Dispersion parameters for undisturbed partially saturated soil

Journal of Hydrology, 143 (1993) 19--43 Elsevier Science Publishers B.V., Amsterdam

19

[3]

Dispersion parameters for undisturbed partially saturated soil K e i t h J. B e v e n a, D . E d H e n d e r s o n a a n d A l i s o n D . R e e v e s b

~Centrefor Research on Environmental Systems, Institute of Environmental and Biological Sciences, Lancaster University, Lancaster LA1 4YQ, UK bGeography Department, University of Dundee, Dundee DD1 4HN, UK

ABSTRACT Beven, K.J., Henderson, D.E. and Reeves, A.D., 1993. Dispersion parameters for undisturbed partially saturated soil. J. Hydrol., 143: 19-43. The advection--dispersion equation (ADE) does not necessarily provide a good description of solute transport in undisturbed soil cores and field plots, as a result of heterogeneities in pore water velocities related to variability in soil hydraulic properties, and non-equilibrium preferential flow and 'immobile' pore water effects. However, it is often used as a functional description of solute transport and this paper reviews the results of applying the ADE to undisturbed soil core and field plot experiments. Very wide ranges of dispersion coefficients and dispersivities have been found. Information available on the spatial variability of the solute transport parameters is also reviewed and suggests that local dispersion parameters may commonly be log normally distributed.

THE PROBLEM OF DISPERSION IN UNDISTURBED SOIL

A number of previous studies have shown that the spatial heterogeneity of flow velocities and dispersion parameters and the likelihood of preferential flow pathways in undisturbed soil mean that the advection-dispersion equation (ADE) may not always provide a good description of solute transport in the field (e.g. Schulin et al., 1987; Beven and Young, 1988; Ellsworth andJury, 1991; Hornberger et al., 1991). The ADE is a version of the classical partial differential diffusion equation, with the one-dimensional form a(Oc)/at

=

-

O(v®c)/Ox + O(DOOc/Ox)/Ox

(1)

where c is solute concentration, 0 is the volumetric moisture content, v is pore water velocity, D is the effective dispersion coefficient, x is distance in the Correspondence to: K.J. Beven, Centre for Research on Environmental Systems, Institute of Environmental and Biological Sciences, Lancaster University, Lancaster LA1 4YQ, U K .

0022-1694/93/$06.00

© 1993 - - Elsevier Science Publishers B.V. All fights reserved

20

K.J. BEVEN ET AL.

direction of flow and t is time. If, and only if, ® and v are constant, this can be rewritten as O(Oc/Ot)

=

-

vO(Oc/Ox)

+

D(O,v)OO 2¢/¢~x2

or

8c/8t

=

D 8 2 c / ~ x 2 -- v S c / ~ x

(2)

For steady flows a number of analytical solutions to (2) are available for various initial and boundary conditions (e.g. Parker and Van Genuchten, 1984b). For arbitrary changes in v, it is generally necessary to resort to numerical solutions. The dispersion coefficient D is expected to vary with the mean flow velocity v. A simple functional relationship is often assumed, of the form D =

Do + ~v"

(3)

where Do [L 2T- t ] is the diffusion coefficient for a solute in that medium which can often be neglected relative to the magnitude of the second term, and a and n are coefficients. The parameter a [l/n] is called the dispersivity of the medium, and is assumed to be a characteristic parameter of the medium structure. The problems that arise in using the ADE are, in part, a problem of scale. Work on the effect of heterogeneity of porous medium properties on dispersion in saturated groundwater systems has shown that it may be possible to use an equation equivalent to the ADE, but that the dispersion coefficient is scale dependent (see, e.g. Gelhar, 1986; Mishra and Parker, 1990; Gelhar et al., 1992). This scale dependence arises from the way in which individual solute 'particles' will gradually sample more and more of the velocity fluctuations associated with the heterogeneity of an aquifer. For steady flow in a large enough system with variability that is second-order stationary in space, eventually every particle will effectively sample the range of the velocity distribution in the aquifer and the dispersion coefficient should approach a constant value. The distance required to reach this point is the Lagrangian length scale of the system, and realistic estimates for unsaturated transport suggest that this scale may be up to the order of 100m (see Fig. 1 of Mishra and Parker (1990)). For unsaturated soil water systems, the same principle applies but in this case may be exacerbated by the non-equilibrium effects of preferential flow, resulting in the bypassing of parts of the pore space, and the vertical heterogeneity of the soil because of layering or horizons which may result in very distinct changes in soil water and transport characteristics of the soil over short distances. Indeed, Germann (1991) argued that the Lagrangian length scale of heterogeneous structured soils should be expected to be greater than

DISPERSIONPARAMETERS

21

the soil depth, and concluded that the application of the ADE is consequently invalid. Theoretical work has suggested, however, that the dominant mechanism of effective dispersion in this case is also the variation in local pore water velocity arising from variability in the hydraulic conductivity of the soil, and that variations in local dispersion coefficients may have only a secondorder effect (e.g. Russo, 1991; Russo and Dagan, 1991). Modified forms of the ADE can also be used to take some account of the effects of preferential flow or bypassing in the system in a similarly functional way. For example, it may be assumed that the bulk of the soil water and solute transport takes place in only part of the water-filled pore space (the 'mobile' pore volume), with an additional exchange term taking account of the retardation of solute in the remaining storage (the 'immobile' pore volume) (see, e.g. Coats and Smith, 1964; Parker and Van Genuchten, 1984b). The mobileimmobile ADE for a conservative solute is defined by the equations: (~(OmCm)/63t -+- t~(OimCim)/~l =

®imdcim/dt = CO(Cm-- Cim)

O((~mDO¢m/63Z)/Oz -- O(VOmCm)/(~Z

(4a)

(4b)

where subscripts m and im refer to mobile and immobile regions respectively. Om/im is the mobile/immobile water content, ¢m/imis the solute concentration in the mobile/immobile water, and o~ is an exchange coefficient, which will be expected to vary with pore structure and perhaps water content and pore water velocity. There are examples of observed solute transport characteristics that cannot be properly described with the ADE (e.g. Beven and Young, 1988). However, functionally, the ADE provides the mean transport and dispersive characteristics necessary to predict solute transport, provided that it is possible to estimate appropriate 'effective' values for the dispersion coefficient. It is possible to draw an analogy here with the necessity of estimating the hydraulic conductivity of unsaturated soil. Following the work of Richards (1931), water flow in unsaturated soil is described by a diffusive equation similar to eqn. (1) but in which the parameters are assumed to vary with the moisture content of the soil. Many functional relationships have been suggested to describe the way in which hydraulic conductivity varies with moisture content, some purely functional, some based on theoretical arguments, some attempting to take account of hysteresis where the value of hydraulic conductivity varies with the history of local wetting and drying. The use of such relationships is accepted in soil physics, even though they reflect the lack of a rigorous theory for the description of water movement at scales of interest because of the general impossibility of knowing complex pore geometries and heterogeneous structures in detail for a particular application. Consequently, it can also be argued that the ADE may be applicable in a

22

K.J. BEVEN ET AL.

functional sense in which the mean transport velocity reflects the mass flux of water averaged over some unit area in the system and the 'effective' dispersion coefficient will reflect the complexities of the flow pathways and heterogeneity in local fluid velocities in the direction of flow. This view then concentrates attention on deriving values for the effective dispersion coefficient under various flow conditions and at various scales. There are two ways of approaching this problem. One is to make some assumptions about the underlying structure of the porous medium and theorising about the effects on the effective dispersion coefficient. A number of studies have taken this approach (e.g. Bresler and Dagan, 1981, 1983; Simmons, 1982; Amoozegar-Fard et al., 1982; Russo, 1991; Beven, 1993). The problem is greatly simplified if only the variation in local transport velocity along each flow pathway is taken into account, neglecting the local dispersion and exchanges with any 'immobile' pore water. These types of models, therefore, are essentially parallel non-interacting piston displacement flow pathway models, differing in the way in which the velocity variations in each flow pathway are defined and calculated. Indeed, under certain conditions, this approach may be shown to be equivalent to the explicitly functional and non-mechanistic Jury Transfer Function Model (see Sposito et al., 1986b). An alternative approach is to identify the effective parameters of the ADE, if necessary in its modified two-component (mobile/immobile pore water) form, directly from tracer experiments without making any explicit assumptions about the structure of the medium itself. Although less theoretically attractive, this approach has the advantage that it may be much easier to carry out a large-scale tracer experiment at a site to determine the effective transport parameters, than to obtain the data necessary to determine, for example, the spatial statistics of hydraulic conductivity required by a theory of transport in heterogeneous soil. The parameters determined from the tracer experiments will also be real values in the sense that, if it is shown that the ADE can indeed predict the behaviour of the tracer, the fitted parameters do reproduce the transport behaviour of the solute. However, it will normally be possible to carry out only a very small number of tracer experiments, and this direct approach will be of limited value if the parameters determined in this way vary markedly under different flow conditions, without also having some way of predicting such changes similar to the relationships used for unsaturated hydraulic conductivity. For example, it could be argued that the degree of dispersion of a solute pulse in an undisturbed structured soil might be expected to increase more markedly than predicted from a constant dispersivity, as a result of the increasing effect of bypassing at higher flow rates. Similarly, it might be expected that the effective proportion of immobile water might increase with an increase in preferential flow, although no clear pattern

DISPERSION PARAMETERS

23

has emerged from published data in this respect (see the discussion by Reeves and Beven (1990)). This paper attempts to summarise the results of directly fitting transport parameters to tracer experiments, concentrating on those studies that have used either undisturbed soil columns in the laboratory or field plots. There have been many different studies of techniques for fitting the transport parameters to tracer data (e.g. Parker and Van Genuchten, 1984b; Jury and Sposito, 1985; Kool et al., 1987; Yamaguchi et al., 1989), including studies of the identifiability of the parameter values (e.g. Abeliuk, 1989; Mishra and Parker, 1989). These studies have shown that the problem is not simple: there is a limited amount of information in a single breakthrough curve and the parameters of the mobile/immobile ADE may interact in their effect on the predicted outputs, so that there may be a large degree of uncertainty associated with the fitted values. The data presented include tracer studies carried out at Lancaster University using undisturbed soil cores of 30 cm in diameter. In common with a number of other studies the parameters of the ADE have been fitted to the observed concentrations using the program CXTFIT of Parker and Van Genuchten (1984b) in which a non-linear least-squares algorithm is used to optimise the parameters of the mobile/immobile ADE under various specified boundary conditions. The CXTFIT program also allows models involving sorption or retardation parameters to be fitted, which has been found to be necessary in fitting some breakthrough curves. PREDICTIVE MODELLING AND THE PROBLEM OF SCALE The functional use of the ADE in this way, with effective parameters derived directly from measurements, implies that those parameter values reflect all the processes affecting dispersion at that scale, including the withinsample velocity fluctuations and exchanges with any 'immobile' pore water. However, it is clear that those values will be valid only for that particular sample, under the same flow conditions. For most purposes, predictions are required at scales much greater than the scale at which it is possible to make measurements. It is therefore necessary to consider the problem of scale in making use of information derived from tracer experiments, as well as the problem of extrapolating values of the ADE parameters to other flow conditions. In particular, the use of a distributed numerical transport model (e.g. SHETRAN-UK, as described by Ewen (1990)), which is based on solutions to the mobile/immobile ADE on a finite difference grid, requires the specification of effective dispersion parameters at the grid scale, typically a 50 m square

24

K.J. BEVEN ET AL.

or greater in plan, with a much finer resolution in the vertical. This poses two problems if the experimental data come from smaller-scale experiments: one is the expected spatial sampling variability of the experiments at the sub-grid scale and its integration to derive grid-scale values (see Beven, 1993); the second is the way in which the effective parameters might be expected to change with the length scale of the predictions (which may be greater or less than the length scale of the original experiments), when it is expected that the Lagrangian length scale will not be reached (see Germann, 1991). DISPERSION PARAMETERS DERIVED FROM UNDISTURBED CORE EXERIMENTS

There are a number of problems associated with carrying out tracer experiments on columns of undisturbed soil in the laboratory. The first is to ensure that the sampling procedure results in soil that is relatively undisturbed and that minimises the possibilities of flow along the sides of the core (see, e.g. Bouma and W6sten, 1979; Germann and Beven, 1981; Henderson et al., 1991). A second problem is one of scale. Unless special handling equipment is available, most cores will be restricted to the order of 30 cm in diameter and 1 m in length. At this scale, it cannot be ensured that an adequate sample of the structural flow pathways is included; indeed, the act of taking the core may cut some of the structural flow pathways with lateral links to adjacent soil. This may be important in properly assessing the impact of preferential flow pathways on solute transport. Analysis of such experiments will normally also be limited to a one-dimensional analysis in the direction of the flow, although dye tracing experiments show that flow patterns within such cores can be highly complex. Third, most tracer experiments are aimed at assessing the transport characteristics of the soil without the additional effects of chemical interactions with the soil materials, such as adsorption and anion exclusion. Thus, an ideal conservative tracer should be used. However, it is common that some tracer is apparently retained within the column in a typical tracer experiment, but it may be very difficult to determine whether this retention is due to diffusion and dispersion into immobile pore volumes, adsorption or some other retention mechanism (see Reeves and Beven, 1990). A further problem arises from the sampling of the tracer solution. In most studies, sampling takes place where outflow from the core is collected, but in some studies samples are taken from within the core using either zero-tension or porous cup lysimeters. Both methods of internal sampling can be expected to distort the flow field and may give misleading apparent concentrations. Zero-tension lysimeters require that the pore space must be locally saturated, or that the lysimeter intersects some preferential flow path, for any flow to be

25

DISPERSION PARAMETERS

TABLE ! Summary of available data of dispersion measurements on undisturbed soil cores Study

Cross-sectional Length area (m) (m2)

Lancaster (a) Abdulkabir (1989)

0.066

(b) Lancaster core (c) Slapton Wood Elrick and French (1966) Anderson and Bouma (1977a) Seyfried and Rao (1987) Dyson and White (1987) Smettem (1984)

0.073 0.073 0.0013 0.00785 0.00933 0.0380 0.03

Wierenga and Van Genuchten (1989)

0.7013 0.0020

Soil type

0.14-0.19 Silty clay loam to sandy loam 30.0 Silty clay loam 55.0 Silty clay loam 0.15 Silt loam 55.0 Silt loam 30.0 Clay loam 0.164 Clay loam 23-28 Loam to silty loam 6.00 Loamy fine 0.30 sand

No. of cores

6 l 1 4 10 5 18 4 1 large core, 'several' small cores

collected. The imposition of a vacuum in a porous cup sampler, on the other hand, may result in water being drawn out from the soil matrix and possibly from any 'immobile' pore volume (see, e.g. Starr et al., 1978). Again, the sampler would need to intersect a preferential flow pathway to reflect the concentrations in the flowing water, but in doing so would disrupt that pathway. The review of published data presented here concentrates on studies using 'conservative' tracers, such as chloride, bromide and tritium. It is recognised that these tracers may not always move in the same way as the water in the column, because of factors such as ion exclusion that m a y affect chloride transport (Thomas and Swodoba, 1970) and isotopic exchange that may affect tritium. Table 1 provides a summary of published undisturbed core experiments, and Table 2 the derived values of dispersion coefficients, dispersivities and, where available, proportions of mobile pore water. In reporting these values no account has been taken of how they were derived from the observations of solute concentrations in each study, so some care is required in interpretation. D a t a from breakthrough curve experiments are reported in various forms: as volume-averaged concentrations at an outflow collection point, as fluxaveraged concentrations at an outflow point or as resident concentrations from sampling points within the column. All can be used in fitting the A D E

26

K.J. BEVEN ET AL.

TABLE 2 Representative values of dispersion parameters from published undisturbed core experiments Study

Elrick and French (1966) undisturbed Anderson and Bouma (1977) medium sand/sub-angular blocky clay loam/prismatic Cassel et al. (1974) Aberdeen undisturbed Boetia undisturbed Smettem (1984) Seyfried and Rao (1987) Jury and Sposito (1985) Core - - highest value Core - - lowest value Dyson and White (1987) Abdulkabir (1989) Grass soil Forest soil Lancaster core

Slapton Wood core Upward flux

Downward flux Wierenga and Van Genuchten (1989) Small cores Large core

Dispersion coefficient (cm 2 h-1 )

Dispersivity (m)

Mean pore Om/19 water velocity (cm h - t )

1.404

-

1.52

4.17 0.417

-

0.0417 0.0417

0.197 0.161 24.0 0.077 (102.8/1.2) (0.218/0.013) 4.04 150.6 (6.44/1.65) (284.4/95.3) 5.642 3.367 23.075 (75.5/2.55)

0.251 0.197 0.066 (0.139/0.029)

0.230 (0.57/0.06) 5.55 (7.58/2.72) 160.4 54.2 18.6

-

0.526 0.357 0.243

370.0 31.5 15.5 13.5 8.4

1.217 0.206 0.203 0.177 0.220

20.9 (46.2/3.9) 2.71

-

0.0151 0.0151 2.3 (5.8/0.6) 2.709 (4.1/1.7) 0.225 0.171 2.941 (6.34/0.74) 1.409 (1.76/0.77) 52.4 (57.6/46.7) 3.04 1.52 0.76 3.04 1.53 0.76 0.76 0.38

0.41 (0.58/0.32) 0.38 (0.59/0.27) 0.68 0.95 i.0 0.61 0.44 0.33 0.42 0.47

0.0087 1.01 (0.0094/0.0077) (2.22/0.18) 34.0 0.08

Figures in parentheses are highest and lowest quoted values for each parameter when available.

DISPERSION PARAMETERS

27

model. Differences between volume-averaged and flux-averaged outflow concentrations were discussed by Parker and Van Genuchten (1984a). Using flux-averaged concentrations can mean that only a single 'mobile' component need be considered (e.g. Jardine et al., 1989) at least for conservative tracers. Figure 1 shows some typical results of fitting the ADE to measured breakthrough curves, in these cases from a pulse input of tracer under steady flow conditions. All the curves show features characteristic of undisturbed soil columns, with a rapid initial breakthrough and steep rising limb. Some extreme cases of such behaviour were reported by Anderson and Bouma (1977a, b), who found that they could fit dispersion coefficients to only the rising limb of the breakthrough curves for some of their columns. It is worth noting, however, that there is not much information in each individual curve; the ADE is essentially providing a simple transfer function from input to output curves. Given the variety of parameters that can be fitted using, for example, CXTFIT (mean pore water velocity, dispersion coefficient, proportion of mobile pore water, mobile/immobile water exchange coefficient, retardation coefficient and effective pulse length), there is the danger of using an overparameterised model. The curves of Fig. 1 have all been fitted using only two parameters, D and ®m/®, but even then in some cases there is a strong intercorrelation between the fitted values. That means that different pairs of values would give almost equally good results. It is also clear from Fig. 1 that some of the detail in the information in the curves is being treated by the non-linear least-squares fitting program as errors. The model is an approximation and its value will lie particularly in the extent to which the derived parameter values can be used or extrapolated for use under other initial, boundary and flow conditions. The extent to which this will be successful remains in doubt, as there has been very little validation of model predictions of transport in undisturbed soil. Certainly, some evidence suggests that the continuum assumptions of the model may be violated. For example, Jardine et al. (1989) reported very high values of D in a forest soil (implying very long mixing length scales). Destructive sampling of their columns after the application of a marker dye at the end of the experiments showed distinct preferential flow pathways; this technique has also bee used to similar effect in undisturbed soil columns by Seyfried and Rao (1987), Hornberger et al. (1991) and others (see Fig. 2). However, some studies have reported strong relationships between transport parameters and variables such as flow rate. Dyson and White 0987), reporting on the results of breakthrough curve experiments in large columns of undisturbed Evesham clay soil, calculated values of D ranging from 2.81 to 90 cm 2h -~ , with a strong dependence on flow rate. Regression

28

K.J. BEVEN ET AL Observed sad almulmtnd Bromide output concentrations during steady state return flow at 250 ml/hoor through the L a ~ t e r Unlverllty soil oore. 100 " Observed

~irr,uict e d

80~ 80 -~ 70

50 2

4O-" 30~

10

(a)

20

30

40 50 80 70 80 90 100 Time from start of Bromide Input (hours)

110

120

130

140

120

180

140

Observed and slmuklted Bromide OUtPUt concentration during steady stlto ralnfal st 250 ml/hoor through the 8lepton soil core. 100

90i 80 70 ~ 80 ~ 80 ~ 4o 3o~

2o! 0

(b)

10

20

30

40

50

Time from

80 70 80 90 100 start of Bromide Input (hours)

110

Obeerved and sknulated o-TIFMBA otltput concentreUon durkng steady stets return flow at 280 ml/hoor through the 81apron ~ core. 25 l

Otserved

Simuo:ed

22.5 l

17,5

, "" '.

~

~

'

157

|

1

,281

/S

'G

7.5

(¢)

0

10

20

30

4O 50 60 ?0 80 90 100 Time after start Of o-TFMBA Input (hours)

110

120

130

140

Fig. 1. Examples of the use of C X T F I T to fit A D E parameters to breakthrough curves under steady-state flow conditions at 0.764 cm h '. (a) Lancaster University core, upward flux with tracer input of one pore volume over 44 h. (b) Slapton Wood core, downward flux with tracer input of one pore volume over 67 h. (c) Slapton Wood core, upward flux with tracer input of one pore volume over 78 h.

29

DISPERSION PARAMETERS

r;avil cnvily

5cm

20cm

25cm

30cm

Fig. 2. Core sections showing dye staining, Lancaster University core. Distances from base of the core. Black indicates recognisable macropores with dye staining, hatched areas indicate dye staining in areas without obvious macropores.

30

K.J. BEVEN ET AL.

analysis suggested a relationship with a more than linear slope of the form D =

4.1 vl42cm2h -J

(R 2 =

0.889)

Dyson and White noted that the dispersivity value of 4.1 cm (0.074) is generally greater than values given in the studies reviewed by Sposito et al. (1986a), but is much lower than the values for the Lancaster experiments listed in Table 2. High values of apparent dispersivity have also been calculated by Smettem (1984) for undisturbed columns of a brown earth soil with subangular blocky peds, ranging from 1.3 to 21.8cm, corresponding to D values of between 1.15 and 102.8cm2h -1 . Undoubtedly, the variation in calculated A D E parameter values reported from these studies must depend in part on the conditions of the experiments as well as the soil type. Table 1 shows that these studies have used a wide variety of column sizes, flux rates and boundary conditions. There has been very little systematic study of these effects, although one study (Parker and Albrecht, 1987) has looked at sample size effects, comparing the data from columns of three sizes taken from two horizons at 19 sites along a 150m transect. Small cores were of 0.054 m length and 0.04 m in diameter, mediumsized cores were of 0.1 m length and 0.06 m in diameter and the large cores were of 0.15 m length and 0.1 m in diameter. The largest cores were therefore considerably smaller than some of the cores in the studies reported above. Parker and Albrecht reported their results in terms of the mean and variance of dispersivity for the various core sizes and horizons, and showed that the mean dispersivity consistently increases with increasing sample size. For the largest samples, mean dispersivities were again very large - - 149 cm for the Ap horizon and 38 cm for the Bt horizon. A number of other studies have reported the results of breakthrough curve experiments but without the calculation of the ADE model parameters. In some cases, this was because it proved impossible to reproduce the observed breakthrough curve, at least using a single domain A D E model (e.g. for some of the curves reported by Anderson and Bouma (1977a) and Smettem (1982)). In all cases the undisturbed core breakthrough curves show consistently rapid initial breakthrough and large dispersion, sometimes with apparently very long tailing even for tracers considered to be conservative (see, e.g. M c M a h o n and Thomas, 1974; Bouma and W6sten, 1979; Tyler and Thomas, 1981; Bouma et al., 1983; Jury et al., 1986). It is worth comparing the results for these undisturbed columns with the data collected by Nkedi-Kizza et al. (1984) on columns made up of various sizes of aggregates from a Californian Oxisol. Both 36C1 and 3H20 tracers were used. It was found that the single domain ADE fitted only the data from the columns of small (0.5-1 mm) aggregates, whereas good fits were obtained

31

DISPERSION PARAMETERS

TABLE 3 Summary of published data of dispersion measurements on undisturbed field plots Study

Plot size Plot area

No. of sub-plots

Depths (m)

20

Up to 1.824

Soil type

Sub-plot area

(m2) Biggar and Nielsen (1976) Van de Pol et al. (1977) Jury and Sposito (1985) Bowman and Rice (1986) Rice et al. (1986) Ellsworth and Jury (1991) Schulin et al. (1987) Jaynes et al. (1988) Jaynes (1991)

150 ha 64m 2

42.3 -

0

0.75

-

0

1.2-1.8

37.2 m 2

3.2

4

3.0

0.62 ha

446.5

14

2.7

0.64 ha

2.25 m 2 4.00m 2 94 m 2

-

5 3 0

4.0 4.0 3.0

37.2 m 2

3.35

4

3.0

37.2m 2

3.35

4

3.0

Horizons of clay, silty clay Loamy sand Horizons of silty clay loam, silt loam, sandy loam Sandy loam Loam/loamy sand Sandy loam/regosol As Bowman and Rice (1986) As Bowman and Rice (1986)

to all the data sets using the two domain ADE. Values of the proportion of mobile pore volume ranged from 0.44 to 0.96, values of D from 0.16 to 8.49cm2h -1 and dispersivity values from 0.32cm for large aggregates to 0.70 cm for the smallest aggregates, with a generally strong linear relationship between D and v. DISPERSION PARAMETERS DERIVED FROM FIELD PLOT EXPERIMENTS

Table 3 gives a summary of published tracer experiments carried out in situ on field plots, with a summary of the parameter values derived in Table 4. In 1976, Biggar and Nielsen published an extensive study of solute transport in a 150ha field, with application of a chloride tracer on 20 randomly chosen plots, each 6.5 m square. In each plot two tensiometers and two porous cup samplers were installed at each of six depths between 0.30 and 1.8 m. Each plot was ponded with water until a steady flow regime was

32

K.J. BEVEN ET AL.

TABLE 4 Representative values of dispersion parameters from published field plot experiments Study

Dispersion coefficient (cm2h- l )

Dispersivity (m)

Mean pore water velocity (cm h- 1)

0.0547 (0.257/0.017) 0.0942 (2.72/0.04)

1.826 (15.28/0.008) 0.1625 (0.218/0.130)

16.14 (47.9/6.5) 21.759 (107.9/1.558)

0.524 (1.41/0.208) 0.445 (0.952/0.094)

0.290 (0.367/0.197) 0.293 (0.533/0.162)

0.0604 0.0471 2.229 (6.67/1.13) 88.90 (99.9/79.2) 149.6 (920.8/1.9) 2.488 (3.00/1.313)

0.0393 0.0214 0.165 (0.21/0.14) 0.178 (0.228/0.138) 0.288 0.658/0.045) 0.173 (0.257/0.197)

0.0163 0.0220 0.146 (0.46/0.071) 5.00 (5.75/4.38) 4.75 (29.92/0.42) 0.146 (0.179/0.117)

-

0.102 0.177

-

Biggar and Nielsen (1976)

8.776 (66.5/0.35) 1.531 (3.54/0.925)

Van de Pol et al. (1977) Bowman and Rice (1986) Water applied semi-weekly Water applied bi-weekly Schulin et al. (1987) Br- as tracer CI- as tracer Rice et al. (1986) Jaynes et al. (1988) Jaynes (1991) Jury and Sposito (1985) Butters and Jury (1989) Field scale Local scale

Figures in parentheses are highest and lowest quoted values for each parameter when available.

established, after which a pulse o f t r a c e r was a d d e d , a n d t h e n p o n d e d infiltration c o n t i n u e d . T h e m e a n value o f D, o v e r all d e p t h s a n d plots, was 15.32 c m 2 h - l , with a variance o f 9.9 x 104 c m 4 h - 2 . T h e range o f average D values for the individual plots was f r o m 0.6 to 312.4 c m 2 h - t . A v e r a g e values o v e r all plots at the individual d e p t h s r a n g e d f r o m a m i n i m u m o f 9.87 c m 2 h-1 at 0 . 3 0 m to a m a x i m u m o f 2 5 . 6 c m 2 h -I at 1.52m, t h e n fell to 1 . 0 6 c m 2 h -~ at 1.83 m. E a c h o f these m e a n values disguises c o n s i d e r a b l e variability between depths a n d plots, s o m e t h i n g t h a t will be discussed f u r t h e r in the next section. T h e d a t a also s h o w e d a relationship between D a n d p o r e w a t e r velocity, o f the f o r m (including d a t a f r o m all plots a n d all depths) D

=

0.025 + 0 . 1 2 2 v l l l c m 2 h -1

(N

=

359, R 2 =

0.632)

33

DISPERSION PARAMETERS

Van de Pol et al. (1977) carried out a similar experiment on a single 8 m by 8 m plot with continuous irrigation by trickle lines for 42 days with unlabelled water, 36 days of chloride and tritium tracers and a further 52 days of unlabeUed water. They sampled for the tracer at three sites and eight depths. The data revealed some interesting behaviour, including an apparent delay in the response to the tracer at a depth of 45 cm, compared with that at both 33.5 and 63.5 cm depths. Jury and Sposito (1985) described the A D E analysis of two earlier field plot experiments, one involving porous cup sampling at five depths on a 4 × 4 grid (Jury et al., 1982) and the other 36 core samples within a 6 m × 6 m area with measurements at 0.1 m depth increments on two days. They showed how the calibrated values of D and v vary with the optimisation method used, and evaluated the uncertainties associated with the parameter values arising from the model error and the error associated with trying to estimate the mean concentration at a given depth from a small number of samples. Jaynes (1991) reported on a bromide tracing experiment carried out on a 6.1 m by 6.1 m plot of clay loam soil, with seven replicate porous cup samplers installed at depths from 0.3 to 3.0 m. His results show that there is considerable variability of the apparent pore water velocity between sites, with the ratio to an expected pore water velocity based on the input flux and mean water content ranging from 0.143 to 3.86. In an earlier study at the same site, Bowman and Rice (1986) and Jaynes et al. (1988) studied transport on a 37 m 2 plot, with solution sampling at depths from 0.3 to 3 m under either weekly or semi-weekly irrigation regimes. Local D values showed a weak relationship with pore water velocity of the form D =

0.11

v E l l c m 2 h -1

(N =

184, R 2 =

0.552)

Two years later the same plot was subjected to a flood irrigation, and Jaynes et al. (1988) reported a much higher mean dispersion coefficient of 88.92 c m 2 h-I and only a poor correlation between pore water velocity and D: D

=

2.55v°77cmZh

-1

(N =

40, R 2 =

0.235)

Other studies have shown that correlations between D and pore water velocity can be highly variable and in some cases apparently negative (Van Ommen et al., 1989). Jaynes et al. (1988) also found that under this flood irrigation regime, there appeared to be little bypassing of the pore space or immobile volume, except close to the surface. This conclusion contrasted with that of the earlier, partially saturated study on the same plot which showed high bypassing ratios, but was similar to the conclusions of Shuford et al. (1977). However, Rice et al. (1986) in a study of 14 12.2m by 9.15m plots subjected to eight

34

K.J. BEVEN ET AL.

flood irrigations, suggested that the proportion of mobile water was only 0.25 of the water-filled pore space. Schulin et al. (1987) studied the transport of surface-applied bromide and chloride solutions within a 94 m E plot under natural rainfall and evapotranspiration conditions on a sloping site. After 399 days, a 15 m trench was dug downslope through the plot and the concentrations of solute in the non-stony soil matrix was measured. A further sample collection was carried out some 5 months later after another summer evaporation period. Some very irregular patterns of solute movement were recorded. However, it was still possible to model the pattern of average concentrations for each depth using the ADE and the stochastic pore water velocity model of Parker and Van Genuchten (1984b), which assumes a constant local dispersivity at the field scale. For the bromide data, the estimated dispersivities for the two models were very similar, showing that the calibration process of the stochastic model had resulted in a value of dispersivity representative of the field rather than local scale. Extrapolation forwards in time of the two models, based on parameters fitted to the data of the first sampling, failed completely. The observed concentrations at the second sampling period showed that the mass of tracer had not moved deeper into the profile at all and in fact had moved upwards as a result of evapotranspiration losses during an extremely dry summer. The model assumptions of a constant mean pore water velocity over time are clearly violated under such conditions. Analysis then requires a fully dynamic model. Butters and Jury (1989) observed a linear growth of the dispersion coefficient over the first 2 m of the areally averaged bromide concentration in a 64ha field. Below that depth, however, the apparent dispersion coefficient showed some erratic behaviour in response to changes in soil texture. At the same site, Ellsworth and Jury (1991) looked at the dispersion of applications of tracer under controlled flux conditions on l0 sites (between 2.25 and 4 m 2) and during the leaching of a resident solute concentration at an additional two sites. Solute concentrations were sampled by destructive sampling to give three-dimensional patterns on the 2.25 m 2 plots, with multiple samples taken on the 4 m 2 plots. An average vertical dispersivity was estimated for the plots at 9.1 cm, with a field scale average dispersion coefficient of 2.86 cm 2h - t. This data set shows some interesting scaling behaviour in that the position of the centre of mass of the moving solute over time shows a very high relationship with the net applied volume of water (R 2 = 0.96). They showed that using average pore water velocity and dispersion coefficients scaled by the net flux rate of water, generally good predictions of solute transport could be obtained at most sites, despite the problems of sampling associated with data derived from small soil cores on large plots. They concluded that 'a laboratory

DISPERSIONPARAMETERS

35

TABLE 5 Summary of means and variances of local scale dispersion parameters assuming a log normal distribution Study

Biggar and Nielsen (1976) Bowman and Rice (1986) Schulin et al. (1987) BrCIParker and Albrecht (1987) Ap horizon, large samples Bt horizon, large samples Weirenga and Van Genuchten (1989) - - small cores Jaynes (1991)

Dispersion coefficient (cm2h-I )

Dispersivity (m)

Mean

Mean

Variance

247.358 179.09

0.0547 0.524

0.0032 0.165

1.69 0.68

0.0393 0.0214

-

1.116 0.281

3.158 2.293

0.0087 4.75

0.3 × 34.99

8.776 16.37 0.0604 0.0471 20.9 149.6

Variance

275.6 37262.0

-

10 -3

experiment performed on an "undisturbed" soil column (which was large enough to characterise the variability represented in the soil cores taken at each sample event) could be used to predict most of the field scale solute transport, if combined with field measurements of evaporation and volumetric water content' (Ellsworth and Jury, 1991, p. 978). SAMPLING VARIABILITY OF DISPERSION PARAMETERS

There have been a number of studies in which sufficient data have been available to evaluate the local spatial variability of the transport parameters, either by multiple column studies or multiple plot studies (see Table 5). It is necessary here to be very careful about problems of scale, as tracer concentrations, especially resident concentrations, may be evaluated from small samples within larger-scale column or plot studies, which may then be replicated at the scale of the field. In what follows we will attempt to draw attention to the variability of the transport parameters derived from data at the local scale of sampling, remembering that the concentration data from which the parameter values are derived will be affected by the local heterogeneity of flow processes af the scale of the small plot. It is worth noting that some studies have found a surprising lack of transverse dispersion within a predominantly vertical flux (e.g. Ellsworth and Jury, 1991), whereas others (e.g. Schulin et al., 1987) have commented on the marked transverse heterogeneity of measured concentrations.

36

K.J. BEVEN ET AL.

Perhaps the earliest study involving a detailed evaluation of spatial heterogeneity of solute transport was that of Biggar and Nielsen (1976; see above). Their data revealed highly skewed distributions for both pore water velocity and apparent local dispersion coefficient, which they suggested could be represented by log normal distributions, a conclusion supported by the studies of Van de Pol et al. (1977), Bowman and Rice (1986) and Jaynes et al. (1988). Distributions of local values were also evaluated in the stony soil plot experiments of Schulin et al. (1987). In this case, dispersion coefficients were log normally distributed but pore water velocities were normally distributed or fell between normal and log normal distributions for the various tracers. Van Ommen et al. (1989) found that their estimated dispersion coefficients were log normally distributed, whereas Parker and Albrecht (1987) found a similar result for the distributions of local dispersivity from their multiple undisturbed column experiments. Finally, Van Wesenbeeck and Kachanoski (1991) recently attempted to evaluate the spatial dependence of dispersion by examining the change in the variance of breakthrough curves averaged over various spatial scales. Their data were based on porous cup samplers placed at depths of 0.4 m at a spacing of 0.2 m in a well-drained, coarse-textured Typic Hapludalf soil with welldeveloped horizons. In a cultivated site, 48 samplers were placed along a single transect, and 50 m away in a forested site there was a transect of 32 samplers. In both cases, breakthrough curves were measured for each sampler after the surface application of a chloride tracer. Van Wesenbeeck and Kachanoski suggested that the variance of the travel times indicated by the spatially averaged breakthrough curves reaches an upper limit at a range of about 2.5 m for the forested site and about 4 m for the cultivated site. This is the only study we know of that has attempted to evaluate the spatial dependence of solute transport in undisturbed soil directly. SUMMARY AND DISCUSSION

The analyses of solute transport data from undisturbed soil columns and field plots reviewed above have revealed that, with some few exceptions, the various forms of the convection-dispersion equation can be fitted to the observed concentrations, whether resident concentrations in the case of field plots or the outflow breakthrough curves of column experiments. This fitting process, however, results in a very wide range of dispersion coefficients and dispersivities, which tend towards high values in these undisturbed materials (maximum values of 7525 cm 2h - ~ for cores and 920 cm z h - ~ for field plots, respectively). In that the calibration method (and particular model structure

DISPERSION PARAMETERS

37

Log dispersion versus log pore water velo~ty from avdable data

7

I

v Core Studies

x Pot Studies

] v e

v

0

~ x

7

r7

-2 --t

,2 f

r - ~ F ~ -5

-4

-3

R ~ r v ~ -2 -1 In(v) (em/hour)

- ,~ ~

~ 0

-

~

~w-~ ~ 1

2

Fig. 3. Summary of pore water velocity and dispersion data from published papers and studies on cores from the University of Lancaster and the Slapton Wood catchment. Triangles indicate core studies, crosses field plot studies. Lines represent the range of values for a particular study. 1, Cassel et al. (1974), Boetia undisturbed soil core; 2, Cassel et al. (1974), Aberdeen undisturbed soil core; 3, Schulin et al. (1987), data for mean of local fits to five large soil samples with bromide tracer; 4, Anderson and Bouma (1977b), data for cores with prismatic soil structure; 5, Anderson and Bouma (1977b), data for cores with blocky soil structure; 6, Wierenga and Van Genuchten (1989); 7, Rice et al. (1986); 8, Jury and Sposito (1985); 9, Van de Pol et al. (1977); 10, Bowman and Rice (1986), data for semi-weekly irrigations; 11, Bowman and Rice (1986), data for once-weekly irrigations; 12, Slapton Wood grass core with constant rainfall at 0.382 cm h-~; 13, Slapton Wood core with constant rainfall at 0.764 cm h-~; 14, Slapton Wood core with constant return flow at 0.764cmh-~; 15, Lancaster core with constant return flow at 0.764cmh-~; 16, Wierenga and Van Genuchten (1989); 17, Abdulkabir (1989), data for grassland cores with steady-state saturated flow; 18, Elrick and French (1966); 19, Slapton Wood core with constant return flow at 1.53 cm h-J; 20, Lancaster core with constant return flow at 1.52cm h-~; 21, Biggar and Nielsen (1976); 22, Smettem (1984); 23, Seyfried and Rao (1987); 24, Dyson and White (1987); 25, Slapton Wood core with constant return flow at 3.04 cm h - ~; 26, Lancaster woodland core with constant return flow at 3.04 cm h - ~; 27, Jaynes (1991); 28, Jaynes et al. (1988).

used) also affects the values fitted, great care needs to be taken in making any mechanistic interpretations of the parameter values. The values quoted here have also been derived from a variety of ways of presenting the values in the various papers and should therefore only be taken as order of magnitude estimates. • The data for all the studies are summarised in Fig. 3, which shows that even though individual studies may demonstrate a reasonable relationship between pore water velocity and dispersion coefficient at a particular site or range of flow conditions, there is little consistency between studies, including in one case different studies at the same site (Bowman and Rice, 1986; Jaynes et al.,

38

K.J. BEVEN ET AL. Published D against v relationships plotted across the range of mean pore water velocities reported for each study. 7 ~C

5~_ 4 j~

3 :/'

× v ->

Dys,Drl ond White 11987) Joyr!es ( 1 9 8 8 Dose IrricotiorBiggo, onJ r,,elsen 1975)

~ 1~lcp,t Y ' Petorn :c',*, F~ Loq, uster F~'tJr n Flow .

.

.

.

2Ji

. ///

1:! J

0

c

~z _

_ _

- - - - - .

~:x

j,/

tC> ?~ _

j"

.z

j

_ _

_

.1L

,

I

-2 d -3 ~ .4 E -5

-4

-3

-2 -1 In(v) [cm/hr]

0

1

2

Fig. 4. Published regression relationships for the variation of dispersion coefficient (D) with pore water velocity (v).

1988; Jaynes, 1991). The generally higher flow rates used in core experiments are apparent. The relationship between pore water velocity and dispersion coefficient has been shown to be also highly variable, with a range of dispersivity values from 1.3 to 149 cm ~/"and values of the exponent n of eqn. (2) from 0.77 to 2.11. The published relationships quoted above are summarised in the plot of Fig. 4, which shows the range (but not the variability) of the original data from which they were developed. Theoretical work suggests that field scale dispersion can be dominated by the variations in v, although in fitting the ADE model to data some of this variation may be compensated by larger values of D. In fact, distributed transport models will predict a mean pore water velocity over an area or grid square such that the dispersive effects of field scale variations in v must all be expressed in terms of an appropriate 'effective' value of D. The derivation of such effective values, and sensitivity to the underlying local variations in the parameter values, has been addressed, for example, by Russo et al. (1989) in a number of hypothetical cases. In a companion paper (Beven, 1993), the value of a single measurement in conditioning the estimates of such 'effective' values is considered. There is much less information available about the range of variability to be expected in the mobile water fraction, and less still about exchange coefficients for mobile/immobile pore water transfers. It is expected that such

DISPERSION PARAMETERS

39

parameters might become more important in attempting to predict the transport of reactive solutes. Several studies have suggested that the spatial variability of local dispersion coefficients may be represented by log normal distributions. However, mean values vary by orders of magnitude between studies, and the range of values for the variance of In D across all the studies reviewed was 0.37 to 4.22 (D in cm2h-l). In one study that attempted to evaluate the transition in transport characteristics from local to field scale in undisturbed soil (Van Wesenbeeck and Kachanowski, 1991), it was suggested that the variance of breakthrough curve travel times, when averaged over all curves within a given spatial scale, reaches a maximum at comparatively short distances (2.5-4 m at the site studied). In the studies reviewed here, the various forms of the ADE have been used as a lumped conceptual model in the sense of Beven (1989) rather than a mechanistic description of the processes of dispersion taking place in the field plots or soil cores. Consequently, it is difficult to place any mechanistic interpretation on the values of the various parameters; the model simply has enough degrees of freedom to be able to fit the modal behaviour in the data, and many other models might equally be able to fit the data (see, e.g. Jury and Sposito, 1985; Dyson and White, 1987; Schulin et al., 1987; Beven and Young, 1988). Intercorrelation between the parameters would also suggest that many other combinations of values may fit the data almost equally well, with resulting uncertainty in extrapolation of predictions to other time periods, flow conditions or different reactive solutes. The problem of parameter identifiability is likely to be worse for the case of reactive solutes unless it is possible to fix at least some parameters a priori. In consequence, it may be difficult to estimate the parameter values for a predictive model for any particular site given the reported variability between experiments, soil type, and in space and time. It is therefore important that the parameters of the model be, wherever possible, based on experimental data for a particular site of interest. However, both core and field plot experiments can be expensive and time consuming to carry out and still give only a small sample of the possible range of conditions at the field, hillslope or catchment scale of interest. Thus it will be important to take account of uncertainty in parameter values when making predictions of solute transport in soils. ACKNOWLEDGEMENTS This work has been funded in part by EEC Grant EV40091 and by U K N I R E X Ltd as part of their Disposal Safety Assessment Research Programme Grant UX]96/242. Thanks are due to Jack Parker of VPI, Blacksburg, Virginia, who supplied us with his CXTFIT parameter optimisation program.

40

K.J. BEVEN ET AL.

REFERENCES Abdulkabir, M.O., 1989. The movement of water and solutes through saturated and unsaturated structured soil. Ph.D. Thesis, Lancaster University, UK, 189 pp. Abeliuk, R., 1989. Identification of unsaturated solute transport parameters. IAHS Publ., 188: 261-270. Amoozegar-Fard, A., Nielsen, D.R. and Warrick, A.W., 1982. Soil solute concentration distribution for spatially varying pore water velocities and apparent diffusion coefficients. Soil Sci. Soc. Am. J., 46: 3-8. Anderson, J.L. and Bouma, J., 1977a. Water movement through pedal soils. I. Saturated flow. Soil Sci. Soc. Am. J., 41: 413-418. Anderson, J.L. and Bouma, J., 1977b. Water movement through pedal soils. II. Unsaturated flow. Soil Sci. Soc. Am. J., 41: 419-423. Beven, K.J., 1989. Changing ideas in hydrology: the case of physically-based models. J. Hydrol., 105: 157-172. Beven, K.J., 1993. Estimating transport parameters at the grid scale: on the value of a single measurement. J. Hydrol., in press. Beven, K.J. and Young, P.C., 1988. An aggregated mixing zone model of solute transport through porous media. J. Contain. Hydrol., 3: 129-143. Biggar, J.W. and Nielsen, D.R., 1976. Spatial variability of the leaching characteristics of a field soil. Water Resour. Res., 12: 78-84. Bouma, J. and W6sten, J.H.M., 1979. Flow patterns during extended saturated flow in two undisturbed swelling clay soils with different macrostructures. Soil Sci. Soc. Am. J., 43: 16-22. Bouma, J., Belmans, C., Dekker, L.W. and Jeurissen, W.J.M., 1983. Assessing the suitability of soils with macropores for subsurface liquid waste disposal. J. Environ. Qual., 12: 305-311. Bowman, R.S. and Rice, R.C., 1986. Transport of conservative tracers in the field under intermittent flood irrigation. Water Resour. Res., 22: 1531-1536. Bresler, E. and Dagan, G., 1981. Convective and pore scale dispersive solute transport in unsaturated heterogeneous fields. Water Resour. Res., 17: 1683-1689. Bresler, E. and Dagan, G., 1983. Unsaturated flow in spatially variable fields, 3. Solute transport models and their application to two fields. Water Resour. Res., 19: 429-435. Butters, G.L. and Jury, W.A., 1989. Field scale transport of bromide in an unsaturated soil. 2. Dispersion modeling. Water Resour. Res., 25: 1583-1589. Cassel, D.K., Kreuger, T.H., Shroer, F.W. and Norum, E.B., 1974. Solute movement in disturbed and undisturbed soil cores. Soil Sci. Soc. Am. Proc., 38: 36-40. Coats, K.H. and Smith, D.B., 1964. Dead end pore volume and dispersion in porous media. Soc. Petrol. Eng. J., 4: 73-84. Dyson, J.S. and White, R.E., 1987. A comparison of the convection-dispersion equation and transfer function model for predicting chloride leaching through an undisturbed structured clay soil. J. Soil Sci., 37: 157-171. Ellsworth, T.R. and Jury, W.A., 1991. A three-dimensional field study of solute transport through unsaturated layered porous media. 2. Characterisation of vertical dispersion. Water Resour. Res., 27: 967-981. Elrick, D.E. and French, L.K., 1966. Miscible displacement patterns on disturbed and undisturbed soil cores. Soil Sci. Soc. Am. Proc., 30: 153-156.

DISPERSION PARAMETERS

41

Ewen, J., 1990. Basis for the subsurface contaminant migration components of the catchment water flow, sediment transport and contaminant migration modelling system. SHETRANUK, Rep. NSS/R229, UK Nirex Ltd, Harwell, 85 pp. Gelhar, L.W., 1986. Stochastic subsurface hydrology from theory to applications. Water Resour. Res., 22: 135s-145s. Gelhar, L.W., Welty, C. and Rehfeldt, K.R., 1992. A critical review of data on field-scale dispersion in aquifers. Water Resour. Res., 28: 1955-1974. Germann, P.F., 1991. Length scales of convection-dispersion approaches to flow and transport in porous media. J. Contain. Hydrol., 7: 39-49. Germann, P.F. and Beven, K.J., 1981. Water flow in soil macropores. 1. An experimental approach. J. Soil Sci., 32: 1-12. Henderson, D.E., Beven, K.J. and Reeves, A.D., 1991. Flow separation of water movement through a large soil core using bromide and fluorobenzoate tracers. Proc. 3rd National Hydrology Symposium, Southampton, September 1991. British Hydrological Society, Institute of Hydrology, Wallingford, UK, pp. 3.25-3.30. Hornberger, G.M., Germann, P.F. and Beven, K.J., 1991. Throughflow and solute transport in an isolated sloping soil block in a forested catchment. J. Hydrol., 124: 81100. Jardine, P.M., Wilson, G.V., Luxmoore, R.J. and McCarthy, J.F., 1989. Transport of inorganic and natural organic tracers through an isolated pedon in a forest watershed. Soil Sci. Soc. Am. J., 53: 317-323. Jaynes, D.B., 1991. Field study of Bromacil transport under continuous-flood irrigation. Soil Sci. Soc. Am. J., 55: 658-664. Jaynes, D.B., Bowman, R.S. and Rice, R.C., 1988. Transport of a conservative tracer in the field under continuous flood irrigation. Soil Sci. Soc. Am. J., 52: 618-624. Jury, W.A. and Sposito, G., 1985. Field calibration and validation of solute transport models for the unsaturated zone. Soil Sci. Soc. Am. J., 49: 1331-1341. Jury, W.A., Stolzy, L.H. and Shouse, P., 1982. A field test of the transfer function model for predicting solute transport. Water Resour. Res., 18: 369-375. Jury, W.A., Elabd, H. and Resketo, M., 1986. Field study of napropamide movement through unsaturated soil. Water Resour. Res., 22: 749-755. Kool, J.B., Parker, J.C. and van Genuchten, M.Th., 1987. Parameter estimation for unsaturated flow and transport models - - a review. J. Hydrol., 91: 255-293. McMahon, M.A. and Thomas, G.W., 1974. Chloride and tritiated water flow in disturbed and undisturbed soil cores. Soil Sci. Soc. Am. Proc., 38: 727-732. Mishra, S. and Parker, J.C., 1989. Parameter estimation for coupled unsaturated flow and transport. Water Resour. Res., 25(3): 385-396. Mishra, S. and Parker, J.C., 1990. Analysis of solute transport with a hyperbolic scale-dependent dispersion model. Hydrol. Processes, 4: 45-70. Nkedi-Kizza, P., Biggar, J.W., Selim, H.M., van Genuchten, M.Th., Weirenga, P.J., Davidson, J.M. and Nielsen, D.R., 1984. On the equivalence of two conceptual models for describing ion exchange during transport through an aggregated Oxisol. Water Resour. Res., 20: 1123-1130. Parker, J.C. and Albrecht, K.A., 1987. Sample volume effects on solute transport predictions. Water Resour. Res., 23: 2293-2301. Parker, J.C. and van Genuchten, M.Th., 1984a. Flux-averaged and volume-averaged concentrations in continuum approaches to solute transport. Water Resour. Res., 20: 866-872.

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K.J. BEVEN ET AL.

Parker, J.C. and van Genuchten, M.Th., 1984b. Determining solute transport parameters from laboratory and field tracer experiments. Va. Agric. Exp. Stn. Bull., 84--3, 96 pp. Reeves, A.D. and Beven, K.J., 1990. The use of tracer techniques in the study of soil water flows and contaminant transport. NSS Rep. 233, UK NIREX Ltd, Harwell, 65 pp. Rice, R.C., Bowman, R.S. and Jaynes, D.B., 1986. Percolation of water below an irrigated field. Soil Sci. Soc. Am. J., 50: 855-859. Richards, L.A., 1931. Capillary conduction of liquids through porous mediums. Physics, 1: 318-333. Russo, D., 1991. Stochastic analysis of simulated vadose zone solute transport in a vertical cross-section of heterogeneous soil during non-steady water flow. Water Resour. Res., 27(3): 267-283. Russo, D. and Dagan, G., 1991. On solute transport in a heterogeneous porous formation under saturated and unsaturated water flows. Water Resour. Res., 27: 285-292. Russo, D., Jury, W.A. and Butters, G.L., 1989. Numerical analysis of solute transport during transient irrigation. 1. The effect of hysteresis and profile heterogeneity. Water Resour. Res., 25: 2109-2118. Schulin, R., van Genuchten, M.Th., Fluhler, H. and Ferlin, P., 1987. An experimental study of solute transport in a stony field soil. Water Resour. Res., 23: 1785-1794. Seyfried, M.S. and Rao, P.S.C., 1987. Solute transport in undisturbed columns of an aggregated tropical soil: preferential flow effects. Soil Sci. Soc. Am. J., 51: 14341444. Shuford, J.W., Fritton, D.D. and Baker, D.E., 1977. Nitrate-nitrogen and chloride movement through undisturbed field soil. J. Environ. Qual., 6: 255-259. Simmons, C.S., 1982. A stochastic-convective transport representation of dispersion in onedimensional porous media systems. Water Resour. Res., 18:1193-1214. Smettem, K.R.J., 1982. Soil water movement in relation to soil structure and input conditions. Ph.D. Thesis, University of Sheffield, 299 pp. Smettem, K.R.J., 1984. Soil water residence time and solute uptake. 3. Mass transfer under simulated winter rainfall conditions in undisturbed soil cores. J. Hydrol., 67: 235-248. Sposito, G., Jury, W. and Gupta, V.K., 1986a. Fundamental problems in the stochastic convection-dispersion model of solute transport in aquifers and field soils. Water Resour. Res., 22: 77-88. Sposito, G., White, R.E., Darrah, P.R. and Jury, W.A., 1986b. A transfer function model of solute transport through soil, 3. The convection-dispersion equation. Water Resour. Res., 22: 255-262. Starr, J.L., de Roo, H.C., Frink, C.R. and Parlange, J.-Y., 1978. Leaching characteristics of a layered field soil. Soil Sci. Soc. Am. J., 42: 306-391. Thomas, G.W. and Swoboda, A.R., 1970. Anion exclusion effects on chloride movement in soils. Soil Sci., 110: 163-167. Tyler, D.D. and Thomas, G.W., 1981. Chloride movement in undisturbed soil columns. Soil Sci. Soc. Am. J., 45: 459-461. Van de Pol, R.M., Weirenga, P.J. and Nielsen, D.R., 1977. Solute movement in a field soil. Soil Sci. Soc. Am. J., 41: 10-14. Van Ommen, H.C., van Genuchten, M.Th., van der Molen, W.H., Dijksma, R. and Hulshof, J., 1989. Experimental and theoretical analysis of solute transport from a diffuse source of pollution. J. Hydrol., 105: 225-251.

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Van Wesenbeeck, I.J. and Kachanoski, R.G., 1991. Spatial scale dependence of in situ solute transport. Soil Sci. Soc. Am. J., 55: 3-7. Wierenga, P.J. and van Genuchten, M.Th., 1989. Solute transport through small and large unsaturated soil columns. Ground Water, 27(1): 35--42. Yamaguchi, T., Moldrup, P. and Yokoski, S., 1989. Using breakthrough curves for parameter estimation in the convection--dispersion model of solute transport. Soil Sci. Soc. Am. J., 53: 1635-1641.