two-site (SOTS) model
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The original and modified SOTS models were utilized to describe the metolachlor kinetic behavior during adsorption and desorption. We followed the method described by Ma and Selim (1994a) to achieve the simulations shown in Fig. 1 for the various initial (input) concentrations (C0). Specifically, model simulations were not carried out independently for each initial concentration of our batch experiment. Rather, each model was fitted to the entire data set; i.e., all seven kinetic curves simultaneously. As a result, only one set of model parameters for our batch results were obtained (Table 2)
and should be independent of initial metolachlor concentration. Furthermore, two ways of fitting the batch results were invoked. One way was to fit metolachlor concentrations for the adsorption kinetic results only (up to 504 h; dashed lines in Fig. 1), and the other was to fit both adsorption and desorption concentration results simultaneously (solid lines in Fig. 1). Best-fit model parameters that are based on adsorption, as well as on both adsorption and desorption results, are given in Table 2 for the original and the modified SOTS models.
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Table 2 Goodness-of-fit and parameter estimates based on the original second-order two-site (SOTS) model (with f) and modified SOTS (without f) for describing metolachlor adsorption and adsorption–desorption kinetics in a Sharkey soil
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As illustrated in Fig. 1, improved description of adsorption kinetic results was obtained when the adsorption data were only used in model simulation. However, adsorption kinetics based on best-fit parameters clearly overpredicted metolachlor concentrations in the soil solution during desorption (Table 2). Although parameters based on the entire batch data set (both adsorption and desorption results) did not provide adequate description of adsorption kinetics, they offered consistently improved descriptions for metolachlor concentrations during desorption (for all different C0 results). Furthermore, the modified SOTS model (without f) offered slight improvement compared with the original SOTS (with
). Higher Ke and Kk values were obtained when adsorption and desorption results were fitted simultaneously. However, the kinetic rate coefficients k1 and k2 were 5 to 10 times lower than that from adsorption data only. Such low k1 and k2 values are indicative of strong desorption hysteresis of metolachlor in the soil (Ma and Selim, 1996). The predictive capability of our model to describe the observed hysteretic behavior is illustrated in Fig. 2 for several desorptions (dashed curves) following 504 h of adsorption (solid curve). These simulations were obtained using modified SOTS; one set of model parameters based on the entire retention data set (adsorption and desorption) was implemented. The use of parameters derived solely from adsorption results provided poor prediction of desorption isotherm results (data not shown). Such findings are consistent with the simulations illustrated by the concentration vs. time curves in Fig. 1.
Based on Eq. [10], a value of the parameter can be obtained if Ke and Kk for the modified SOTS model (without f) are known. As shown in Table 2, we obtained an estimated value of
for the adsorption results and
based on the entire retention data set (adsorption and desorption). These values compare well with of 0.0085 that is based on a simple estimation from the equilibrium form of the Langmuir equation for 192 h adsorption data, which lends credence to our SOTS approach. Moreover, the higher value of for adsorption than for desorption is consistent with observed higher affinity of metolachlor during desorption vs. adsorption; i.e. strong hysteretic behavior.
Transport
Metolachlor BTCs from the miscible displacement soil column experiments are shown in Fig. 3 through 8
. Higher peak concentrations were observed for BTC when flow interruption was not imposed compared with those when flow was interrupted. The first flow interruption resulted in a sudden decrease in metolachlor concentration in the effluent, and the second caused an initial decrease followed by an increase in concentration. Such concentration changes are indicative of the dominance of either physical (Reedy et al., 1996) and/or chemical nonequilibrium (Selim and Ma, 1995) behavior during metolachlor transport. Total mass recoveries from the effluent after 20 pore volumes were higher for BTCs without flow interruption (Table 1). A slight increase in effluent metolachlor concentration during leaching at large times (>20 pore volumes) was observed for BTCs with flow interruptions (Fig. 6 and 8). Such an increase may be due to the leaching-off of metolachlor metabolites (microbial degradation). This is likely since we measured 14C activity only in the effluent and since degradation products have higher solubility than their parent compounds (Chesters et al., 1989). Such a finding may also indirectly suggest that microbial degradation was not taking place before 20 pore volumes of continuous flow and that degradation of 14C in the effluent mainly involved metolachlor.
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Fig. 3 Predicted metolachlor breakthrough curves (BTCs) for soil column No. I-1 (<2-mm aggregates) using the original (top) and modified (bottom) second-order two-site (SOTS) models. Model parameters were based on adsorption (ad) or adsorption–desorption (ad–des) batch data sets
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Fig. 4 Predicted metolachlor breakthrough curves (BTC)s for soil column no. I-2 (<2-mm aggregates), with flow interruption using the original (top) and modified (bottom) second-order two-site (SOTS) models. Model parameters were based on adsorption (ad) or adsorption–desorption (ad–des) batch data sets
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Fig. 5 Predicted metolachlor breakthrough curves (BTC)s for soil column no. II-1 (2–4 mm aggregates) using the original (top) and modified (bottom) second-order two-site (SOTS) models. Model parameters were based on adsorption (ad) or adsorption–desorption (ad–des) batch data sets
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Fig. 6 Predicted metolachlor breakthrough curves (BTC)s for soil column no. II-2 (2–4 mm aggregates) with flow interruption, using the original (top) and modified (bottom) second-order two-site (SOTS) models. Model parameters were based on adsorption (ad) or adsorption–desorption (ad–des) batch data sets
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Fig. 7 Predicted metolachlor breakthrough curves (BTC)s for soil column no. III-1 (4–6 mm aggregates) using the original (top) and modified (bottom) second-order two-site (SOTS) models. Model parameters were based on adsorption (ad) or adsorption–desorption (ad–des) batch data sets
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Fig. 8 Predicted metolachlor breakthrough curves (BTC)s for soil column III-2 (4–6 mm aggregates), with flow interruption, using the original (top) and modified (bottom) second-order two-site (SOTS) models. Model parameters were based on adsorption (ad) or adsorption–desorption (ad–des) batch data sets
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The BTCs were predicted with both the original and modified SOTS models (Fig. 3–8). In addition, the two SOTS chemical nonequilibrium models were coupled with the MIM model to evaluate the importance of physical nonequilibrium in the different aggregate sizes. To simplify our presentation with the least confusion in notation, we restrict the use of SOTS–CDE for model predictions when physical nonequilibrium was not considered, and the use of SOTS–MIM in transport studies for coupled physical–chemical nonequilibrium models. The best predictions for the <2-mm aggregates (column I-1) were obtained with the modified SOTS model when independently estimated parameters derived from the adsorption and desorption batch experiments were used (Fig. 3). Moreover, the coupling of SOTS with the mobile–immobile concept (SOTS–MIM) provided inferior BTC predictions compared to those that did not consider physical nonequilibrium (SOTS–CDE). Therefore, for soil aggregate size range of <2 mm, physical nonequilibrium may not be important, and the differentiation of mobile from immobile water phases may not be necessary. This finding is consistent with an earlier study for atrazine transport in Sharkey soil (Selim and Ma, 1995).
As illustrated in Fig. 3, predicted BTCs based on the SOTS–MIM model were shifted to the right of SOTS–CDE model predictions. Such a right shift suggests a delay in solute transport due to the role of the immobile water phase. In addition, the tailing of BTCs was best predicted using parameters from both adsorption and desorption retention kinetic (batch) results. This finding is significant because both adsorption and desorption processes take place in column (miscible displacement) experiments. The predicted BTCs for the <2-mm aggregated soil (column I-2) are shown in Fig. 4. Similar to those shown in Fig. 3, predicted BTCs based on the SOTS–MIM model were shifted to the right of the SOTS–CDE model. The modified SOTS model and parameters from adsorption and desorption batch results provided the best overall BTC predictions. In fact, the decrease in metolachlor concentrations during flow interruption was equally well described using the modified SOTS model.
Unlike those of the <2-mm aggregates (Fig. 3 and 4), for the 2- to 4-mm soil aggregates (column II-1) the BTC predictions based on the mobile–immobile (SOTS– MIM) concept provided improved predictions vs. the SOTS–CDE model. This was the case for both the original and modified SOTS–MIM model formulations. This finding lends credence to the use of physical nonequilibrium (MIM) in our models in order to describe BTCs from the 2- to 4-mm aggregate soil column. Best prediction was achieved for the modified SOTS–MIM with parameters derived from the adsorption and desorption batch results (Table 3) . Parameters based on adsorption kinetics alone overpredicted the overall tailing of the BTC. For the case with flow interruption for the same column (2–4 mm aggregate), the mobile–immobile concept (SOTS–MIM) provided best overall predictions, including peak concentration, BTC front and tailing, and concentration changes during flow interruption when compared with the SOTS–CDE models (Fig. 6). Improved prediction was also obtained when parameters based on adsorption and desorption batch results rather than adsorption alone were utilized (see Table 3). Moreover, the modified SOTS model provided better predictions than the original SOTS model. For the sake of comparison, values for the root mean square errors (RMSE) for all cases considered in this study are listed in Table 3.
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Table 3 Root mean square errors (RMSE) of metolachlor breakthrough curve prediction using parameters derived from batch experiments
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The BTCs from soil column (III-1) packed with the largest aggregate sizes (4–6 mm) show obvious early breakthrough. Some data points were missing from the BTC front due to malfunction of the fraction collector (Fig. 7). The modified SOTS provided improved BTC predictions compared to the original SOTS. Similarly to Fig. 5 and 6, BTC predictions based on the SOTS–MIM provided improved description of metolachlor transport in this soil. In addition, best predictions were achieved based on the modified SOTS–MIM with parameter estimates from the adsorption and desorption batch results (Table 3). Similar conclusions can be drawn based on BTC predictions for column III-2 with flow interruptions (Fig. 8).
Thus far comparison of the SOTS models was based on RMSE. Other characteristics specific to each BTC may be of interest, however, such as peak concentration (peak discharge), mean residence time, and variance. Therefore, we further examined the first (M1, mean residence time) and second temporal moment (M2, variance) of the measured and predicted BTCs. Although our estimated temporal moments are based on truncated BTCs, they can be used to compare how well the various models predict metolachlor transport in terms of mean residence time and variance.
 | (15) |
where T is time expressed in pore volumes, Cr(T) is relative concentration of metolachlor (C/C0), and the integral of Cr(T) over T pulse duration (pore volumes). The calculated first and second temporal moments are listed (Tables 4 and 5)
. Calculated moments showed improved predictions and were achieved for both the original and the modified SOTS when parameters that were derived from adsorption and desorption kinetic retention experiments were utilized. This finding is consistent with RMSE results given in Table 3. However, best predictions based on the first and second moments were not always in agreement with those based on RMSE. Similarly, model evaluation based on peak concentration and peak position are also different from that of RMSE (Fig. 3–8). Therefore, the goodness of a model depends on selected criteria. Root mean square errors takes into account the deviation of each data point. It also represents best overall indicator for model comparison and is in agreement with overall observation of BTCs.
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Summary and conclusions
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Two models based on SOTS approaches were evaluated for their capability to predict the movement of metolachlor in a Sharkey clay soil. The two models were also coupled with the mobile–immobile physical nonequilibrium approach. The capability of such nonequilibrium approaches for predicting the transport of metolachlor in Sharkey clay soil columns under different experimental conditions (aggregate sizes and flow interruption) was examined. Kinetic retention (adsorption and desorption) experiments for a range of input concentrations were carried out to obtain independent estimates of necessary model parameters. Based on the total RMSE, parameters based on both adsorption and desorption data provided improved BTC predictions more than parameters from adsorption kinetic isotherms only. Furthermore, the modified SOTS–CDE and SOTS–MIM provided improved description of metolachlor transport in the Sharkey soil and are thus recommended over the original SOTS approaches.Brusseau Rao Jessup Davidson 1989
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NOTES
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Published as manuscript no. 99-09-0311.
Received for publication May 14, 1998.
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