Published online 3 October 2006
Published in Vadose Zone J 5:1086-1092 (2006)
DOI: 10.2136/vzj2005.0101
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Quantifying Minimum Monolith Size and Solute Dilution from Multi-Compartment Percolation Sampler Data
Gerrit H. de Rooija,*,
Olaf A. Cirpkab,
Frank Stagnittic,
Saskia H. Vuurensa and
Jan Bolld
a Wageningen Univ., Dep. of Environmental Science, Sub-Dep. Water Resources, Soil Physics, Ecohydrology, and Groundwater Management Group, Nieuwe Kanaal 11, 6709 PA The Netherlands
b Swiss Federal Inst. of Aquatic Science & Technology (Eawag), Dep. of Water Resources and Drinking Water, Überlandstr. 133, 8600 Dübendorf, Switzerland
c School of Ecology & Environment, Office J228 Warrnambool Campus, P.O. Box 423, Warrnambool, Victoria 3280, Australia
d Dep. of Biological & Agricultural Engineering, Univ. of Idaho, P.O. Box 442060, Moscow ID 83844-2060
* Corresponding author (ger.derooij{at}wur.nl)
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ABSTRACT
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Preferential flow affects solute transport in natural soils, leading to high spatiotemporal variation of concentration. A multicompartment solute sampler (MCS), yielding multiple breakthrough curves at a given depth, can monitor tracer movement in a heterogeneous soil. We present a technique to estimate from MCS data whether a soil monolith is sufficiently large to capture preferential flow, which is a necessity for tracer breakthrough curves to be representative. For several soils, we estimate that an MCS should be larger than 0.1 to 0.2 m2. We also expand dilution theory to analyze the concentration variations of a tracer passing the control plane monitored by the MCS, in addition to the conventional plume spreading analysis. We characterize the set of locally observed breakthrough curves by the entropy-based dilution index. For given first and second-central moment, the spatially uniform log-normal breakthrough curve maximizes the dilution index. The ratio between observed and maximum dilution index is denoted reactor ratio. For a 300-compartment solute sampler, covering an area of 0.75 m2, we compute a reactor ratio of 0.665, compared with 0.04 for stochastic-convective and 1 for convective-dispersive transport. With a single, large collector the reactor ratio would be 0.958, severely underestimating concentration variations. Large collector areas are clearly inadequate to estimate dilution. Values of the dilution index and the reactor ratio for individual sampling compartments indicate efficient longitudinal mixing in most but not all cases, and considerable spatial variation of the leaching process.
Abbreviations: BTC, breakthrough curve • MCS, multicompartment sampler • SSDC, spatial solute distribution curve
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INTRODUCTION
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QUANTITATIVE ASSESSMENT of solute transport in soils requires measuring the spatial and temporal distributions of water and solute fluxes. Multicompartment samplers are a promising technique developed for this purpose. Here, the vertical fluxes in a control plane are sampled by multiple collectors which may be buried underneath undisturbed soil in the field (e.g., Boll et al., 1997) or installed at the outflow of large undisturbed soil monoliths, collected from field sites and brought into the laboratory (Poletika and Jury, 1994; Quisenberry et al., 1994; de Rooij, 1996; Stagnitti et al., 1998; Strock et al., 2001). A key feature of an MCS is its ability to observe fluxes at two adjacent scales: that on the scale of an individual sampling compartment (about 10–3– 10–2 m2) and that over a larger scale of the entire MCS (about 10–1– 100 m2). These scales envelop a considerable portion of soil variability within a field (about 104 m2) (van Es, 2002), which explains their increasing deployment in solute transport studies.
Multicompartment samplers are challenging to operate, and still only sample a limited section of the field. Various authors have therefore opted to take a collection of standard monoliths at various locations in a field, and subject them to identical initial and boundary conditions in the laboratory (e.g., Sassner et al., 1994; Mallants et al., 1994, 1996; Lennartz and Kamra, 1998). The breakthrough curve (BTC) on the field scale is assumed to be composed of the BTCs of all individual monoliths. This essentially is the experimental equivalent to the parallel-column model of Dagan and Bresler (1979), who viewed a field as an ensemble of non-interacting columns.
There is a considerable risk in experimenting with soil monoliths, whether equipped with a multicompartment collector or a single outlet. The column radius of the monolith must be large enough so that preferential flow paths can drain microcatchments of their natural size. In too narrow columns, the flow field becomes unnaturally uniform, resulting in biased breakthrough curves. At the same time, averaging concentrations over large cross-sectional areas reduces the observed peak value. The spread of the BTC reflects both the variability of mean arrival time within the observation area and the spread of BTCs that would be observed at single points (e.g., Cirpka and Kitanidis, 2002). To capture peak concentrations, small measurement areas are needed. In contrast to a set of multiple standard soil columns, multi-compartment samplers allow both a large cross-sectional area of the monoliths, facilitating naturally fluctuating flow, and small sampling areas. Because of high spatial and temporal resolution, MCS data allow quantifying both solute dispersion (characterizing the non-uniformity of solute flux over the entire cross-section of the monolith) and dilution (characterizing the local solute concentrations within the plume; Kitanidis, 1994). Especially for concentration-dependent processes (e.g., nonlinear sorption, microbiological processes) dilution is a more relevant characteristic of solute distribution than dispersion (e.g., Janssen et al., 2006). For unsaturated flow, dilution has only been quantified in the laboratory, in a two-dimensional artificial porous medium with well-defined heterogeneity (Ursino et al., 2001).
The number of analytical tools to process MCS data is expanding. Stagnitti et al. (1999) and de Rooij and Stagnitti (2000, 2004) developed the spatial solute distribution curve (SSDC) to characterize the spatial redistribution across a reference plane of uniformly applied solutes and quantify the severity of preferential flow. They applied the beta distribution to parameterize the SSDC. De Rooij and Stagnitti (2002) proposed the leaching surface to unify the BTC and the SSDC in a single description of the spatiotemporal redistribution of uniformly applied solutes.
While the leaching surface provides information about the solute flux density at any given time and location in a reference plane, it does not give the distribution of local flux concentrations. Thus, it cannot be used to quantify dilution. For the latter purpose, Kitanidis (1994) introduced the concept of the dilution index, which was interpreted by the latter author as the effective volume occupied by the solute cloud. The original definition of the dilution index requires local resident concentrations over the entire extent of the solute plume at a given time. This information cannot be obtained by an MCS. Consequently, the analysis by Ursino et al. (2001) cannot be applied directly to MCS data.
Several attempts have been made to characterize dilution by other quantities. Kapoor and Kitanidis (1998) related the total concentration variance in the domain to the lack of dilution. This quantity, however, also relied on information on the resident concentration throughout the domain. Cirpka and Kitanidis (2002) took the width of concentration BTCs, obtained at single points, as a measure of dilution, whereas the width of BTCs, obtained after averaging over a large control plane, was seen as a measure of combined dilution and spreading. The advantages of the latter approach are that it relies on measurable quantities, and that predictions can be made by linear stochastic theory. However, it lacks the rigor of the dilution index, which is derived from the entropy of the concentration distribution.
For the dilution of solutes undergoing vertical solute transport in soils, we may distinguish two extremes. One is stochastic-convective transport (e.g., Simmons, 1982), which neglects diffusive processes altogether so that the degree of dilution does not increase with travel time or distance. The other one is one-dimensional convective-dispersive transport, which presumes that lateral mixing is so effective that lateral concentration fluctuations, caused by velocity variations, are readily smeared out (Kitanidis, 1994). For given first and second central travel-time moments, this is the regime with highest dilution. Solute transport in real systems is somewhere between those two extremes. Lateral mixing is effective, but incomplete.
This paper aims to increase the amount of information that can be extracted from an MCS experiment. To avoid the pitfall of insufficiently large soil monoliths, we provide a method to validate whether the sampling area of an MCS is adequate to guarantee representative flow behavior. The method is tested on field and laboratory data. Furthermore, we extend the concept of the dilution index (Kitanidis, 1994) to the spatiotemporal distribution of flux concentrations in a reference plane to make it applicable to MCS data. This method is applied to the laboratory data only. We demonstrate the importance of solute sampling at nested scales (the scale of the individual compartment and that of the MCS) by comparing the results with those that would have been obtained when only a single large sampler had been used.
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THEORY
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Minimal Column Radius
In natural soils, which are characterized by spatial non-uniformity and preferential flow, the probability density function of solute travel time will be skewed: preferential flow paths carrying a large fraction of the water produce an early peak while the low flow regions between the preferential flow paths produce an elongated tail of large travel times. In extreme cases (macropores), the distribution may be multimodal. Using undersized columns in solute transport studies reduces the peak height (reporting less water through preferential flow paths) and shifts the modal value pertaining to the preferential flow paths toward larger travel times (receiving smaller amounts of water generally reduces the flow velocity). The modal value of the low flow regions, if present, will shift toward smaller travel times (because of increased flow rate), and more probability mass will be attributed to the low flows. Overall, the range of the probability density function will reduce, and its symmetry will increase.
Column experiments provide reliable information about solute leaching under field conditions only if the column radius is sufficient to avoid forced one-dimensional flow. The SSDC can help determine the minimum radius required. Stagnitti et al. (1999) obtained the SSDC by sorting the drainage sampling compartments of a MCS from high to low amounts of cumulative leaching and plotted the fraction of total leaching as a function of the fraction of the sampling area z (0 
z 1). The beta distribution often gives a good fit to the SSDC (Stagnitti et al., 1999; de Rooij and Stagnitti, 2000, 2004). Denoting the cumulative beta distribution (Abramowitz and Stegun, 1964, Eq. [26.5.1]; Gupta and Nadarajah, 2004) by P(z) (0 P(z) 1), the combined catchment area of the strongest leaching fraction z of the sampling area is AP(z), where A (L2) is the combined area of all sampling compartments (de Rooij and Stagnitti, 2000). In the worst case, the fraction z is clustered into neighboring compartments (one preferential flow path), and its water stems from a single microcatchment within the soil column: The inflow of an area AP(z) leaves the soil volume at the observation depth through an area Az (Fig. 1). Assuming that both the catchment area and the outflow area of the preferential flow tube are approximately circular, the difference between the radii yields an estimate of the maximum lateral flow distance in the soil portion from which the MCS received the drainage. This is based on the assumption that the macroscopic flow direction is predominantly vertical, which is realistic for most soils but may be incorrect if soil layers are consistently inclined in the same direction. Dye tracers can reveal this (e.g., Kung, 1990a,1990b).
The difference between the microcatchment radius and the outflow area radius is:
 | [1] |
where Rs (L) is the microcatchment radius, and Ro (L) the radius of the corresponding outflow area at the observation depth. We determine the value z which maximizes Rs – Ro by setting the derivative of Eq. [1] with respect to z to zero:
 | [2] |
where p(z) = dP(z)/dz is the beta probability density function (Abramowitz and Stegun, 1964, Eq. [26.1.33]; Gupta and Nadarajah, 2004). The catchment size responsible for maximum lateral flow can thus be found by iteratively determining the value of z for which the following equality holds:
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If zc denotes the root of Eq. [3], the corresponding microcatchment radius Rc (L) is:
 | [4] |
The monolith radius should be significantly larger than Rc. If Rc is estimated from MCS data, and the calculated Rc is close to the actual (equivalent) radius of the MCS, the sampler possibly is too small to capture the flow heterogeneity caused by the largest flow paths. This can be verified in part by plotting the spatial distribution of leaching: Patterns with length scales 
the sampler size should result in an apparent spatial trend in the leaching data (For instance, if an MCS is installed in a location where it intersects only half of a preferential flow path of roughly the same size as the sampler, the sampling compartments on the side within the preferential flow path will consistently yield more drainage than the compartments on the other side). If the collective catchment area of the stream tubes delivering solutes to an MCS buried in a field differs strongly from the sampling area of the MCS, the instrument is too small to capture the scale of the flow patterns. This can be verified by the total amount of solute captured by the MCS during the experiment. Hence, spatial trends in drainage and a large difference between the sampler area and its catchment area preclude reliable estimates of Rc.
In large samplers, the assumption of a single preferential flow path is unrealistic, and Rc can easily be overestimated. This can be verified by dividing the sampling area into smaller subsections and fitting P(z) for the individual subsections. The smallest subsection size for which the fitted P(z) do not substantially differ from those for larger subsections gives a reliable minimal size for the MCS. Values for Rc calculated from the beta-distributions of these subsections should be considered representative for that soil.
Dilution across a Plane
Kitanidis (1994) analyzed the dilution of spatial concentration distributions by their entropy. The analysis required exact knowledge of the spatial distribution of resident concentration at a fixed time, which is difficult to obtain. In our analysis, we consider the solute flux concentration of percolating water crossing a control plane instrumented with a MCS. From the multiple BTCs, we analyze the probability density p(x1, x2, t) (L–2 T–1) that a solute particle passes the control plane at a given horizontal location (x1, x2) and time t:
 | [5] |
in which cf denotes the flux concentration (ML–3), q3 is the vertical specific-discharge component (L T–1), and mtot is the total mass passing the MCS (M):
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The dilution index E (L2 T) is the exponential of the distribution's entropy. For continuous concentration distributions of solutes passing a control plane it is (compare Kitanidis, 1994):
 | [7] |
Our measured data are not continuous. Instead, we sample certain volumes of leached water in the MCS. Thus, the integrals are replaced by a weighted sum:
 | [8] |
where Aj (L2), 
tj (T), cj (M L–3), and Vj (L3) are the cross-sectional area, time increment, concentration, and sampled volume of sample j, respectively, and n is the total number of samples.
For E, we can determine a lower and an upper limit, Emin and Emax, respectively. The lower limit is the volume of the injected tracer solution divided by the mean infiltration rate 
3. A meaningful upper limit is given by the dilution index maximizing the entropy, given the mean and variance of the BTC in the total flux passing the control plane. In the spatial coordinates bounded on A, the uniform distribution is maxentropic, whereas in the semi-infinite space of time t, the log-normal distribution is maxentropic for given first and second-central moments in the log-space (e.g., Kapur, 1989). This results in the following maxentropic distribution pme(x1, x2, t):
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with the mean µ and variance 
2 of log-time. Note that the uniform distribution over the sampling area A results in pme being a function of t only. The dilution index Emax of this distribution is (compare Kapur, 1989, Table 3.8):
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Given the mean m and variance s2 of the BTC, measured over the entire cross-section of the sampler, the following identities hold:
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 | [11b] |
resulting in the maximum dilution index Emax of:
 | [12] |
The reactor ratio M is the ratio between the actual dilution index and the maximum value, given the moments of the distribution (Kitanidis, 1994):
 | [13] |
For one-dimensional convective-dispersive transport, the reactor ratio M is essentially unity, because the analytical solution of flux concentration undergoing one-dimensional convection and dispersion hardly differs from the log-normal distribution, which determines Emax (Kreft and Zuber, 1978). For stochastic-convective transport, the reactor ratio M is Emin/Emax, because the volume containing solute cannot increase if advection is the only transport mechanism considered. Thus, stochastic-convective transport is characterized by a constant E with increasing depth x3, while convective-dispersive transport has a uniform value of M (
1).
For stochastic-convective transport, the mean breakthrough time is m = x3/
in which = 
is the mean seepage velocity, and the variance of the transfer function s2 is s2 = (msrel)2 with the coefficient of variation srel which is assumed uniform and has to be determined experimentally (Jury and Roth, 1990, p. 35–41). From Eq. [12] follows:
 | [14] |
in which we have added the term Emin describing the dilution at the injection plane. Consequently, the theoretical reactor ratio MSC for stochastic-convective transport is:
 | [15] |
For convective-dispersive transport, the following identities hold: m = x3/ and s2 = 2Dx3/3, in which D is the dispersion coefficient (Jury and Roth, 1990, p. 35–41). A relation between the theoretical dilution index ECD and x3 for convective-dispersive transport again follows from Eq. [12], noting that M 1:
 | [16] |
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MATERIALS AND METHODS
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De Rooij (1996) placed a soil monolith (1.00 m2) on an MCS with 300 drainage-sampling compartments of 2.50 x 10–3 m2 each, surrounded by an outer ring of 60 compartments of different sizes. The average distance between the soil surface and the sampling depth was 0.85 m. With this set-up, a leaching experiment was performed (de Rooij and Stagnitti, 2000, 2002a), in which a uniformly applied 6.4 x 10–3 m pulse of CaCl2 at a concentration of 1 mol L–1 was leached by artificial rain showers of 2.00 x 10–2 m each (rainfall rate: 0.246 m d–1) on every Monday, Wednesday, and Friday. In this study, we only use the data from the inner compartments to eliminate boundary effects (A = 0.750 m2).
The second data set is by Boll et al. (1997) who buried MCSs with 25 compartments of 3.60 x 10–3 m2 in various eastern U.S. soils. They kept the soil above the samplers undisturbed, and monitored spatial drainage variation under natural or artificial rainfall. They applied tracer pulses, but ceased sampling before complete breakthrough. We used the cumulative leaching (de Rooij, 1996) or drainage data (Boll et al., 1997) of these experiments to determine the SSDC or equivalent spatial drainage distribution curves. After fitting the beta distribution to these curves (Stagnitti et al., 1999; de Rooij and Stagnitti, 2000) we first determined the root of Eq. [3] and then Rc from Eq. [4]. In case of de Rooij's (1996) data, we also determined Rc for quadrants and octants of the sampler.
We processed the volumes and concentrations of the 8700 samples of the inner compartments of de Rooij's (1996) chloride leaching experiment to obtain the dilution index according to Eq. [8]. In contrast to the experiments of Boll et al. (1997), the breakthrough was complete in the data of de Rooij (1996), which means that the measured concentrations were below the detection limit at the end of the experiment. The latter is a prerequisite to compute the dilution index.
As minimum value Emin of the dilution index, needed for the theoretical considerations (see Eq. [14] and [15]), we took the time needed to infiltrate the solute pulse at the mean infiltration rate, times the cross-sectional area.
To determine the effect of spatial resolution of solute sampling, we computed the mean drainage rate and the flux-weighted concentrations at the lysimeter scale. This breakthrough curve corresponds to a measurement with a single-compartment sampler. Here too, we calculated E, as well as m and s for the lysimeter-scale BTC. These were used to compute Emax and M. In addition, we compared the range of solute flux concentrations observed in all 300 compartments with the range resulting from the lysimeter-scale data.
For each sampling compartment of de Rooij's (1996) lysimeter, we calculated scaled concentrations from its individual BTC, as well as m and s. From this, we determined E, Emax, and M for each compartment.
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RESULTS AND DISCUSSION
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The data of de Rooij and Stagnitti (2000) gave Rc = 0.28 m (their monolith radius was 0.565 m). The Rc values for the quadrants hardly varied (average 0.15 m, 8 x 10–5 m difference between largest and smallest value), while for octants the difference was clearly larger: 2.9 x 10–4 m (average Rc = 0.10 m). For the octants, the catchment area corresponding to the calculated Rc expressed as a percentage of the total sampling area also became more variable. We therefore conclude that the quadrant value of Rc = 0.15 m is a reliable value for this soil, and can be obtained from a sampled area of 0.19 m2.
The cracked soils of Boll et al. (1997) (clay and clay loam) had average Rc values of 0.12 (clay–Willsboro) and 0.13 m (clay loam–Ithaca), comparable to that of the structureless sand of Georgetown (0.12 m), but larger than that of the silt loam in Freeville (0.09 m) (locations refer to Boll et al., 1997). The individual values for Rc range from 51 to 78% of the equivalent sampler radius of 0.17 m; the sampler (0.09 m2 sampling area) should not have been smaller, and preferably somewhat larger.
During de Rooij's (1996) chloride-leaching experiment, 0.305 m3 of water drained from the 300 inner compartments (A = 0.75 m2), leaching 3.9 mol CaCl2. The vast bulk of this (99.5%, corresponding to 3.79 L, or 5.05 x 10–3 m, of tracer solution) left the lysimeter in a period during which 0.219 m3 of drainage was collected from the inner compartments, between 11 and 53 d after the chloride application. Since the lysimeter size was adequate, the dilution index could be calculated with confidence. In view of the size of the MCS, and the fact that not all of Boll et al.'s (1997) experiments yielded a complete BTC, we calculated dilution for de Rooij's (1996) experiment only.
The mean breakthrough time m of solute leaching from the entire sampling area was 22.96 d, with a standard deviation s of 7.13 d. Table 1 lists the minimum dilution index Emin, that is, the time of tracer injection, the maximum dilution index Emax, calculated from Eq. [12] using the values of m and s2 given above, as well as the actual dilution index E according to Eq. [8], which was calculated from compartment-scale and lysimeter-scale data. The dilution indices are scaled by the total area of all compartments considered.
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Table 1. Extreme values for the dilution index E/A, scaled by the area of the considered compartments, and the reactor ratio M, as well as their values for the data from 300 drainage compartments below de Rooij's (1996) lysimeter, and for lysimeter-scale data (all compartments combined).
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The minimum dilution index Emin and reactor ratio Mmin (Table 1) are for stochastic-convective transport, the maximum, Emax and Mmax, are for convective-dispersive transport. As Kitanidis (1994) pointed out, large values of M (close to 1) signify a high degree of mixing: it indicates that the solute, originally present in a limited volume, is distributed over a considerable portion of the total water volume that envelops the plume.
The observed value of M of 0.596 (Table 1) is in between the convective-dispersive value (1; Table 1) and the stochastic-convective value (0.04; Table 1). However, its value should be considered an overestimation: The non-zero sample size causes unavoidable additional dilution in the sampling procedure. For the same reason, the theoretical stochastic-convective value of M in Table 1 principally eludes experimental verification. Given the uncertainty about the practically observable value of M for a stochastic-convective system being sampled with samples of non-zero size, a clear dominance of either convective-dispersive or stochastic-convective transport cannot be inferred from the intermediate observed value of M. Nevertheless, the data indicate that a solute transport experiment with leaching monitored only at the m2 scale and not at the 10–3 m2 scale will clearly not allow an accurate estimate of dilution, and therefore will misrepresent the true distribution of solute concentrations.
Figure 2 gives theoretical curves for E and M as a function of sampling depth for stochastic-convective and convective-dispersive transport. The curves according to Eq. [14] and [16] were constrained on the values of M (Table 1; minimum value) and E (Table 1; maximum value) at the sampling depth of 0.85 m. We could determine the mean seepage velocity = 0.037 m d–1, which corresponds to a volumetric water content 
of 16.9%. The dispersion coefficient, needed for the convective-dispersive model, is 1.76 x 10–8 m s–1, which corresponds to a dispersivity of 0.041 m. The coefficient of variation srel, needed for the stochastic-convective model, is 0.31. All of these quantities were determined from the temporal moments of the breakthrough curve at the lysimeter scale.
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Fig. 2. Theoretical dependence of the area-normalized dilution index E/A (dashed lines) and reactor ratio M (solid lines) on the sampling depth for stochastic-convective transport (black) and convective-dispersive transport (red). The values for the observed plume at the bottom of the lysimeter are shown by markers.
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A convective-dispersive transport regime requires strong lateral mixing, resulting in small lateral concentration gradients. By lumping the compartment data of each sampling round into a lysimeter-scale drainage rate and flux concentration, we effectively erased any lateral variation. The high reactor ratio of 0.945 for the lysimeter-scale data (against the benchmark of a lognormal distribution of solute with the same first and second moments as the lysimeter-scale data) reflects this.
We further investigated the artificial dilution caused by non-zero sample sizes by identifying the peak concentrations for the compartment-scale and the lysimeter-scale data. The peak concentration at the lysimeter scale was 0.337 times the maximum observed in the 8700 drainage samples. The lysimeter-scale concentration peak represented a water volume of 9.6 L, while the drainage sample with the highest concentration contained 0.0850 L. A more complete understanding of the effect of ignoring small-scale variations is offered by Fig. 3, which shows the distribution of observed concentrations in the soil samples and in the lysimeter-scale data. Lateral averaging made the concentration distribution more uniform over a much narrower range: the very low concentrations but especially the high concentrations vanish by averaging. This corroborates the need for multiple small sample collectors if solute concentrations rather than just solute spreading are deemed important.
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Fig. 3. The distribution of chloride concentrations in the drainage samples collected from the 300 inner compartments of de Rooij's (1996) lysimeter, and the distribution that resulted after combining all samples of each sampling round (entire lysimeter). Sample volumes were converted to millimeter drainage over the entire sampling area to facilitate interpretation. Please note the logarithmic scale on the vertical axis.
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Table 2 gives key values of the distributions of E/A, Emax/A, and M derived from the BTCs of individual sampling compartments. The means and standard deviations are weighted by the water flux collected by the compartments. Due to normalization by the collection area, the values are comparable to the lysimeter-scale parameters. Figure 4 shows the cumulative distribution function of M determined on the compartment scale. The BTCs of individual sampling compartments tend to be fairly close to lognormal (average M of about 0.88; Table 2, Fig. 4), but with some exceptions (significant left tail in the distribution shown in Fig. 4). The large values of the standard deviations of E and Emax, as well as the large range of these variables (Table 2), indicate considerable variation in shapes and the first two moments of the individual BTCs. The average dilution index for individual compartments (Table 2) is larger than that of the entire lysimeter, based on data from all compartments (Table 1). This indicates that mixing within individual flow tubes is better than the overall mixing; longitudinal mixing apparently is more effective than lateral mixing. Since the individual BTCs clearly vary considerably (see also de Rooij and Stagnitti, 2002, Fig. 4), it is not surprising that spatial averaging (Table 1, lysimeter scale) overestimates E by overestimating lateral mixing. This is confirmed by the reactor ratios in Fig. 4: The reactor ratio for the entire lysimeter is comparable to the minimum M of the individual compartments, and only 68% of the average M for individual compartments. The lysimeter-scale M is much larger (108% of the average M).
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Table 2. Properties of the distribution of dilution parameters derived from breakthrough curves (BTCs) of the 300 individual sampling compartments.
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Fig. 4. Flux-weighted cumulative distribution function of reactor ratios M calculated from breakthrough curves of individual sampling compartments. For comparison, the mean value (Table 2) is plotted together with the value for the entire plume (calculated from all 300 compartments, Table 1) and the value for a single collector under the entire lysimeter (lysimeter scale, Table 1).
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CONCLUSIONS
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If the distribution of solute concentration in soil water is important, MCSs can provide the required information for an undisturbed soil volume (either in the field or in the laboratory) for prolonged periods of time. We have provided a means of estimating the minimal required size of MCSs, which needs to be 0.1 m2 (minimal value) to 0.2 m2 (safe value) for a range of soils investigated by us. We have extended the dilution theory to make it applicable to MCS data, thus complementing the BTC of the entire device, SSDC, and the leaching surface, all of which describe plume spreading. This additional tool facilitates a quantitative analysis of concentration variations within a solute plume passing the sampling depth of an MCS. By doing so, we clearly demonstrated the inadequacy of sampling devices with large collector areas (about 1 m2 or more) for this purpose. The theory indicates that stochastic-convective transport results in a constant dilution index with solute travel distance, while convective-transport gives a constant reactor ratio of 1. Thus, the dilution index and reactor ratio determined from MCS data provide valuable information about the solute transport regime in an investigated soil volume. We hope this work will stimulate the further deployment and development of multicompartment solute samplers, and improve our understanding of solute transport and its implications in the field.
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REFERENCES
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INTRODUCTION
