Published online 24 January 2007
Published in Vadose Zone J 6:29-52 (2007)
DOI: 10.2136/vzj2006.0096
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
REVIEWS AND ANALYSES
Review of Dispersivities for Transport Modeling in Soils
Jan Vanderborght* and
Harry Vereecken
Agrosphere ICG-IV, Forschungszentrum Jülich GmbH, D-52425 Jülich, Germany
* Corresponding author (j.vanderborght{at}fz-juelich.de)
Received 6 July 2006.
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ABSTRACT
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The one-dimensional convection–dispersion equation is often used to estimate the risk of nonpoint source groundwater contamination and the dispersivity in this equation is known to be a sensitive parameter for predicting the mass that leaches through the vadose zone to the groundwater. We derived a database of dispersivities from leaching studies in soils. Besides dispersivities, the database contains information about experimental parameters: transport distance, scale of the experiment, flow rate, boundary conditions, soil texture, pore water velocity, transport velocity, and measurement method. Dispersivities were found to increase with increasing transport distance and scale of the experiment. Considerably larger dispersivities were observed for saturated than for unsaturated flow conditions. No significant effect of soil texture on dispersivity was observed, but the interactive effects of soil texture, lateral scale of the experiment, and flow rate on dispersivity were significant. In coarse-textured soils, lateral water redistribution may take place across relatively larger distances, which explains the larger dependency of dispersivity on lateral scale of the experiment in coarse- than in fine-textured soils. The activation of large interaggregate pores may explain the increase in dispersivity with increasing flow rate in fine-textured soils, which was not observed in soils with a coarser texture. The distribution of dispersivities was positively skewed and better described with a lognormal than a normal distribution. Different experimental factors explained 25% of the total variability of loge-transformed dispersivities. The unexplained variance of the dispersivity was large and its coefficient of variation was 100%.
Abbreviations: 1-D, one-dimensional • BTC, breakthrough curve • CDE, convection–dispersion equation • TDR, time domain reflectometry
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INTRODUCTION
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MODELS to calculate chemical transport in soils are increasingly being used in practice. Risk assessment on the basis of model calculations of leaching of surface-applied chemicals (fertilizers and pesticides) or of pollutants from contaminated sites toward the groundwater becomes more and more legally prescribed. Examples are the FOCUS groundwater scenarios (FOCUS, 2000), which must be used in European pesticide registration procedures to estimate the risk of groundwater contamination by surface-applied pesticides. The reliability of these model calculations depends on the accuracy with which relevant processes for contaminant transport are implemented in transport models. An important process is transport in the water phase. Reviews of different model approaches to describe the transport of dissolved substances have been given by Feyen et al. (1998), Jury and Flühler (1992), Nielsen et al. (1986), and Vanclooster et al. (2005). Besides an appropriate model choice, the parameterization of the model also plays an important role. To parameterize the water flow model, databases of soil hydraulic parameters, e.g., the HYPRES database (Wösten et al., 1999) and the UNSODA database (Nemes et al., 2001), and pedotransfer functions relating soil hydraulic properties with spatialized information about other soil properties have been derived (Schaap et al., 1998; Vereecken et al., 1989, 1990). Using pedotransfer functions in combination with soil databases, hydraulic properties and model predictions can be spatialized, as well as soil management and soil and groundwater protection policies. An example of combining modeling of pesticide leaching with spatialized information about soil properties and land use to derive spatial distributions of pesticide groundwater concentrations was given by Tiktak et al. (2004). In their study, hydraulic properties and soil properties that are related to pesticide sorption and decay (organic C content and soil pH) were spatialized, whereas transport parameters were chosen to be constant. Boesten (2004) showed that the dispersivity, which is a transport parameter describing spreading or dispersion of a surface-applied solute pulse, has an important impact on predicted yearly averaged pesticide concentrations, especially for substances with a low leaching potential. Data sets of transport parameters such as those existing for hydraulic parameters are missing, however. Gelhar et al. (1992) reviewed dispersivities derived from groundwater tracer studies and found a scale dependency of the dispersivity, which increases with increasing transport distance. Since the transport distance in groundwater tracer studies is much larger than the typical transport distance in soils, it is questionable whether parameters derived from groundwater tracer studies can be applied to soils. Furthermore, the structural properties of soils and aquifers are substantially different. In soils, flow and transport are perpendicular to the layering, whereas in aquifers they are mostly aligned with stratification. The origin of stratification in soils and aquifers is also substantially different. In aquifers, stratification is mostly the result of a sedimentation process, whereas in soils it results from leaching and precipitation processes. Finally, flow and transport processes in aquifers occur under saturated and more or less steady flow conditions. In soils, flow and transport are highly dynamic processes that change in magnitude and direction due to the continuously changing boundary conditions. Also, the water content and the pore volume in which transport takes places change continually. Therefore it is highly questionable whether transport properties obtained from groundwater tracer studies can be translated to soils.
The objective of this review is to give an overview of transport properties, more specifically the dispersivity, that were derived from tracer studies in soils. We focus on the dispersivity since the convection–dispersion model is the most widely used model to predict transport in soils and to interpret tracer experiments, especially those that were performed under natural boundary conditions at the field plot scale and in undisturbed soil, which are most relevant for practical applications. Beven et al. (1993) made a review of tracer experiments and dispersivities in soils. The number of studies, however, especially the number of field-scale studies, that could be reviewed at that time was relatively small. Overviews of parameters of other transport models, such as mobile–immobile model parameters (van Genuchten and Wierenga, 1976), were restricted to experiments that were performed in relatively small soil columns, which were often filled with disturbed soil or artificial media such as glass beads, and which were performed at relatively high flow rates (Goncalves et al., 2001; Griffioen et al., 1998; Haggerty et al., 2004).
In this review, we first define the dispersivity and how it is derived from leaching experiments. Then we present a database of dispersivities derived from leaching experiments in soils. Specific objectives of this review are to provide, on the basis of available data in the literature, answers to the following questions:
- What is the range of dispersivities observed in leaching experiments and how are they distributed?
- How does dispersivity depend on experimental factors such as scale of the experiment, leaching rate, and transport distance?
- Is there a relation between soil texture and dispersivity?
Information of this type is needed for a realistic parameterization of dispersivity and its uncertainty in modeling studies.
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DEFINITION OF DISPERSIVITY
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Flow in the vadose zone is generally in the vertical direction. A common simplification is that of one-dimensional (1-D) transport in which solute fluxes and concentration gradients in the horizontal direction are neglected. This assumption follows from the generally widespread application of chemicals or pollutants at the soil surface (i.e., diffuse or nonpoint source pollution). For several practical applications (prediction of transport of agrochemicals and salts), the convection–dispersion equation (CDE) is used:
 | [1] |
where 
is the volumetric water content, C [M L–3] the concentration in soil water, 
b [M L–3] the soil bulk density, S [M M–1] the concentration of the sorbed phase, v [L T–1] the pore water velocity, D [L2 T–1] the hydrodynamic dispersion coefficient, F(C,S) a function describing reactions of the substance in the solid and liquid phases (e.g., decay, kinetic sorption–desportion, precipitation–dissolution), and z the vertical coordinate.
Hydrodynamic dispersion accounts for a solute flux due to a concentration gradient that leads to a decrease of peak solute concentrations with time and a smoothing of concentration gradients. Two mechanisms are responsible for the dispersive solute flux: molecular diffusion and hydromechanical dispersion. The former represents the effect of thermal agitation and molecular collision, while the latter represents the effect of variations in advection velocity that exist at a smaller scale than the scale of the averaging volume. Hydrodynamic dispersion is related to the molecular diffusion constant of the substance in bulk water, D0, and the pore water velocity, v, as
 | [2] |
where 
[L] is the dispersivity, 
a tortuosity coefficient, and D0 [L2 T–1] the molecular diffusion coefficient. Data from leaching experiments generally do not contain information allowing discrimination between molecular diffusion and mechanical dispersion; however, the effective molecular diffusion coefficient [()D0] is in the order of 0.5 cm2 d–1 and its contribution to the hydrodynamic dispersion D observed in leaching experiments is often very small (for the experiments considered in this review, it was on average 5% of D). Therefore, was simply derived from the ratio D/v, assuming that molecular diffusion can be neglected.
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DERIVATION OF DISPERSIVITY
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Dispersivities are typically derived either from observed depth profiles of inert tracer concentrations, or from breakthrough curves (BTC) of tracer concentrations that are measured in the effluent of columns or in a soil profile using solution samplers or other devices suchs as time domain reflectometry (TDR) probes. In general, the hydrodynamic dispersion coefficient, D, and the pore water velocity, v, are derived from these profiles, assuming that v and D are constant in the soil profile and do not change with depth and time (i.e., a hydrodynamically homogeneous soil profile and steady-state flow conditions). For such situations, analytical solutions of the transport equation can be derived and fitted to the observed depth profiles or BTCs.
An overview of analytical solutions of the 1-D CDE is given in Toride et al. (1999). Alternatively, time moments of BTCs or depth moments of concentration depth profiles may be calculated and used to derive the dispersion coefficient and pore water velocity (Jacques et al., 1998; Jury and Sposito, 1985; Jury and Roth, 1990; Russo, 2002). The calculation of time or depth moments does not require the specification of a certain process model. Time moments can also be related to parameters of other models that were fitted to BTCs, such as the mobile–immobile model (Valocchi, 1985), the convective lognormal transfer function model (Jury and Sposito, 1985), or a stream tube model that accounts for dispersion within stream tubes (Toride and Leij, 1996). From the time moments that are calculated from parameters of other models, an apparent dispersivity can be derived using the relation between the time moments of a BTC and the dispersivity. In some field-scale transport experiments, a salt tracer was applied to a large surface while BTCs were measured locally at several locations (Biggar and Nielsen, 1976; Bowman and Rice, 1986; Vandepol et al., 1977). The solution of a 1-D CDE can then be fitted to the locally measured BTCs and distributions of "local" CDE parameters, pore water velocities, and "local" dispersivities, (local), derived. Using time moment analysis, the apparent dispersivity of the field-scale averaged BTC, (field), can be derived from the distribution of the "local" CDE parameters (Toride and Leij, 1996):
 | [3] |
where < (local) > is the arithmetic average of the local dispersivities and 
lnv2 is the variance of the loge-transformed local velocities.
The analytical solutions of the CDE hold for constant flow rates. When the flow rate is not constant, the time coordinate is often transformed to a cumulative infiltration or drainage coordinate I(t) as follows:
 | [4] |
where Jw(z,t) is the water flux at depth z and time t. This transformation leads to a similarly smooth course of concentrations as during steady-state flow conditions, and allows the analytical solution of the steady-state CDE, in which the time coordinate t is replaced by I(t), to be fitted against the measured concentrations. Equation [4] involves a transformation of the dimensions of the fitted parameters, v(I) and D(I). The fitted velocity, v(I) [L L–1] represents the distance across which the solute is leached per unit of leached water depth so that v(I) = –1. The dispersivity (I) has the same dimension as the dispersivity obtained under steady-state flow conditions. In general, (I) is larger than (Beese and Wierenga, 1980; Vanderborght et al., 2000a; Wierenga, 1977), depending on the temporal fluctuations of the water content during the leaching experiment. Beese and Wierenga (1980) report that (I) may be a factor 3 larger than in the topsoil, where the dynamics of the water flux and water content are large. The difference between (I) and decreases with increasing depth and for smaller fluctuations in the water content (Vanderborght et al., 2000a).
For soil profiles with the water content variable with depth, the following depth transform has been proposed (Ellsworth and Jury, 1991):
 | [5] |
When depth is transformed using Eq. [5], the pore water velocity v(z*) corresponds with the water flux Jw, while the dispersivity (z*) is expressed in terms of transformed length units. Although a theoretical basis for the use of the depth transform in Eq. [5] is basically missing, BTCs in soil profiles with vertically varying soil water contents were often fitted with a solution of the CDE for a homogeneous soil profile after replacing the real depth coordinate by z*. The dispersivity (z*) that is derived from a BTC at the transformed depth z* can be expressed in real depth coordinates as
 | [6] |
where (z) is an "apparent" dispersivity in real depth coordinates that predicts the breakthrough at depth z assuming a soil profile with a vertically constant water content . The direct effect of vertical variability in on (z) can be assessed using the following formula (Vanderborght et al., 2000b):
 | [7] |
where CV() is the coefficient of variation of the water content. According to Eq. [7], vertical variability in the water content leads to larger apparent dispersivities. Since CV() is generally <1, the direct effect of vertical variability of the water content on (z) is not so large.
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DATABASE OF DISPERSIVITIES
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The database we developed contains 635 entries derived from 57 publications in scientific journals. Since soil structure has an important impact on solute transport, only experiments in undisturbed soils were considered, thereby excluding experiments in repacked or refilled soil cores or columns. Besides dispersivities, several experimental factors were also included in the database so that relations between experimental factors and dispersivities can be inferred.
The following information was included:
- dispersivity (cm)
Experimental Scale
- transport distance z (cm), i.e., the vertical distance that the applied tracer travelled—the transport distance corresponds with the length of the soil column, the depth where a BTC was measured, or the center of mass of a concentration depth profile
- scale of the leaching experiments—three classes were considered: core-scale (undisturbed soil cores with a length <30 cm), column scale (undisturbed soil monoliths with a length >30 cm), and field scale
Boundary and Flow Conditions during the Leaching Experiment
- transport velocity v (cm d–1) derived from the tracer breakthrough or concentration depth profiles
- pore water velocity vp (cm d–1) derived from the flow rate divided by the volumetric water content
- ratio of v/vp, which is a measure for preferential solute transport (v/vp > 1) or solute retardation (v/vp < 1)
- average flow rate Jw (cm d–1), which is the net infiltrated water depth during the leaching experiment divided by the duration of the experiment
- effective flow rate Jw eff (cm d–1), which is a measure for the flow rate intensity in the soil during the experiment (for a definition, see Vanderborght et al., 2000a)
- flow boundary condition type: steady (steady, unsaturated flow obtained from steady irrigation), ponding or flooding (steady saturated flow), intermittent (periodic flow under unsaturated conditions induced by intermittent irrigation), intermittent ponding (periodic ponding or flooding of the soil surface), climatic (natural rainfall and soil evaporation), and intermittent-climatic (natural rainfall and soil evaporation with intermittent additional irrigation)
Soil Properties
- USDA soil texture class: clay, silty clay, sandy clay, clay loam, silty clay loam, sandy clay loam, silt loam, silt loam gravel, loam, sandy loam, loamy sand, sand, and sandy gravel
- soil depth—depth from which soil cores were taken: A (topsoil, 0–30 cm), B horizon (30–60 cm), or C (subsoil, deeper than 60 cm).
Concentration Measurements
- type of concentration that was measured: volume-averaged or resident vs. flux-averaged concentrations
- measurement type: direct (in the effluent from soil columns or cores), coring (analysis of soil samples), samplers (extraction of soil solution in the soil profile using suction samplers or suction plates), TDR (concentrations derived from bulk soil electrical conductivity measured with TDR), tile drains, dye tracers (image analysis of photographic recordings of dye-stained patterns on excavated soil surfaces), or calculated (average concentrations calculated from the average of local concentration measurements)
Miscellaneous
- experiment number—the experiment number groups all dispersivity values that were obtained for the same field plot under the same leaching conditions
- name of the field site where the experiments were performed or from which the soil samples were taken
- author and year of publication
A complete list is given in the appendix.
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EFFECTS OF EXPERIMENTAL CONDITIONS
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Flow Rate, Scale of the Experiment, and Transport Distance Effects on Dispersivity
The scale of the study, flow boundary condition type, and soil texture are considered to be important experimental factors influencing solute dispersion. Figures 1
, 2
, and 3
show the number of data entries in the different factor classes, together with the mean dispersivity, mean flow rate, and mean transport distance in the factor classes. With increasing scale of the leaching experiment, the average transport distance was found to increase, whereas the flow rate decreased (Fig. 1). The effect of the experimental scale on the dispersivity can therefore not be derived without considering the effects of flow rate and transport distance on the dispersivity. About two-thirds of all dispersivities were derived from leaching experiments performed under steady-state flow conditions (Fig. 2). Although the mean flow rate in experiments performed under continuous and intermittent flooding boundary conditions was quite different, the mean dispersivities in these classes were similar and much larger than for the other boundary condition classes. The mean dispersivity was smallest in experiments that were performed under steady-state unsaturated flow conditions, whereas the mean flow rate was the second largest in these experiments. The degree of saturation of the soil surface (i.e., continuously or periodically saturated [ponded] vs. unsaturated) seems to have a larger impact on the dispersivity than the mean flow rate during the experiment. The larger dispersivities observed for leaching experiments with saturated soil surface conditions clearly reflect the effect of flow and transport through larger pores (i.e., macropores), which are activated under saturated conditions. Looking at the combination between soil texture and flow boundary condition class (Fig. 3), it is remarkable that in clayey soils (clay, silty clay, sandy clay, clay loam, silty clay loam, and sandy clay loam) most experiments were performed under saturated flow conditions. Experiments with climatic boundary conditions were mainly performed in coarse-textured soils. This correlation between boundary condition and soil texture needs to be considered when the effects of soil texture and boundary condition on the dispersivity are investigated.
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Fig. 1. Number of observations (bars), mean flow rate (blue line), mean transport distance (black line), and mean dispersivity (red line) in the experiment scale classes.
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Fig. 2. Number of observations (bars), mean flow rate (blue line), and mean dispersivity in the flow boundary condition classes.
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Fig. 3. Number of observation in different soil texture classes (c: clay, sic: silty clay, sc: sandy clay, cl: clay loam, sicl: silty clay loam, scl: sandy clay loam, sil: silt loam, silg: silt loam gravel, l: loam, sl: sandy loam, ls: loamy sand, s: sand, sg: sandy gravel) and flow boundary condition classes.
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For further analysis, all experiments performed under flooding and intermittent flooding boundary conditions were grouped in a separate class. For the other group of experiments, the soil texture classes were grouped into two texture classes: a "coarse" texture class that lumps the sand, loamy sand, and sandy loam classes and a "fine" texture class, lumping the other texture classes. Experiments that were performed on soils with a large stone content (texture classes sandy gravel and silt loam gravel) were excluded.
Although flow rate, Jw, and transport distance are continuous variables, their effect on the dispersivity was investigated through flow rate and transport distance classes. If available, the effective flow rate, Jw eff, rather than the time-averaged flow rate was considered. For transient flow conditions, the flow rate intensity in the soil column, which is quantified by Jw eff, was shown to be better correlated to the dispersivity than the time-averaged flow rate (Vanderborght et al., 2000b). Four flow rate classes were defined: flow rates <1 cm d–1, between 1 and 10 cm d–1, and >10 cm d–1, and experiments performed with flooding and intermittent flooding boundary conditions. Most of the experiments that were performed with climatic conditions or climatic conditions with intermittent irrigation fell into the flow class with flow rates <1 cm d–1. Exceptions were studies in which a large amount of water infiltrated during a relatively short time (rainfall events of >10 cm d–1). These studies fell into the class with flow rates >10 cm d–1.
For the transport distances, three classes were defined: studies with a transport distance 
30 cm, between 31 and 80 cm and between 81 and 200 cm. To give the same weight to experiments in which dispersivities were determined for several travel distances (e.g, in a soil column or a field plot), the data entries in a travel distance class that correspond to the same experiment or experiment number were averaged and further treated as a single entry.
In Fig. 4
, the distribution of dispersivities in different flow rate and experimental scale classes are shown for the different transport distance classes. Dispersivities derived from experiments that were performed using a flooding boundary condition were consistently larger than dispersivities that were derived from other experiments. For the 0- to 30-cm travel distance class, a clear increase in dispersivity with increasing flow rate was present in the core- and column-scale experiments. This increase was not or not so clearly apparent for field-scale experiments, neither for the 31- to 80- nor the 81- to 200-cm travel distance classes.
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Fig. 4. Effect of flow rate (Jw) class and scale of the experiment on the dispersivity for the different transport distance (z) classes. The boxes span the 25 and 75% percentiles, the thick black line is the median, and the 0 and 90% percentiles correspond with the extremities of the vertical bars. The numbers above or below the boxes correspond with the number of observations in the class. (Data from the same experiment in a travel distance class were averaged and treated as a single entry).
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Most leaching studies were performed using artificial leaching rates. To reduce the duration of the leaching experiment, the average imposed flow rate in leaching experiments was mostly considerably larger than under natural boundary conditions. For instance, a leaching experiment performed with a flow rate of 10 cm d–1 would correspond with a yearly precipitation amount of 36 500 mm, which is one to two orders of magnitude larger than the yearly precipitation amount. On the other hand, rainfall and soil water flow are highly dynamic processes, with high rainfall or flow intensities occurring during only a short period of time, and with long intermittent periods without rainfall or significant downward flow. Therefore, close to the soil surface, vertical movement occurs during relatively short pulses with a high flow rate. These high flow rates become sensibly buffered with depth, depending on the hydraulic buffer capacity of the soil. From rainfall intensity records, the amount of rain that falls with intensities smaller or larger than a certain threshold can be derived. As an example, in Jülich (Germany), 10% of the total yearly precipitation occurs with an intensity >13.2 cm d–1, whereas 50% of the total yearly precipitation occurs with an intensity >3.6 cm d–1. These data indicate that leaching experiments in the flow rate class 1 < Jw < 10 cm d–1 may well be considered realistic also. Because flow rates >10 cm d–1 were not considered to be realistic for natural boundary conditions, dispersivities from this flow rate class were excluded from further analyses. Since the dispersivities in the 10 cm d–1 < Jw class, except for the core- and column-scale experiments and travel distances <30 cm, were not very different from dispersivity distributions in other flow rate classes (Fig. 4), however, their exclusion did not considerably influence the results of further analyses.
The effect of experimental scale and travel distance on dispersivities derived from experiments with a flow rate <10 cm d–1 is shown in Fig. 5
. Both the transport distance and the lateral scale of the transport experiment had an impact on the dispersivity. Generally, dispersivity increased when the lateral scale of the experiment increased. Therefore, field-scale experiments are expected to be more representative for the dispersion process under real conditions than experiments in soil columns or lysimeters that reduce lateral redistribution of water flow, and hence the overall dispersion process. The difference between field- and column-scale experiments, however, was smaller for larger travel distances where the two distributions tended to converge. Furthermore, solute fluxes can readily be measured in a column experiment, but not so easily in a field experiment. In field experiments, concentrations are measured locally at a number of points and the actually sampled area is generally only a small fraction of the total cross-sectional area of the field plot, and sometimes may be even smaller than the area of a soil column or lysimeter.
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Fig. 5. Effect of transport distance and scale of the experiment on the dispersivity (). The boxes span the 25 and 75% percentiles, the thick black line is the median, and the 0 and 90% percentiles correspond with the extremities of the vertical bars. The numbers above the boxes correspond with the number of observations in the class. (Data from experiments with ponding boundary conditions and from experiments with a flow rate >10 cm d–1 were excluded; data from the same experiment in a travel distance class were averaged and treated as a single entry.)
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Looking at the effect of the transport distance on the dispersivity, the column-scale experiments clearly showed an increase in dispersivity with transport distance. For the field-scale experiments, the dispersivity distribution in the 0- to 30-cm travel distance class was similar to that in the 31- to 80-cm class. The larger dispersivities in the 0- to 30-cm transport distance class for the field-scale experiments may also be a result of transient flow conditions at the soil surface, and the transformation of the time coordinate to a cumulative infiltration or drainage coordinate (Eq. [4]) that is often applied to field data before fitting the solution of the steady-state CDE. The increase in dispersivity with increasing travel distance in soils is in line with the generally observed trend for dispersivities derived from groundwater tracer studies (Gelhar et al., 1992). From those data, a rule of thumb that the dispersivity is approximately 1/10 of the travel distance was inferred. Applying this rule of thumb to soil data, the median dispersivity would be overestimated. Considering the large difference between the transport distances of the tracer experiments on the basis of which this rule was derived and the transport distances in soils, the rule of thumb seems to be quite universal and also roughly applicable to soils.
In Fig. 6
, median values of dispersivities in the different experimental scale classes are plotted vs. the median value of the transport distance for the various distance classes, together with linear and power law model fits. The plot suggests that the rate of increase in dispersivity with travel distance is larger for smaller than for larger travel distances. The rate of increase in the median dispersivities between the first and second and between the second and third travel distance classes is similar. A constant rate of increase in dispersivity with travel distance can be explained by assuming that velocities of individual solute particles remain perfectly correlated with travel distance. When the spatial scale across which particles travel with a constant velocity is smaller than the transport distance, the rate of increase in the dispersivity decreases with travel distance and the dispersivity reaches an asymptotic value (e.g., Jury and Roth, 1990). Figure 6 suggest that, in general, the asymptotic regime was not reached within the first 1 m of the soil profile. This means that regions with higher or lower water fluxes and particle velocities are vertically continuous over a distance of at least a few decimeters in soils. Since several examples of soil profiles in which the dispersivity reached an asymptotic value exist, however, this statement cannot be applied to each individual soil profile. When the increase in the dispersivity with travel distance is explained by particle velocities that remain constant along their trajectory, it is presumed that the variance of the particle velocity does not change with depth in the soil profile. The increase in with travel distance can alternatively be explained by an increase in the particle velocity variance with depth, whereas the particle velocities are correlated only across a microscopic distance. One could presume that the topsoil is more homogenized due to tillage than the subsoil soil, so that the variance of the particle velocity is smaller in the top- than in the subsoil; however, there are two arguments against using only this hypothesis to explain the increase in with travel distance. An indirect argument is the increase in dispersivity with lateral scale of the experiment. This implies that velocity variations exist on a macroscopic lateral scale and therefore must extend over a macroscopic vertical distance. The second argument is that dispersivities are not larger in soil cores from deeper soil horizons than in cores from the topsoil (Fig. 7
).
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Fig. 6. Effect of transport distance and scale of the experiment on the dispersivity: median values of dispersivities () vs. median values of the transport distance in a transport distance class. Lines are best fits of a linear and a power law model.
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Fig. 7. Dispersivities () in soil cores taken from topsoils and subsoils. The boxes span the 25 and 75% percentiles, the thick black line is the median, and the 0 and 90% percentiles correspond with the extremities of the vertical bars. The numbers above the boxes correspond with the number of observations in the class.
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Effect of Soil Texture on Dispersivity
The effect of flow rate class on dispersivities in the coarse- and fine-textured soil classes is shown in Fig. 8
. For both soil texture classes, dispersivities were larger for saturated than for unsaturated flow conditions. It should be noted that the opposite was observed for leaching experiments involving repacked soil columns (e.g., De Smedt et al., 1986; Elrick and French, 1966; Maraqa et al., 1997). For unsaturated flow experiments, dispersivity distributions did not depend on flow rate class in soils with a coarser texture, whereas dispersivities increased with increasing flow rate in soils with a finer texture. In soils with a finer texture, the pores in the soil matrix or the intraaggregate pores are small so that the soil matrix has a low hydraulic conductivity. When the leaching rate exceeds the conductivity of the intraaggregate pores, interaggregate pores, which are continuous across a much larger distance than the intraaggregate pores, are activated, leading to an increase in the dispersivity with increasing flow rate. In soils with a coarser texture, the pore sizes and hydraulic conductivity of the soil matrix are much higher and more soil matrix pores are activated with increasing flow rates so that the water-filled pore network becomes better connected and the tortuosity of the flow paths decreases. This may even lead to a decrease in dispersivity with increasing flow rate.
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Fig. 8. Effect of flow rate (Jw) class and texture on the dispersivity (). The boxes span the 25 and 75% percentiles, the thick black line is the median, and the 0 and 90% percentiles correspond with the extremities of the vertical bars. The numbers above or below the boxes correspond with the number of observations in the class. (Data from the same experiment in a travel distance class were averaged and treated as a single entry).
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In the remainder of our analysis, only data from unsaturated leaching experiments with a flow rate <10 cm d–1 were considered. Figure 9
displays the effect of the transport distance and texture on dispersivity and Fig. 10
the effect of the scale of the experiment and texture. Soils with a coarse texture seemed to have a smaller dispersivity than soils with a finer texture. The dispersivity for both soil classes increased with travel distance. For the fine-textured soils, the distributions of disperivities in column- and field-scale experiments were similar, whereas for coarse-textured soils, larger dispersivities were observed in field- than in column-scale experiments. This suggests that the lateral spatial scale of the transport variability is smaller for fine-textured than for coarse-textured soils. In coarse-textured soils, lateral redistribution of water and funnelling of water toward "preferential flow regions" may take place in the soil matrix. This lateral redistribution may be strongly reduced by imposing no-flow lateral boundary conditions in column-scale experiments. Solute spreading and dispersion due to rapid transport in large interaggregate pores occurs on a smaller lateral scale, which may explain why no increase in dispersivity from the column to the field scale was observed for the fine-textured soils.
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Fig. 9. Effect of transport distance and texture on the dispersivity (). The boxes span the 25 and 75% percentiles, the thick black line is the median, and the 0 and 90% percentiles correspond with the extremities of the vertical bars. The numbers below the boxes correspond with the number of observations in the class. (Data from experiments with ponding boundary conditions and from experiments with a flow rate >10 cm d–1 were excluded; data from the same experiment in a travel distance class were averaged and treated as a single entry.)
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Fig. 10. Effect of the scale of the experiment and texture on the dispersivity (). The boxes span the 25 and 75% percentiles, the thick black line is the median, and the 0 and 90% percentiles correspond with the extremities of the vertical bars. The numbers below the boxes correspond with the number of observations in the class. (Data from experiments with ponding boundary conditions and from experiments with a flow rate >10 cm d–1 were excluded; data from the same experiment in a travel distance class were averaged and treated as a single entry.)
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VARIABILITY OF DISPERSIVITIES AND ANALYSIS OF VARIANCE
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The box plots in Fig. 5 and Fig. 7
to 10 suggest that the dispersivities are lognormally distributed (the box plots are symmetric around the median when the y axes of the plots are logarithmically scaled). In Fig. 11
, histograms of untransformed and loge-transformed dispersivity distributions in the different travel distance classes are shown. The histograms qualitatively indicate that dispersivities are lognormally distributed. The logarithmic transformation also suggests that the large dispersivity values in the tails of the untransformed distributions should not be considered as outliers, i.e., observations with a probability of exceedance much lower than (no. of observations)–1. An ANOVA was performed to investigate the significance of the different factors and those parts of the variance of the loge-transformed dispersivity distribution that can be explained by these factors. To reduce the number of factors and to have a sufficient number of observations within a factor combination, only the factors transport distance, texture, and experimental scale were investigated. Experiments with a flow rate <10 cm d–1 were considered, while the effect of flow rate class was not further investigated. Core-scale experiments were excluded to avoid factor combinations without data. Since the data set was not balanced, the data were analyzed within the framework of generalized linear models using the GLM procedure of the SAS software (SAS Institute, Cary, NC). The outcome of the ANOVA is shown in Table 1. The total variance of loge-transformed dispersivities in the data set, loge2 :total, was 0.878 and the coefficient of variation of the untransformed dispersivities CVtotal was 119% (CVtotal = 100
[exp(loge2 :total) – 1]). The model-explained variance was 25% (R2) of the total variance and the unexplained variance loge2 :total was 0.753 and CVerror was 105%. The variability of dispersivities within a factor class combination remained, therefore, relatively large and unexplained. The factor explaining most of the variance was the transport distance. Also, the interaction between the scale of the experiment and the texture, and the variability explained by the scale of the experiment, were significant (at the 5% significance level). The effect of soil texture was not significant.
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Fig. 11. Histograms of (a–c) untransformed and (d–f) loge-transformed dispersivities in different transport distance classes: (a) and (d) from 0 to 30 cm, (b) and (e) from 31 to 80 cm, and (c) and (f) from 81 to 200 cm; µ and are the mean and standard deviation of the loge-transformed dispersivities, respectively. (Data from experiments with ponding boundary conditions and from experiments with a flow rate >10 cm d–1 were excluded; data from the same experiment in a travel distance class were averaged and treated as a single entry.)
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Table 1. Analysis of variance of loge-transformed dispersivities measured in column- and field-scale experiments for fluxes <10 cm d–1.
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MEASUREMENT METHOD
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The measurement method was evaluated in terms of the ratio of the transport velocity derived from the tracer movement, v, and the pore water velocity that was predicted from the flow rate and the water content, vp. Deviation of the ratio v/vp from 1 should be an indication that the transport process was not well described by a model that presumes that solute transport takes places in the entire water-filled pore space, or an indication that the observed solute transport was not representative for the overall transport process within the soil sample or field site. The v/vp ratios are shown in Fig. 12
for different measurement types and transport distance classes, and for saturated conditions or high flow rates (Jw > 10 cm d–1; Fig. 12a) and unsaturated flow conditions (Fig. 12b). The spreading of the v/vp ratios was the largest for soil solution samplers, whereas direct measurement of the concentration breakthrough in the effluent of a soil core or column produced the smallest variability in v/vp. Time domain reflectometry and soil coring led to an intermediate spreading in v/vp. The spreading in v/vp can be related to the soil volume that is actually sampled by the method. Soil samplers only sample soil water locally, so that only a small fraction of the cross-sectional area is sampled, even when a large number of samplers is used. As a consequence, the BTC obtained from averaging local measurements may deviate considerably from the averaged BTC of local concentrations or solute fluxes at that depth (e.g., Weihermuller et al., 2005). Furthermore, suction samplers distort the flow field to a certain extent, which also leads to deviations of v estimated from BTCs measured with suction samplers. The cross-sectional area sampled with TDR probes and the soil volume sampled with soil coring are larger, which explains the smaller variability of v/vp when these methods are used. If the concentration is measured directly in the effluent of a soil core or column, then the entire cross-sectional area is sampled and the BTC is obviously representative; however, the variability in v/vp may be quite large also for the direct method. For example, for saturated conditions or high flow rates and small transport distances, v/vp was found in several experiments to be considerably >1. These results may be attributed to preferential flow and an earlier arrival of the peak concentration than expected based on the average flow rate and the volumetric water content. If the total water-filled porosity is still accessible to the solute by slow diffusion, a fast breakthrough of the peak concentration may be followed by a long tail of the BTC due to slow release of solutes from the bypassed pore region. This tailing cannot be described with the CDE, so that a CDE fit leads to an overestimation of the average pore water velocity in the total solute-accessible pore volume (i.e., inclusive of the pore volume in which water flow is very slow). In the case of effluent from a soil core or column, flux concentrations or flux-weighted averages of local concentrations are measured. By contrast, TDR and soil coring methods produce volume-averaged or resident concentrations. Time domain reflectometry measures time series of resident concentrations, whereas concentration depth profiles are derived from soil coring. For small travel distances, v derived from TDR measurements seemed to underestimate vp, especially for saturated flow conditions. If preferential flow occurs through a small part of the total pore volume, early breakthrough in the preferential flow region is difficult to detect from volume-averaged concentrations as measured with TDR. For larger travel distances and transport times, the opportunity for mixing and exchange of solutes between preferential flow paths and bypassed regions increases, in which case the deviation between v and vp decreases. Another explanation for the smaller estimates of v is that solute fluxes and pore water velocities are derived from a BTC of resident concentrations assuming that the transport process can be predicted using a convection–dispersion model with a constant dispersivity. When dispersivity scales with travel distance, this approach leads to an underestimation of the pore water velocity (Jacques et al., 1998).
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Fig. 12. Effect of measurement type and transport distance class on the ratio v/vp, which is a measure for preferential solute transport (v/vp > 1) or solute retardation (v/vp < 1), derived from tracer experiments with (a) saturated conditions at the soil surface or a flow rate >10 cm d–1 and (b) unsaturated conditions and a flow rate <10 cm d–1. The boxes span the 25 and 75% percentiles, the thick black line is the median. The numbers correspond with the number of observations in the class.
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DISCUSSION AND CONCLUSIONS
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In agreement with reviews of dispersivities observed in groundwater tracer studies, dispersivity in soils scaled with travel distance. Scaling of dispersivity with transport distance implies that the transport distance in leaching experiments should be similar to the range of transport distances for which predictions of solute concentrations must be made. Scaling of dispersivity with transport distance could not be explained by an increase in soil heterogeneity in the subsoil, which is not regularly homogenized by soil tillage. This implies that solute particle velocities are correlated across a macroscopic distance, or that regions with a higher or lower transport velocity extend across a certain macroscopic vertical distance. Dispersivities are always derived assuming a vertically homogeneous soil profile with a constant or depth-independent dispersivity. Therefore, dispersivities in this review are "equivalent" parameters that parameterize transport in an equivalent vertically homogeneous soil profile. How to combine this vertically homogeneous equivalent soil profile, in terms of the dispersion parameter, with vertical variations in soil biological and chemical properties, requires further investigation. Since no general asymptotic regime was observed within the range of transport distances that was covered by leaching experiments in soils, dispersivities in this review cannot be extrapolated to larger transport distances in deep vadose zones. Information about larger scale transport in the vadose zone is scarce and difficult to obtain because of the long travel times (e.g., Javaux and Vanclooster, 2004a, 2004b). Dispersivity also scaled with lateral scale of the experiment. Considering that solute dispersion is caused by spatial variations in local water velocities, that this spatial variability is predominantly caused by spatial variations in soil water fluxes, which outweigh the effect of soil water content variability, and that the soil water flux at the soil surface is relatively homogeneous, the lateral scale of the "representative" volume is related to the distance across which water can be laterally redistributed within the soil profile; however, this lateral scaling of dispersivity and the scale across which water is laterally redistributed is related to soil texture. In fine-textured soils, similar distributions of dispersivities were derived from column- and field-scale studies, indicating that column-scale studies may be representative for field-scale transport. For coarse-textured soils, however, lateral redistribution may take place across a larger distance so that field-scale dispersion was larger than the dispersion observed in column-scale studies. Therefore, column-scale studies in coarse-textured soils, which are, for instance, performed to evaluate the risk of pesticide leaching, may not be representative for field-scale dispersion and, since pesticide leaching is positively correlated with dispersion, may underestimate field-scale pesticide leaching. Larger scale structures such as compacted areas under wheel tracks alternating with homogenized seed beds (Coquet et al., 2005a, 2005b), sand lenses (Hammel et al., 1999), soil horizons with spatially variable thickness (van Wesenbeeck and Kachanoski, 1994), or layers with different texture that extend across large distances (Kung, 1990) may also play an important role and redistribute water across relatively large lateral distances. Lateral redistribution of water at the soil surface and within the soil profile on sloping terrain may lead to lateral variations in vertical infiltration or leaching. Finally, variability in plant water uptake at both the local and larger scales may lead to additional variability in vertical water fluxes. Since most leaching experiments were performed in bare soils, the effect of plants on solute spreading requires further investigation. The fact that dispersion is caused by macroscopic variations in pore water velocities also has important implications for the prediction of nonlinear transport processes. Due to these macroscopic variations in pore water velocities, concentrations vary in the horizontal direction. If local processes (e.g., decay or sorption) depend in a nonlinear way on the local concentrations, then it is trivial that the lateral average of the local process cannot be described by implementing the averaged concentration in the nonlinear process model (e.g., Janssen et al., 2006; Vanderborght et al., 2006).
Dispersivities seem to be larger in fine-textured soils but the difference between dispersivities in fine- and coarse-textured soils was not found to be significant. Besides the interaction between the lateral scale of the leaching experiment and soil texture, there is also an interactive effect of soil texture and flow rate on dispersivity. In fine-textured soils, dispersivity also increases with flow rate for unsaturated flow conditions. In soils with a coarser texture, dispersivity distributions were not found to be different for different flow rate classes, except when the soil surface was saturated.
Dispersivity distributions were better described by a lognormal than a normal distribution. The lognormal shape of the distribution in combination with a relatively large variance implies that large dispersivity values (i.e., much larger than the median value of the distribution) must be expected and considered in sensitivity studies. A large part of the variance of the dispersivity distribution could not be explained, however, by the above-mentioned parameters or factors. Since solute dispersion is a parameter that quantifies the effect of the water velocity variability on transport, it should be related to soil properties that embody information about this variability. In the field of stochastic continuum modeling of unsaturated flow and transport processes, dispersivity is derived or "predicted" from the spatial variance and spatial covariance of soil hydraulic parameters (e.g., Rubin, 2003). Unfortunately, obtaining information about the spatial variability of soil hydraulic parameters is at least as elaborate as carrying out a leaching experiment. Therefore, relying on indirect information about soil heterogeneity such as soil structure and soil classification seems to be more realistic to improve predictions of the dispersivity, and eventually for spatializing dispersivity parameters.
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APPENDIX
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Dispersivities () and experimental factors
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Travel distance
|
Scale
|
v
|
vp
|
v/vp
|
Jw
|
Jw eff
|
Type of flow
|
Texture
|
Horizon
|
Conc.
|
Measuring technique
|
Exp. no.
|
Experimental field
|
Source
|
| cm |
|
cm d–1 |
|
cm d–1 |
|
|
|
|
|
|
|
|
| 4.5 |
80.0 |
column |
1.0 |
1.0 |
1.00 |
0.36 |
|
intermittent |
cl |
|
flux |
direct |
195 |
Aberdeen soil series, North Dakota |
Cassel et al. (1974) |
| 10.2 |
13.0 |
core |
70.1 |
41.3 |
1.70 |
13.40 |
13.40 |
ponding |
sl |
C |
flux |
direct |
137 |
Alentejo region, Portugal |
Goncalves et al. (2001) |
| 20.7 |
20.0 |
core |
102.7 |
30.8 |
3.33 |
12.90 |
12.90 |
ponding |
l |
A |
flux |
direct |
138 |
|
|
| 7.9 |
20.0 |
core |
47.5 |
27.8 |
1.71 |
10.70 |
10.70 |
ponding |
l |
B |
flu | |
TOP
ABSTRACT
INTRODUCTION
