Soil Science Society of America Journal 64:499-504 (2000)
© 2000 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
Spatial Variability of Solute Leaching
Experimental Validation of a Quantitative Parameterization
Gerrit H. de Rooija and
Frank Stagnittib
a Wageningen Univ., Dep. of Environmental Sci., Sub-dep. Water Resour., Nieuwe Kanaal 11, 6709 PA Wageningen, The Netherlands
b Deakin Univ., School of Ecology and Environ., Center for Applied Dynamical Systems and Environmental Modelling, P.O. Box 423, Warrnambool Victoria, Australia 3280
ger.derooij{at}users.whh.wau.nl
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ABSTRACT
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Soil heterogeneity, soil structure, and fingered flow affect solute leaching from the vadose zone to the ground water. Recently, the spatial variation of cumulative solute fluxes at a given depth was characterized by fitting the two-parameter beta distribution to sorted amounts of solute leaching at different sampling points. We tested this parameterization on data from a chloride tracer experiment performed on a monolith lysimeter, below which drainage was collected from 300 compartments with a combined area of 0.75 m2. The effect of total sampling area, sample size, and the number of samples and their spatial distribution (random locations vs. clustered) on the fitted parameters was examined. Sixteen or more sampling locations of 25 cm2 each (5% of the total area) resulted in adequate representation of parameter values. Increasing the sample size underestimated the degree of heterogeneity. We therefore recommend that the fitted parameter values of the beta distribution be reported together with the sample size. In solute-transport experiments, collecting many small samples will give more accurate results than taking fewer but larger samples.
Abbreviations: cdf, cumulative distribution function • EC, electrical conductivity • HI, heterogeneity index
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INTRODUCTION
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SOLUTE MOVEMENT IN SOILS is strongly affected by irregular flows in structured soils (Flury et al., 1994; Quisenberry et al., 1994; Shuh et al., 1997; Stagnitti et al., 1998) and preferential flow in soils with unstable wetting fronts (Starr et al., 1978, 1986; Ghodrati and Jury, 1992; Ritsema et al., 1993). Heterogeneity of soil properties further limits our capability to predict solute leaching (Roth et al., 1991; Van Wesenbeeck and Kachanoski, 1991; Jury and Flühler, 1992; Snow et al., 1994). Several approaches have been developed to model the effect of macroporosity (van Genuchten and Wierenga, 1976; Gerke and van Genuchten, 1993; Steenhuis et al., 1994a) and fingered flow (Steenhuis et al., 1994b; de Rooij, 1995; de Rooij and de Vries, 1996). Soil heterogeneity has spawned numerous modeling approaches (e.g., Jury et al., 1987; Bresler and Dagan, 1983; Mantoglou and Gelhar, 1989; Jury and Roth, 1990; Simmons et al., 1995; de Rooij and de Vries, 1999).
In contrast, the quantitative characterization of the effects of these mechanisms on the variability of solute fluxes has received little attention. While the temporal variation of solute leaching is captured by the breakthrough curve and can be assessed through transfer functions based on the solute travel-time distribution (Jury and Roth, 1990), there is no equivalent concept for the variability in space of solute leaching. Stagnitti et al. (1999) recently advanced a relatively simple approach that might be viewed as a spatial transfer function, since it captures the spatial (instead of temporal) redistribution of uniformly applied solutes. Stagnitti et al. (1999) fitted the two-parameter beta distribution to the cumulative solute leaching in a horizontal plane. They combined the fitted parameter values into a heterogeneity index (HI). With this HI, one can easily compare and evaluate the nature and degree of solute leaching heterogeneity in various soils under different conditions, provided that cumulative solute leaching data are available at a fairly large number of locations having the same sampling area. A complicating factor is the poorly known effect of sample size on the heterogeneity within the sample population (Parker and Albrecht, 1987; Ellsworth and Boast, 1996), which hampers a direct comparison between results of different experimental setups.
The first objective of this paper is to investigate the dependence of the HI on the measurement setup. The variability of the HI within a 0.75-m2 area is investigated, as well as the effect of sample size. The response of the HI to the total number of samples is also determined. The second objective is to explore the use of the descriptive qualities of the beta distribution function to characterize the degree of convergence and divergence of the flow lines between the soil surface and the sampling depth. This information quantifies the nature and severity of flow heterogeneity and preferential flow. The paper uses data from a leaching experiment in a free-draining monolith lysimeter, under which drainage was monitored at 360 locations.
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Materials and methods
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We obtained an undisturbed core (1.13-m diam., 0.55 m high) from the coastal dune area in the southwestern part of the Netherlands. The core consisted of a sandy mesic Typic Psammaquent (Soil Survey Staff, 1988) and was obtained by pressing a large galvanized-steel cylinder into the soil (Belford, 1979; de Rooij, 1996). The soil had a 0.2- to 0.4-m water-repellent top layer in which fingers developed under dry conditions (e.g., Ritsema et al., 1993). To facilitate spatial drainage sampling, the core was placed on a 0.35-m high grid consisting of 50- by 50-mm galvanized-steel compartments. The compartments near the lysimeter wall were shaped differently to accommodate the shape of the soil core. Their size ranged from 25.5 to 53.9 cm2. The grid compartments were backfilled with sand taken from the appropriate depth at the core sampling site. The maximum observed pressure head at 5 cm above the top of the grid was less (i.e., more negative) than -25 cm, while evidence from a multistep outflow experiment showed that the soil's air-entry value was larger than -16 cm. The grid therefore prevented lateral redistribution of water and solutes in the capillary fringe by completely containing it. At the bottom of each of the 360 drainage compartments was a 50-mm long drainage tube with a glass fiber capillary wick (Boll et al., 1992). Full details of the experimental equipment and procedures are described by de Rooij (1996).
A rainfall simulator (consisting of a shallow, closed water chamber with 0.55- by 25-mm hypodermic needles mounted in the bottom at 50-mm rectangular spacing) provided a uniform water input (coefficient of variation, 5–10%). The simulator moved independently in the main horizontal directions with an amplitude of 50 mm to irregularize the drop impact pattern. To avoid bypass flow along the lsyimeter wall, the small gap between the soil core and the lysimeter wall was backfilled with water-repellent sand. Furthermore, we mounted a 35-mm wide gutter on the inside of the lysimeter wall, at 
3 cm above the soil surface. This gutter collected and removed all water flowing downward along the cylinder wall. Ponding did not occur during the experiment. After a prewetting period, 20-mm artificial showers with a rainfall rate of 246 mm d-1 were applied every Monday, Wednesday, and Friday for 114 d. After 57 d, 6.4 mm of a 1.0 M CaCl2 solution was uniformly applied to the soil surface, and the next shower reduced to 13.6 mm to yield a total water application of 20 mm. Drainage was collected in flasks placed below the drainage outlets. Before every shower, the amount of drainage collected since the start of the previous shower and its electrical conductivity (EC) were measured for each drainage compartment. At the time of the chloride application, the temporal variability of the background EC was negligible, and chloride concentrations could be determined by subtracting the background EC of every drainage compartment from the actual EC (a linear calibration relationship between EC and chloride concentration was available). At the completion of the experiment the amount of chloride still leaching was negligible.
Here, we used the cumulative amount of chloride that leached from the individual drainage compartments to fit parameter values for the beta distribution. To avoid geometry effects, we ignored the drainage compartments that are near the lysimeter wall because these have varying shapes and sizes. The remaining 300 compartments have a combined area of 0.75 m2 and span 0.90 m in both horizontal directions.
We described the distribution of cumulative leaching through these compartments by the beta distribution (Abramowitz and Stegun, 1964, p. 930):
 | (1) |
where P denotes a probability density, B is the beta-function (Abramowitz and Stegun, 1964, p. 258), and 
and 
are parameters. The variable z is bounded on [0,1]. If
reduces to the uniform distribution function. Integrating Eq. [1] over z gives the cumulative distribution function (cdf), which is the incomplete beta function (Abramowitz and Stegun, 1964, p. 944). The cdf was fitted to the data through a nonlinear curve-fitting procedure (see Stagnitti et al., 1999, for details).
The parameters and are obtained by fitting the cdf to a plot of the fraction of total solute leached with the fraction of the total sampling area (samples sorted in descending order of amount leached). We quantified the variability of solute leaching by the HI, a scaled form of the coefficient of variation (i.e., a scaled ratio of the standard deviation of the cumulative solute leaching over the mean, both of which are simple expressions in and ). The HI equals 1 if
, and is defined as
 | (2) |
For many solute-transport processes, lateral solute spreading is much smaller than longitudinal solute spreading (Flühler et al., 1996). For experiments involving a uniform, non-ponding solute application at the soil surface, P(z) then gives an approximation of the ratio of the area of infiltration (Ai, m2) over the area of outflow (Ao, m2) at the sampling depth:
 | (3) |
For
, the flow lines on average run parallel. If P(z) > 1, the flow lines converge (indicating a preferential flow path), and P(z) < 1 defines areas with diverging flow. For any given P(z), the value of P gives the degree of convergence or divergence of the flow lines, while z is the fraction of the sampling area where the flow is more convergent or less divergent than P. Solutes in these areas presumably travel faster than those in the streamtube corresponding to P(z). The integral of P between 0 and z (the value of the cdf at z) is the fraction of the flow that passes through these areas with more convergent or less divergent flow. For uniformly applied solutes, this translates into the fraction of the soil surface that supplies water to streamtubes with faster flow than that corresponding to P. Equation [1] thus describes the geometry of the flow pattern between the soil surface and the sampling plane. Figure 1
summarizes the information contained in a fitted cdf.
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Fig. 1 The cumulative distribution function (cdf) given by the integral of the beta distribution function P defined in Eq. [1]. When the cdf is fitted to data of fractional solute leaching (sorted in descending order) with fractional sampling area, the slope P(z) at any point on the curve (represented by the straight line) equals the ratio of the infiltration area (Ai) over the outflow area (Ao). Hence, P(z) quantifies the severity of preferential flow for one stream tube. The coordinates define the fractions of the sampling plane (z) and the soil surface (cumulative probability density) occupied by streamtubes with stronger preferential flow than that corresponding to P(z)
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To study the variability of the HI within the soil, we divided the lysimeter bottom into quadrants (75 compartments each) and subsequently into octants (37 or 38 compartments each), and calculated the HI for each quadrant and octant separately, as well as for the lysimeter as a whole. To reveal the effect of sample size on P(z) and the HI, we formed sets of two (1 x 2), four (2 x 2), nine (3 x 3), and 16 (4 x 4) compartments. The amounts of cumulative leaching of the compartments within a set were added and a set treated as one sample. All 300 compartments were combined in 150 sets of two. Seventy-three sets of four compartments could be formed, leaving eight drainage compartments that could not be included in any set and had to be excluded from the data set. With 32 sets of 9 compartments, 12 compartments were ignored; and with 15 sets of 16 compartments, 60 compartments were omitted. Finally, the effect of sampling density on the HI was studied by randomly selecting 2n compartments (with n = 0,..,8) from the total set of 300, and calculating the HI for each subset. The selection procedure was such that the selected subset for any value of n was included in that for the next higher value.
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Results and discussion
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The leaching pattern that provided the data set is shown in Fig. 2
. The pattern is affected by the microtopography of the soil surface in the lysimeter, with low areas producing high leaching. Note that leaching at the boundary is exaggerated because of the large size of the boundary compartments compared with the inner compartments.
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Fig. 2 Spatial distribution of chloride leaching from the lysimeter bottom. The center of the lysimeter is at
. Each grid point represents a measurement location. The flat areas in the corners are outside the lysimeter cylinder. Average leaching (28.1 mmol Cl-) is indicated by the bold contour line. Cumulative leaching amounts in excess of the average have contour lines at multiples of 25 mmol Cl-
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Despite the erratic distribution of solute leaching, the curve of fractional leaching vs. fractional area (drainage compartments sorted in decreasing order of amounts of leaching) is very smooth (Fig. 3a)
. The shape of this curve varies only mildly among different regions (quadrants and octants) within the lysimeter. The curves of the quadrants and octants with the smallest and largest HI are shown in Fig. 3b and c. These figures show that different regions of the core exhibited similar leaching patterns. The relatively small coefficients of variation in Table 1
for the HI, , and confirm this. Not surprisingly, the range of fitted parameter values is the largest for the smallest lysimeter sections (Table 1).
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Table 1 Fitted parameter values of the beta distribution (Eq. [1]) for different sections of the lysimeter. The resulting values of the heterogeneity index (HI; Eq. [2]) are also given
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Figure 4
shows the dependence of the HI, , and on the size of the individual sampling areas. As indicated previously, various numbers of drainage compartments had to be excluded from the data set for the larger sample sizes, which may explain the slightly erratic behavior of the parameters for large sample sizes. For smaller sample sizes, clearly decreases with decreasing sample size, while and the HI increase. The behavior of the HI can be described by a power relationship:
 | (4) |
where A is the size of the individual sampling areas (m2). The coefficients were obtained by linear regression of ln(HI) vs. ln(A)
. Note that this results in an infinitely large HI for zero sampling size. This result is not surprising. One would certainly expect that the averaging process that results from summing individual compartments into larger aggregates would lead to a lowering in the intrinsic (smaller-scale) heterogeneity. Further, at a microscopic scale, every conceivable flow path is possible and of course a higher heterogeneity is expected. Also of interest, the HI appears to become invariant to changes in sample size for sample sizes >100 cm2. This result suggests that in solute-leaching studies, an intrinsic maximum sample area may exist, above which small-scale variations will be ignored. This is an important consideration in contaminant-transport studies involving very highly toxic materials where very small concentrations or amounts can pollute ground water. In such cases, determining the dependence of the HI on the sample size may provide valuable data in designing a suitable field-monitoring and sampling strategy.
Figure 5
demonstrates the improvement of the fit for increasing sampling population. For
(16 samples;
) and
(32 samples;
), the curves are very similar to that for the entire data set (300 samples;
). This finding is corroborated by Fig. 6
, which shows that the HI, , and are virtually independent of the number of samples if more than 16 samples are used. These results support the use of the multisegment percolation systems in vadose zone experiments advocated by Stagnitti et al. (1999) and many others. Our data suggest that the 25 sampling areas used by Stagnitti et al. (1999) were at a safe margin from the smallest allowable number. Again, the HI proves useful in providing a quantification of the sample number required to achieve adequate representation in solute-leaching studies.
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Fig. 5 Observations and fitted curves for 2n sampling locations randomly selected from a total of 300. The curves for n  5 (including the observed and fitted curves for all 300 samples) are nearly identical
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The value of the HI for 32 samples (1.3133) is in the upper range of that of the octants (1.2067–1.3398), which have a comparable number of sampling locations (37–38). Apparently, the larger spacing between samples for the randomly selected 32 locations, as compared with the adjacent sampling within the octants, increased the HI only slightly, and adjacent sampling gave correct results on the plot scale for this soil. If the heterogeneity within larger areas is of interest, it is probably better to increase the spacing between samples to achieve a better representation of field conditions. A nested setup, as suggested by Stagnitti et al. (1999), for example, by a number of randomly located multisegment percolation systems (consisting of no fewer than 16 sample points), can provide information on the heterogeneity at different scales.
Figure 7a
shows the fitted cdf's for varying sample sizes, as well as hypothetical curves for uniform soils with stable flow or flow occurring only in fingers. The curves for the data (25-cm2 sample size) and the fitted curve for the corresponding sample size are nearly identical. The remaining fitted curves more closely resemble the curve for uniform flow when the sample size increases, consistent with the HI values shown in Fig. 4.
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Fig. 7 Fitted curves of the cumulative beta distribution function (a), and the beta distribution function P (see Eq. [1]) (b), for different sample sizes (cm2). The curves for flow in a uniform soil with 25% of the soil occupied by fingers and a uniform soil without fingering are included for comparison
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To analyze this trend toward uniformity more closely, we present the derivatives of the curves of Fig. 7a in Fig. 7b. These curves are the fitted distribution functions. Here too, the cases of uniform flow and uniform fingers are included. In Fig. 7b, the extreme values of P represent small and large amounts of cumulative solute leaching (indicative of areas with strongly diverging and converging flow). The figure clearly shows the decrease of the area occupied by strongly convergent flows (preferential-flow paths characterized by high values of P) with increasing sample size. Thus, excessively large sampling areas make it impossible to recognize the presence of small areas with strongly preferential flow that can rapidly contaminate ground water with surface-applied substances. Table 2
illustrates this for three levels of P. Doubling the sample size nearly halves the estimate of the area that transports more than twice the amount of solutes it would have transported in a uniform soil (P 2). Increasing the sampling size by an order of magnitude reduces the estimated area with P 2 by an order of magnitude.
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Table 2 The effect of sample size on the estimated degree of preferential flow, expressed as the ratio of area of inflow over area of outflow (P)
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It must be noted that the above is based on a data set with a non-zero (25-cm2) sample size. This data set itself, as any other, inevitably underestimates the degree of flow heterogeneity. Our results suggest that a sampling strategy in which a relatively large number of small samples are collected gives more reliable results than collecting fewer but larger samples. Results obtained from large samples may even be misleading and significantly underestimate the risk to ground water reserves by small concentrations of highly toxic contaminants.
In conclusion, the beta distribution has proven to be a valuable descriptive tool that provides information about the distribution of areas with converging and diverging flow in a soil. The HI, which is easily derived from the beta distribution function, can concisely and effectively characterize the spatial variation of solute leaching in undisturbed soil. The index could adequately describe the variability within a 0.75-m2 area using 16 samples of 25 cm2 each, together representing 5% of the total area. The characteristics of the variability varied only moderately for the 0.75-m2 area. The value of the HI depends on the size of the individual sampling areas, especially if these are smaller than 100 cm2. We therefore recommend to always report the HI values with the size of the samples used. The alternative, to resort to larger sample sizes to eliminate the size dependency, is unappealing because large samples result in a gross underestimation of the degree of preferential flow. In many cases they are not suited to assess the risk of rapid solute leaching. It appears advisable to keep the sampling areas of the individual drainage collection cells as small as possible: many small samples give more reliable results than a few large ones.
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ACKNOWLEDGMENTS
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The work of G.H. de Rooij has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. F. Stagnitti acknowledges support from the Australian Research Council's Large Grant Scheme (Grant No. A89701825), the Department of Industry, Science & Tourism Major Collaborative Grant Scheme, and a fellowship from the OECD Cooperative Research Programme: Biological Resource Management for Sustainable Agricultural Systems. The authors would like to thank three anonymous reviewers for their careful criticisms, which have undoubtedly improved the final version of this manuscript.
Received for publication February 23, 1999.
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TOP
ABSTRACT
INTRODUCTION
