Published online 24 January 2007
Published in Vadose Zone J 6:140-148 (2007)
DOI: 10.2136/vzj2006.0103
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
One-Dimensional Modeling of Transport in Soils with Depth-Dependent Dispersion, Sorption and Decay
Jan Vanderborght* and
Harry Vereecken
Agrosphere, ICG-IV Forschungszentrum Jülich GmbH, D 52425 Jülich, Germany
* Corresponding author (j.vanderborght{at}fz-juelich.de)
Received 21 July 2006.
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ABSTRACT
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Macroscopic spatial variations in advection velocity lead to an increase in dispersion with increasing travel distance or depth. In soils, this increase goes along with a decrease in decay and sorption of organic substances. We used three different one-dimensional models that make different assumptions about the dispersion process to compare predicted leaching in a 1-m-deep soil profile with layers with different sorption and decay parameters. The first two convective–dispersive models assume that dispersion results from microscopic variations in solute particle velocities that are not correlated across soil layer boundaries. The third model, a stream tube model (STM), assumes that the particle velocity remains constant along its trajectory and is perfectly correlated in different layers. The three models were parameterized to predict the same inert tracer breakthrough curve (BTC) at 1-m depth. The first convective–dispersive model assumes a constant dispersion coefficient ("homogeneous" convection–dispersion equation [CDE]). The second model uses different dispersion coefficients in the different layers ("layered" CDE) to predict the same inert tracer BTCs as the STM at the layer boundaries. Despite similar predictions of inert tracer BTCs, the models predicted different BTCs of reactive substances at 1-m depth. The different predictions by the STM and layered CDE illustrate the importance of the correlation of solute particle velocities in different soil layers. They also point to a fundamental problem related to the use of a CDE with a depth-dependent dispersion to mimic a dispersion process caused by macroscopic variations in particle velocities.
Abbreviations: BTC, breakthrough curve • CDE, convection–dispersion equation • STM, stream tube model
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INTRODUCTION
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TO ESTIMATE the risk of contaminant leaching toward the groundwater, model calculations are often used. The reliability of these calculations is determined by the choice of the model and the model parameters, which should represent the crucial processes and soil properties correctly. Transport of areawide surface-applied chemicals is generally predicted using a one-dimensional transport model, assuming that, on average, the transport process is one-dimensional and occurring in the vertical direction.
In practice, the convection–dispersion model is often used to simulate transport in soils. The dispersion parameter is used to describe the spreading of a concentration pulse, which is applied at the soil surface, and lumps all microscopic and macroscopic processes that generate or create this spreading. Microscopic processes leading to solute dispersion are molecular diffusion and microscopic variations in pore water velocity due to pore-scale heterogeneity. At the field scale, macroscopic variations in water velocities may result from lateral redistribution of infiltrating water due to spatial variability of the hydraulic conductivity that exists at a macroscopic scale. This macroscopic variability leads to a distortion of the solute plume and dispersion or spreading of a one-dimensional depth profile of laterally averaged concentrations. In the convection–dispersion model, the dispersive solute flux is assumed to be proportional to a concentration gradient. This assumption holds when solute dispersion is caused by variations in solute particle velocities that extend over a microscopic distance or a distance that is much smaller than the particle travel distance (Dagan, 1989). If spatial variations in particle velocities extend over a macroscopic distance of the same order of magnitude as the particle travel distance or transport distance, the dispersion coefficient increases with travel distance or time (Feyen et al., 1998; Vanclooster et al., 2005; Zhou and Selim, 2003). Reviews of dispersion coefficients derived from tracer experiments in soils show that, in the first meter of the soil profile, the dispersion coefficient generally increases with depth (Beven et al., 1993; Vanderborght and Vereecken, 2007). This increase in dispersion with travel distances suggests that macroscopic velocity variations are important for solute dispersion at the field scale in soils.
Besides spatial variability in soil physical properties, chemical and biological soil properties also vary considerably in soils. The effect of stochastic spatial variability in sorption (Attinger et al., 2003; Bellin et al., 1993; Burr et al., 1994; Yang et al., 1996), sorption kinetics (Cvetkovic and Dagan, 1996; Dagan and Cvetkovic, 1996), and degradation rates (Cvetkovic and Shapiro, 1990; Hu et al., 1997) on transport of reactive substances has been studied mainly in the context of flow and transport processes in aquifers. In aquifers, a trend or systematic change of these properties with location can often be ignored. In soils, these properties and parameters change considerably with depth within the first meter of a soil profile and this trend or deterministic change cannot be neglected compared with stochastic variability. For instance, the soil organic matter content, which is important for sorption and decay of organic molecules, decreases considerably with increasing depth. In soils, this deterministic change with depth goes hand in hand with an increase in the dispersion coefficient with increasing travel distance.
The objective of this study was to investigate the effect of a depth-dependent dispersion on transport of reactive substances in soils with depth-dependent sorption and decay parameters. Therefore, we compared predictions by three different one-dimensional transport models that make different assumptions about the dispersion process. The first two models assume that the solute spreading or dispersion are caused by solute particle velocity variations that extend over a much smaller distance than the transport distance, i.e., a convective–dispersive process, and can be predicted using a convection–dispersion model. The first model assumes a convective–dispersive process in a hydrodynamically homogeneous soil profile, i.e., the mean pore water velocity and dispersion coefficient are constant with depth (homogeneous CDE model). The second model also assumes a convective–dispersive transport process but higher dispersion coefficients in the deeper soil layers (layered CDE) to accommodate the generally observed trend of dispersion with travel distance. The third model assumes that solute spreading is caused by macroscopic variations in particle advection velocities, and velocities of individual solute particles are assumed to remain constant or to be perfectly correlated along their trajectory, i.e., a stochastic–convective process (Simmons, 1982). Since the velocity of a particle remains constant, transport is assumed to occur independently in "stream tubes" and the model is called a stream tube model (STM). The three models were parameterized so that they predicted the same breakthrough of an inert tracer at 1-m depth. The layered CDE and STM models also predicted the same breakthrough of an inert tracer at the boundaries of the different soil layers. To simplify the description of the transport process, we assumed steady-state flow conditions in the soil profile so that analytical solutions of the one-dimensional transport equations could be used.
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THEORY
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For substances undergoing linear reactions, e.g., linear sorption and first-order decay, transport under steady-state flow conditions can be described using transfer functions (Jury and Roth, 1990):
 | [1] |
where zn is the depth of the bottom of the nth soil layer (the first layer is at the inlet or the soil–atmosphere surface), f(zn,t) [T–1] is the transfer function linking the input signal C(z = 0,t), i.e., the concentrations in the infiltrating water at the soil surface, to the output signal, C(zn,t), i.e., the concentration in the effluent at depth zn. Note that the concentrations considered here and below are flux concentrations. For an instantaneous application of a solute pulse, i.e., C(z = 0,t) = (M0/Jw)
(t), where M0 (M L–2) is the applied tracer mass per unit area, Jw (L T–1) is the water flux, and (t) [T–1] the Dirac function, Eq. [1] simplifies to
 | [2] |
To predict transport in a layered soil profile, the transfer function f(zn,t) needs to be related to transfer functions that describe transport through the individual soil layers.
The first two models assume a convective–dispersive transport process. Since particle velocities are correlated only over a microscopic distance along their trajectory, the variance of particle locations and the variance of particle arrival times increase linearly with time and depth, respectively. We further assumed that transport in deeper soil layers does not influence transport in the shallower layers, i.e., solute concentrations and downward fluxes in the upper soil layers are not influenced by concentrations and downward fluxes in deeper layers. As a consequence, transport in each layer is predicted using an analytical solution of the CDE for a semi-infinite soil profile with a solute flux boundary condition at the inlet surface of the soil layer. This assumption is correct when no upward movement of individual solute particles takes place at the local or microscopic scale and plausible when solute dispersion is predominantly caused by local variations in soil water velocities rather than by molecular diffusion. It should be noted that this assumption is also generally used and accepted to predict solute concentrations in the effluent from a soil column with a finite length. Given the definition of a transfer function for a Dirac solute application at the surface of a soil layer (Eq. [2]) and assuming independency from transport in deeper soil layers, the transfer function in soil layer n with thickness 
zn, f(zn,t) (which relates the flux concentrations at the outlet to those at the inlet of the layer), of a substance undergoing linear sorption and first-order decay is, for a convective–dispersive process, the solution of the CDE at x = zn:
 | [3] |
for the following boundary conditions:
 | [4] |
where x [L] is the spatial coordinate with x = 0 at the upper boundary of soil layer n, 
n [T–1] is the first-order decay rate, vn [L T–1] is the pore water velocity, Rn is the retardation coefficient, and Dn [L2 T–1] is the dispersion coefficient of the nth layer. The effect of molecular diffusion on solute dispersion is assumed to be negligible so that the dispersivity 
n [L] can be derived from Dn as
 | [5] |
The retardation coefficient Rn is defined as
 | [6] |
where 
b [M L–3] is the soil bulk density, Kdn [L3 M–1] is the linear sorption coefficient in the nth soil layer, and 
[L3 L–3] is the volumetric soil water content. Solving Eq. [3] for boundary conditions in Eq. [4], f(zn,t) is
 | [7] |
If transport in the upper soil layers is not influenced by the transport in the deeper layers and if the travel time of a particle through the deeper layer is independent of the arrival time of the solute particle at the upper boundary of the soil layer, then the transfer function f(zn,t) can be predicted using a convolution of the transfer function from the soil surface to the top of layer n, f(zn–1,t), and a transfer function, f(zn,t), for layer n:
 | [8] |
Using Eq. [7] in Eq. [8], the transfer function for a convective–dispersive process is obtained. The integral in Eq. [8] must be solved numerically, however.
Since the obtained solution for a convective–dispersive transport process in a layered soil is based on the assumption that transport in a soil layer is not influenced by the transport process in deeper soil layers (i.e., assuming that each soil layer is semi-infinite), it is strictly speaking different from a numerical solution of a CDE with depth-dependent sorption and decay parameters. A numerical solution would imply continuity of concentrations across soil layer boundaries, which in turn leads to an influence of the transport parameters in deeper soil layers on concentrations in shallower soil layers. Whether this influence is physically relevant was questioned above.
Since the travel time through layer n is not correlated to the solute particle arrival time at the inlet of layer n, the variance of arrival times of nondecaying particles (i.e., n = 0 for all layers) at depth zn, var(t;zn), equals the sum of the variance of travel times through the soil layers, var(t;zi):
 | [9] |
The third model, the STM, assumes a stochastic convective transport, i.e., the velocity of an inert solute particle does not change with depth and is perfectly correlated in two different soil layers. Since we assume a constant particle velocity, it is trivial that also the variance and the mean of inert solute particle velocities do not change with depth and are the same in different soil layers. The layering that we consider in the STM is therefore only related to different sorption and decay parameters in the different soil layers. A more general STM that also considers different particle velocity variances in different soil layers was described by Jury and Utermann (1992). As a consequence of the constant particle velocity, the transfer velocity through a deeper layer depends on the arrival of the solute particles at the top of the layer, and an analog equation to Eq. [8] for a stochastic convective process is
 | [10] |
where f(zn,
|zn–1,t – ) [T–1] is the transfer function through layer n of solute particles that arrive at layer n at time t – . If the advection velocity with which a solute particle moves through the soil is constant, this transfer function is
 | [11] |
where Rtot,n–1 = (
i=1i=n–1Rizi)/zn–1 and Rtot,n–1zn–1/(Rnzn + Rtot,n–1zn–1) is a normalization factor so that the 0th time moment of the BTC of a nondecaying substance does not change with depth. Introducing Eq. [11] into Eq. [10], we obtain:
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In the convective lognormal transfer function model (Jury, 1982), a lognormal travel time distribution of an inert tracer (i.e., nonsorbing and nondecaying tracer), finert, is assumed:
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where µ and 
are, respectively, the mean and standard deviation of the loge–transformed arrival times of inert solute particles at a reference depth l.
Making use of the following property of a stochastic–convective transport process:
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and of the relation between the transfer function of a sorbing and decaying substance in a profile with constant sorption and decay parameters, freact, and the transfer function of an inert tracer:
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f(zn,t) (Eq. [12]) for a STM with a lognormal travel time distribution of an inert tracer (Eq. [13]) can be written as
 | [16] |
where tot,n = (i=1n iRizi)/zn, and Rtot,n = (i=1n Rizi)/zn.
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MODEL PARAMETERIZATION
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We simulated transport of a set of dummy substances in a soil profile with three layers: 0 to 0.3, 0.3 to 0.6, and 0.6 to 1.0 m. The half-lives or DT50 [T] values [DT50 = loge(2)/] of the substances in the top soil layer varied between 5 and 200 d. The DT50 values in the deeper soil layers are fractions of the DT50 value in the upper soil layer (see Table 1). The Kd of the substances was assumed to be proportional to the organic matter content, OC [M M–1], with Koc [L3 M–1] as the proportionality factor:
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The Koc values of the different substances ranged from 10 to 200 L kg–1. The OC, b, and in the three soil layers are given in Table 1. The change in OC and DT50 values with depth were based on factors that are used in model calculations of pesticide leaching risk for pesticide registration in the European Union (FOCUS, 2000). The flow rate in the soil profile was assumed to be 0.3 m yr–1, which corresponds for = 0.3 with a pore water velocity, v, of 1 m yr–1.
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Table 1. Dispersivity lengths of the hydrodynamically homogeneous soil profile (hom), variance (2), and mean (µ), of the loge-transformed travel time distribution, dispersivities at several depths predicted by the stream tube model (hom(z), Eq. [18]), dispersivities of the different soil layers (n, Eq. [20]), soil water velocity (v), organic carbon content (OC), ratio of the half-life (DT50) value in the deeper layers and the DT50 value in the top layer, volumetric water content (), and bulk density (b).
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To parameterize the dispersivity, , and the variance of the loge–transformed arrival times, 2, we used dispersivity lengths that were derived from inert tracer experiments in soils. Figure 1
(Vanderborght and Vereecken, 2007), shows distributions of dispersivities derived from inert tracer experiments in small soil cores, larger soil columns, and field plots for different travel distances. The travel distance corresponds with the depth where a breakthrough curve was measured or with the center of mass of a concentration depth profile. These dispersivities were obtained by fitting the solution of the CDE with a constant dispersivity and velocity, i.e., assuming a hydrodynamically homogeneous soil profile. We denote the dispersivities by hom(z) where the subscript hom refers to homogeneous soil profile and z to the travel distance. In the first model, the "hydrodynamically homogeneous" CDE model, we assumed that the dispersivity is constant in the entire soil profile. Based on distributions in Fig. 1, we set the dispersivity in this model, hom(z = 100cm), equal to 6 cm, so that this model gives a realistic prediction of the spreading of an inert tracer BTC at 1-m depth. We will refer to this model as the homogeneous CDE model, although decay and sorption parameters in this model vary with depth. The increase of hom(z) with z (Fig. 1), however, is in fact contradictory to the assumption of a hydrodynamically homogeneous soil profile.
The parameters 2 and µ of the STM (Eq. [16]) were determined so that the STM and the hydrodynamically homogeneous CDE model predict a BTC of an inert tracer at 1-m depth with the same first and second temporal moments. Using relations between model parameters and temporal moments at a certain depth z, the following relations between the parameters of the STM and the homogeneous CDE model are obtained (Jury and Sposito, 1985):
 | [18] |
 | [19] |
For hom(z = 100cm) = 6 cm, 2 = 0.113 (Eq. [18]). Equation [18] implies that a stochastic–convective transport process leads to a linear increase with travel distance of dispersivities of a hydrodynamically homogeneous CDE, which are derived from BTCs at several depths in the soil profile (dashed line in Fig. 1).
In the layered CDE model, the increase in hom(z) with depth was assumed to be caused by higher dispersivities in deeper soil layers due to larger microscopic variability of advection velocities and a larger pore-scale heterogeneity. To compare the effect of microscopic (layered CDE) vs. macroscopic variation in advection velocities (STM) on the leaching of reactive substances, the dispersivities, n, of the different soil layers in the layered CDE model were determined so that first and second time moments of inert tracer BTCs matched with the STM predictions at the layer boundaries. The dispersivities n were derived using
 | [20] |
This equation is based on Eq. [9] and on the following relation between the variance of arrival times for a certain travel distance z, var(t;z), and dispersivity :
 | [21] |
The variances of the arrival times at the layer boundaries predicted by the STM are represented by hom(zn), which are calculated using Eq. [18] for zn = 0.3, 0.6, and 1 m and 2 = 0.113.
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RESULTS
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Predictions of Breakthrough Curves and Leached Mass Fractions
Figure 2
shows breakthrough curves of two substances, one with a relatively low and one with a relatively high leaching potential or risk, that are simulated by the three different one-dimensional models. An application dose of 1 kg ha–1 was assumed. The BTCs are predicted at the boundaries of the layers with different sorption and decay rate parameters.
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Fig. 2. Predictions of breakthrough curves in a layered soil profile of a substance with a low leaching potential (left) and a substance with a high leaching potential (right) at three different depths (z) by three different models: stream tube model (STM, solid black line), hydrodynamically homogeneous convection–dispersion (CDE) model (solid blue line), and layered CDE model (dashed green line), that predict the same breakthrough of an inert tracer at 100-cm depth.
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From Fig. 2, it can be derived that:- the three models that predict the same breakthrough of an inert tracer at 100 cm do not predict the same breakthrough of a reactive tracer at 100-cm depth—the differences are, in relative terms (note the different scales of the y axes), much larger for the substance with the lower leached mass fraction or leaching potential;
- the layered CDE and STM, which predict the same breakthrough of an inert tracer at 30-cm depth, predict the same BTC of the reactive substances at 30-cm depth;
- the layered CDE and the STM, however, do not predict the same breakthrough of the reactive substances at 60- and 100-cm depth although they predict the same breakthrough of an inert tracer at these depths;
- the highest peak concentrations are predicted by the homogeneous CDE model.
Point 1 implies that, for the prediction of leaching of a reactive substance undergoing decay, the depth dependency of dispersion fluxes or dispersion constants are important, especially for substances with a low leaching potential. Point 2, however, demonstrates that these differences are only important when the decay constants and sorption parameters vary with depth. For constant decay and sorption parameters with depth (i.e., in the upper 30 cm of the considered soil profile), the prediction of the BTC of a decaying substance at a certain depth is the same for models that predict the same breakthrough of an inert tracer at that depth. The homogeneous CDE model, which assumes a higher dispersive flux through the first soil layer than the layered CDE and STM, predicts higher peak concentrations of the decaying substances at 30-cm depth. This seems counterintuitive since higher dispersive fluxes smooth out or decrease concentration peaks of nondecaying substances. A higher dispersive flux, however, also implies that a larger fraction of the surface-applied mass is rapidly transferred through the upper soil layer. The fraction that is rapidly transferred has less time or opportunity for decay so that the leached mass fraction is larger and peak concentrations of decaying substances can be larger for a larger dispersive flux. Most of the decay takes place in the upper soil horizon and the larger dispersive fluxes in the deeper soil layers in the layered CDE and STM do not compensate for the larger leached mass fraction that is predicted by the homogeneous CDE model at the bottom of the first layer.
Point 3 implies that the correlation of transfer times of individual solute particles through two different soil layers plays a role. In the STM, the variance of travel times of inert solute particles through soil layers of the same thickness is the same and the model assumes that transfer times of a particle in two different layers are perfectly correlated. The variance of travel times therefore quadratically increases with travel distance. In the layered CDE model, the transfer times of a particle in two different layers are not correlated and the variance of travel times through a set of layers is the sum of the travel time variances through the individual layers (Eq. [9]). A larger increase in the travel time variance with depth in the deeper soil layers is achieved by assuming a larger microscopic variance of particle velocities in the subsoil or a larger dispersivity. For a decaying substance, only the fraction of the applied mass that is rapidly transferred through the first layer reaches the surface of the second layer. In the STM, this fraction is also rapidly transferred through the second layer since a perfect correlation of particle velocities is assumed. In the layered CDE model, this fraction is transferred with an average velocity through the second layer since the particle velocity in the second layer is not correlated to its velocity in the first layer. Therefore, the residence time and the opportunity for decay in the second layer of the fraction that leaches through the first layer are smaller for the STM than for the layered CDE model. This explains why the STM predicts more leaching at the bottom of the second and third soil layers than the layered CDE model.
Figure 3
shows predicted leached mass fractions at 1-m depth for a set of substances with different leaching potential or decay rate and sorption parameters plotted vs. the predicted leached mass fractions by the STM. Despite the fact that the same inert tracer breakthrough is predicted at 1-m depth by the three models, the predicted leached mass fractions of reactive substances are quite different and the relative differences between the model predictions increase with decreasing leaching potential of the substances. Relative differences between leached mass fraction predictions were also reported to increase with decreasing leaching potential of the substance by Boesten (2004), who investigated the effect of the dispersion coefficient on pesticide leaching, and by Larsson and Jarvis (2000), who investigated the effect of macropore flow on pesticide leaching. This indicates that a correct description of transport processes in the water phase becomes more important for an accurate prediction of leaching of substances with a low leaching potential.
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Fig. 3. Leached mass fractions of substances with different sorption and decay rate parameters at 1-m depth in a layered soil profile predicted by a hydrodynamically homogeneous convection–dispersion model (blue crosses) and by a layered convection–dispersion (CDE) model with different dispersivities for different soil layers (green diamonds) vs. predicted leached mass fractions by a stream tube model (STM, solid circles). Dashed lines represent the maximal leached mass fraction for a yearly averaged concentration <0.1 µg L–1, an application dose of 1 kg ha–1, and a deep percolation of 0.3 m yr–1.
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DISCUSSION AND CONCLUSIONS
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In soil profiles with vertically varying decay and sorption constants, transport mechanisms within the soil profile must be correctly described to predict the leached mass fraction of a reactive substance at a certain depth in the soil profile. Different models that predict the same breakthrough of an inert tracer at a given depth but that use a different conceptualization of the transport within the profile lead to different predictions of leached mass fractions. The dispersive solute flux through a soil layer and the correlation of particle velocities across layer boundaries were shown to be important parameters for the leached mass fraction of a decaying substance, especially of substances with a low leaching potential. In the 1-m-deep soil profile that we considered, the highest decay rate and sorption were assumed in the top soil layer. This particular condition may be rather generally true for substances that sorb to soil organic matter and are decayed by microbial activity. For this situation, we found that a model that presumes a constant dispersion coefficient in the entire soil profile and that is calibrated to the BTC of an inert tracer at the bottom of the soil profile predicts higher leached mass fractions than the two models, i.e., the layered CDE and the STM, that presume an increase in dispersivity with travel distance. Of the latter two models, the STM predicts higher leached mass fractions than the layered CDE model that predicts the same breakthrough of an inert tracer as the STM at the soil layer boundaries. Although not explicitly demonstrated, it can be easily shown that the order of predicted leached mass fractions by the three different models is reversed when decay rate and sorption increase with depth.
Although a CDE model that assumes a hydrodynamically uniform soil profile does not describe the transport process correctly, it consistently overestimates the leached mass fraction and may therefore be used as a conservative model. This conclusion, however, must not be extrapolated to substances with a nonlinear sorption behavior. For such substances, a STM predicts an earlier breakthrough than a convection–dispersion model (Vanderborght et al., 2006).
The differences between predicted leached mass fractions of a reactive substance by the layered CDE and the STM point at a fundamental problem related to the parameterization of models using inert tracer BTCs. Both models can reproduce inert tracer BTCs within the profile for the case where dispersivity increases with travel distance while making opposite assumptions about the transport mechanisms, more specifically about the correlation of solute particle velocities across the layer boundaries. This implies that inert tracer BTCs do not contain sufficient information to identify which of the two models best represents the transport mechanisms and best predicts the transport of decaying substances.
The assumption of no correlation of particle velocities in the layered CDE, however, can hardly be maintained considering the large amount of experimental evidence (e.g., obtained from dye tracer experiments [Flury et al., 1994], from concentration measurements in soil cores [Coquet et al., 2005; Ellsworth and Boast, 1996], or from tracer BTCs at various locations in the field [Biggar and Nielsen, 1976; van Wesenbeeck and Kachanoski, 1991]) that flow and transport processes in a field soil are heterogeneous on a macroscopic scale, which implies a spatial correlation of advection velocities. Hamlen and Kachanoski (1992) report spatial correlation of advection velocity across soil boundaries. Such correlation can result from a "hilly" soil layer boundary that generates lateral water redistribution and funneling of water with locally higher vertical water fluxes and velocities in depressions, both above and below the layer boundary (Perillo et al., 1999; van Wesenbeeck and Kachanoski, 1994).
Another argument against the layered CDE is that there is no evidence that the pore-scale heterogeneity increases with depth. An analysis of dispersivities derived from leaching experiments in soil cores taken from the topsoil and deeper soil horizons did not show larger dispersivities in cores taken from deeper soil horizons (see Fig. 7 in Vanderborght and Vereecken, 2007). Therefore, it may be assumed that a correlation of particle velocities across layer boundaries better represents the transport mechanisms than a convection–dispersion model, which postulates no correlation.
On the other hand, a perfect correlation of advection velocities through the soil profile would imply a linear increase in the dispersivity length with increasing travel distance. Figure 1 shows that the increase in dispersivity with increasing travel distance tends to level off with increasing travel distance, which is a sign of an imperfect correlation of particle velocities along their trajectory through the entire soil profile. Horizontal structures within the soil profile that impede flow, such as plow pans, may lead to lateral mixing and reduce the correlation of particle velocities across soil layer boundaries (Forrer et al., 1999). Also, soil homogenization due to tillage may destroy the connectivity of soil structures between the topsoil and deeper soil layers and therefore reduce the correlation of advection velocities between top- and subsoil.
Alternative or "generalized" one-dimensional models have been developed to describe transport processes with imperfect correlation of advection velocities within the soil profile. The generalized form of the CDE is the "fractional convection–dispersion equation" (Benson et al., 2000; Pachepsky et al., 2000). This equation represents a transport process in which the probability distribution of individual particle displacements is heavily tailed as a result of regions of high velocity, which are assumed to be continuous at all scales in soil. Because spatial correlation of particle velocities is explicitly considered, this model is capable of directly reproducing a scale-dependent dispersion process (and not indirectly via a scale-dependent dispersion coefficient, which is implemented in a convection–dispersion model). Whether this model is capable of predicting leached mass fractions of a decaying substance in soils with vertical gradients in decay rate and sorption parameters requires further investigation. To predict a nonlinear increase in dispersivity with travel distance, a generalized transfer function (GTF) has been introduced (Liu and Dane, 1996; Zhang, 2000). The GTF, however, is merely a mathematical extension giving more flexibility to describe the dispersion process in soils but lacking a sound physical basis. It remains unclear, therefore, how this model can be coupled with depth-dependent sorption and decay parameters.
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ACKNOWLEDGMENTS
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This study was triggered by discussions in the FOCUS (FOrum for the Co-ordination of pesticide fate models and their USe) Groundwater work group and we thank the other members of this group, J.J.T.I. Boesten, R. Fischer, B. Gottesbüren, K. Hanze, A. Huber, M. Jandorová, T. Jarvis, R. L. Jones, M. Klein, B. Remy, P. Sweeney, A. Tiktak, M. Trevisan, M. Vanclooster, and Z Varanaviciene for their critical comments.
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ABSTRACT
INTRODUCTION
