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Analysis of partial breakthrough data by a transfer-function method.

Introduction

Assessment of the transport of solutes through the vadose zone is becoming increasingly important to monitor the quality of land and water resources. The transport of solute is usually predicted by mathematical models (van Genuchten and Wierenga 1977; Vanclooster et al. 1994) from the solute-transport parameters. Accurate and efficient determination of these parameters is important for the success of these models (Roth et al. 1990). The experimental and analytical techniques currently used for determining the parameters are, however, often inadequate and complicated. Solutions of solute are added to the top of a soil column at a constant rate of water flow, usually under saturated conditions. The solution moves through the soil with the flowing water and breaks through at the end of the column in the effluent. Breakthrough curves (BTCs) are developed from the concentrations of the effluent as functions of time or number of pore volumes of solution. BTCs can now be constructed from the electrical conductivity (EC) of bulk soil estimated from measurements by time-domain reflectometry (TDR) (Kachanoski et al. 1992; Ward et al. 1994; Mojid et al. 2004). The 1-dimensional convection-dispersion equation (CDE) governs the movement of solute through the soil profile. The solute-transport parameters are usually determined by relatively simple analytical (Lapidus and Amundson 1952; Lindstrom et al. 1967), semi-analytical (Yamaguchi et al. 1994) or numerical solutions (Vanclooster et al. 1994) of the CDE, graphical methods (van Genuchten and Wierenga 1986) or curve-fitting procedures (Parker and van Genuchten 1984), or by time moments (Agneessens et al. 1978). A complete set of solute breakthrough data is needed in each of these methods to determine the solute-transport parameters (van Genuchten and Wierenga 1986). However, the measurement of a complete set of breakthrough data is often time-consuming, especially for fine-textured soils and at low water flow rates. The chemical analysis of effluents is also laborious and expensive. So, it would be invaluable for practical purposes if solute-transport parameters could be determined from partial breakthrough data, especially the earlier portion. But, to date, there has been no attempt to investigate this possibility.

Transfer functions have been used extensively and successfully in soil science for almost 20 years, and their properties and applications have been described by Jury and Roth (1990). However, most workers (e.g. White et al. 1986; Sposito et al. 1986) use transfer functions to fit non-parametric or parametric (e.g. log-normal distributions) functions representing the probability-density functions of the distribution of the residence or travel times of water and/or solute in soil. There are other types of transfer function (Becker and Charbeneau 2000; Eykholt and Li 2000) that predict BTCs by inverting the transfer function to the time domain. Recently, Mojid et al. (2004) described a simple transfer-function technique, which is mathematically equivalent to the solution of the CDE for a narrow pulse, and used this technique successfully to determine solute-transport parameters from measured breakthrough data in repacked soil columns. They demonstrated the robustness of their technique in determining solute-transport parameters by comparing their results with those obtained from the equilibrium model of the CXTFIT program of Parker and van Genuchten (1984). This paper follows that of Mojid et al. (2000) to demonstrate the feasibility of the transfer-function technique to determine solute-transport parameters from partial, rather than complete, breakthrough data, i.e. from the earlier sections of BTCs.

The mathematical technique we employ, curve fitting in the time domain, is somewhat outside current usage in soil science but is widely employed in chemical engineering and systems control. Chapter 1 of Wakao and Kaguei (1982) gives a clear and detailed exposition of the method and its application to flow processes governed by the CDE in both inert and reactive porous media. Curve fitting in the time domain is a method in which observed response signals downstream are compared with those predicted from an input signal that was measured upstream. If the observed and predicted response signals agree well, the parameter values used for its prediction may be regarded as correct. The input signal may take any form and does not depend on the precise boundary condition imposed at the entry to the porous column; the method uses a concentration-time function, which is measured inside the column. In addition, if the response signal is also measured within the column instead of at its exit, the errors incurred in using mathematical solutions to the CDE for a semi-infinite instead of a finite column, which become important when column Peclet numbers fall below 5, are avoided. We follow the nomenclature and terminology of Wakao and Kaguei (1982) in developing the theory of our method and convert to the conventional usage of soil science when presenting our results.

Theory

The transfer-function method

For linear and dynamic systems with continuous variables, the input and response signals are correlated by a transfer function, which is the ratio of the Laplace transform of the response to the Laplace transform of the input. So, for such systems, the response signal can be expressed in terms of its input signal and the transfer function. Mojid et al. (2004), following Wakao and Kaguei (1982), described the application of the technique and its performance.

For the vertical transport of a reactive solute by steady flow of water through a column of soil, the classical CDE is:

R [partial derivative]C/[partial derivative]t = D[[partial derivative].sup.2]C/[partial derivative][z.sup.2] - V[partial derivative]C/[partial derivative]z (1)

where R is the retardation factor of solute, C is the concentration of solute in pore water, D is the dispersion coefficient, V is the average velocity of the pore water, and z is the axial distance. The estimated response concentration [[C.sub.r.est(t)]] at time t is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [C.sub.i([alpha])] is the time-dependent input concentration of solute in the soil column, [alpha] is the time interval between 2 consecutive measurements of the input concentration, and f(t), the Laplace inversion of the transfer function, is the impulse response of a Dirac delta input of solute to the column. The value of f(t) for a transport system governed by the CDE (Eqn 1) is given (Wakao and Kaguei 1982, p. 10) by:

f(t) = [[[pi]N[(t/[tau]R).sup.3].sup.-1/2]/2[tau]R exp [[-(1 - t/[tau]R).sup.2][(4Nt/[tau]R).sup.-1]] (3)

where N is the mass-dispersion number (= D/ZV), which is the reciprocal of the column Peclet number, P (= ZV/D), [tau] is the mean travel time or mean residence time of solute, and Z is the distance between the positions of the measurements of input and response concentrations. The estimated response BTCs are calculated by Eqns 2 and 3, and compared to the measured BTCs in the time domain to determine the solute-transport parameters. The accuracy of fitting by the transfer-function method is evaluated by the root-mean-square error (RMSE) between the measured and estimated BTCs, given by:

RMSE = [square root of ([[integral].sup.[infinity].sub.0]] [[[C.sub.r(t)] - [C.sub.r.est(t)]].sup.2]dt/[[integral].sup.[infinity].sub.0] [[[C.sub.r(t)]].sup.2]dt] (4)

where [C.sub.r(t)] is the time-dependent measured response concentration of solute. Eqn 2 estimates a set of response concentrations from a set of input concentrations. Since the estimated response concentrations are compared to the measured concentrations in the time domain, only a section of accurately measured BTC need be used for reliable comparison. This suggests that the solute-transport parameters might be determined from an early segment of the measured breakthrough data by the transfer-function method.

Methodology

Experiment

In order to test the postulated hypothesis, measurements of solute concentrations at 2 different depths in a column of soil are required. The concentration at the upper depth is the input and that at the lower depth is the response. Measuring concentrations between 2 sensors in the sample rather than from the top of the sample to one sensor has several advantages. First, it allows measurement over a well-defined horizon from a soil without having to remove the upper layers. Second, it is often difficult to supply a small solute pulse correctly in order to satisfy the boundary conditions under unsaturated conditions. Third, it is easier to measure the input correctly in the soil after some mixing that otherwise would be erroneous. Full experimental details are found in Mojid et al. (2004); only a summary is presented here.

A PVC column, 30cm long and 16cm in diameter, was placed vertically and axially over a 1.2-m supporting column uniformly filled with a sandy loam soil. The upper column was filled uniformly and compactly with the air-dried and sieved soil to be used in the solute-transport experiments. Four different porous materials (coarse sand, sandy loam soil, clay loam soil, and clay soil) were used in the experiments. Table 1 lists some physical properties of these soils. Two 3-wire TDR sensors (15cm long), one at 11cm and the other at 24 cm below the top of the upper column (vertical distance between the 2 sensors, Z= 13cm), were inserted horizontally. A sufficient quantity of tap water with an EC of 17 mS/m was leached through the columns. A constant hanging water table maintained at 20 cm above the base of the lower columns created suction in the upper soils. A cartridge pump applied tap water at a constant rate specific to each soil to ensure unsaturated flow. However, because of the very low hydraulic conductivity of the clay soil, it was difficult to attain perfect unsaturated flow with our pump. There was a possibility of mixed saturated and unsaturated flow; the flow might be saturated in some sections and unsaturated in other sections of the soil surface. The applied water was spread uniformly over the surface of the soil in the upper columns. The whole system reached equilibrium between the applied and drainage water in 2-6 weeks. At equilibrium, 5 [cm.sup.3] of a solution of calcium chloride (Ca[Cl.sub.2]) was spread instantaneously uniformly over the surface of the soil in the upper columns. The application of water was continued. A Campbell Scientific CR10 datalogger, programmed with PC208 dataiogger support software, recorded the water content and bulk EC of the soil at suitable intervals. The measurements continued until the whole of the applied salt leached out from the upper columns. The experiment was conducted with the same set-up for 3 different concentrations (equivalent to areal distributions of 0.1, 0.2, and 0.3 g/[m.sup.2] of the soil surface) of the applied salt and measurements at each concentration were repeated 3 times except for the clay soil, in which measurement was conducted only for the lowest concentration without any repetition. Table 2 summarises the physical conditions of the experiments.

Data analysis

The analysis of TDR data is based on the assumption that the concentration of a solute in soil water is linearly related to the EC of soil water, which, in turn, is linearly related to the EC of bulk soil (Kachanoski et al. 1992; Ward et al. 1994). The initial concentration of solute in the soil water before applying the pulse of solute is deducted from the time-dependent measured concentrations. The resulting concentrations are then normalised with respect to the total measured concentration. If the initial and time-dependent concentrations of solute in the soil water are represented by [C.sub.i] and [C.sub.t], respectively, and the corresponding bulk EC by [E.sub.a(i)] and [E.sub.a(t)], respectively, then:

[C.sub.t] = [[alpha].sub.1] + [[alpha].sub.2][E.sub.a(t)] (5)

[C.sub.i] = [[alpha].sub.1] + [[alpha.sub.2][E.sub.a(i)] (6)

where [[alpha].sub.1] and [[alpha].sub.2] are calibration constants. The time-dependent normalised concentration [[C.sub.0(t)]] for the initially relaxed (non-zero initial concentration) system is then computed by:

[C.sub.0(t)] = [C.sub.t] - [C.sub.i]/[[summation].sup.T.sub.t=0] [[E.sub.a(t)] - [E.sub.a(i)]] (7)

where T is the total duration of the experiment. Substituting Eqns 5 and 6 into Eqn 7, the normalised concentration is calculated by:

[C.sub.0(t)] = [E.sub.a(t) - [E.sub.a(i)]/[[summation].sup.T.sub.t=0] [[E.sub.a(t)] - [E.sub.a(i)] (8)

The BTCs are drawn by plotting this normalised concentration, [C.sub.0(t)], against time, t.

Four sets of data were selected from the complete BTC for each soil. The durations were: 5, 7, 10, and 15 h for coarse sand; 30, 50, 70, and 119h for sandy loam; 30, 60, 90, and 119h for clay loam; 100, 160, 260, and 364 h for clay. In sectioning the BTCs, our criteria were that the first section included about half of the rising limb, the second section a little more than the peak, the third section most of the falling limb, and the fourth section the entire BTC including the long tail. The mean travel time, [tau], and mass-dispersion number, N, of solute were fitted from each set of data by the transfer-function method (Eqn 2) using a non-linear least-square fitting technique. Note, however, that we report not N, but the column Peclet number, P = [N.sup.-1]. We fixed the retardation factor, R, at unity assuming Ca[Cl.sub.2] to be a non-reactive solute. The other solute-transport parameters, such as the velocity of pore water, V (= Z/[tau]), dispersion coefficient, D (= VZN = [Z.sup.2]N/[tau]), and dispersivity, [lambda] (= D/V= ZN), were calculated from x, N, and the distance, Z, between the input and response BTCs. The coefficient of determination, [R.sup.2], and the RMSE were estimated for each dataset.

Results and discussion

Figures 1-4 compare typical measured and estimated response BTCs from the 4 sections of the breakthrough data for each porous material for the lowest concentration of the applied solute pulse. The fit is similar in all soils except for the clay in which the fit is poor. The estimated response BTCs from the earliest section of data often fit poorly with the measured response BTCs but those from the subsequent sections of data, however, match the measured BTCs very closely in the coarse sand, sandy loam, and clay loam. These close fits suggest that the transfer-function method can be used to estimate solute-transport parameters from partial breakthrough data without the prolonged tail section. The reason for the poor fit in clay soil might be errors in the measured BTCs. Alternatively, the sharp peak in the response BTC of Fig. 4 might be the result of either a preferential flow of water and solute (possibly a saturated vertical flow path as fine cracks were visible on the surface of the core) or non-uniform initial distribution of solute on the soil column.

[FIGURES 1-4 OMITTED]

Table 3 and Fig. 5 compare the mean solute-transport parameters over all concentrations and their replications (a total of 3 x 3 = 9 data points in each case) for the 4 different sections of the breakthrough data for all soils except the clay. The solute-transport parameters for the clay soil are listed in Table 4. The mean travel time of solute, [tau], estimated from the earliest section of the BTCs is smaller than those estimated from later sections of the BTCs for all materials except the clay soil in which [tau] remained the same. The 3 later sections of the BTCs resulted in stable values oft in each material. Similar behaviour is observed in the values of mass-dispersion number, N = [P.sup.-1], estimated from different sections of the BTCs in all soils. The other solute-transport parameters, derived from [tau] and N, are also consistent for the later sections of the BTCs (Table 3 and Fig. 5). Since the transfer-function method has been shown to provide accurate values of solute-transport parameters from complete BTCs (Mojid et al. 2004), these results demonstrate that partial BTCs can provide equally accurate estimates of the solute-transport parameters. Most of the parameters estimated from the earliest section of the BTCs are, however, not consistent with those from subsequent sections and hence should not be regarded as accurate. Note that the coefficients of variation (= standard error/mean) for all parameters among the 3 later sections of breakthrough data are reasonably small and consistent (Table 3). Consequently, there are no statistically significant differences in the solute-transport parameters in the last 3 segments of BTC. However, all parameters show some variability among the concentrations and their replications as illustrated in Fig. 5. These findings have been confirmed by similar experiments on these materials, except the clay soil, in which 2 other water fluxes were applied.

[FIGURE 5 OMITTED]

The coefficient of determination, [R.sup.2], for the fit between the estimated and measured BTCs is very close to unity (between 0.98 and 1.00) and RMSE is very low (between 0.09 and 0.01) for the sections of breakthrough data in the coarse sand, sandy loam, and clay loam soil. Such values of [R.sup.2] and RMSE indicate good agreement between the measured and estimated response BTCs in these materials, confirming the good visual fits demonstrated in Figs 1-3. For the clay soil, however, [R.sup.2] ranges between 0.81 and 0.92, and the RMSE between 0.23 and 0.24 (Table 4). We ascribe the underestimation of x and N from the earliest sections of the data relative to the later sections to non-uniform flow in the early stages of breakthrough. These errors probably occur because of the non-uniform spreading of the solute pulse and water stream over the surface of our experimental soils. Manually it was very difficult and even impossible to distribute a small solute pulse (5 [cm.sup.3]) and water stream perfectly uniformly. This non-uniformity probably caused variable concentrations of solutions across the cross-sectional plane of the soil, the effect of which is pronounced in the early breakthrough. Similarly, some errors in the tailing portion of the breakthrough may be due to interactions of the solute with the soil minerals. The resulting effect is a slower decrease in concentration than that expected for a non-reactive solute and for an inert porous medium. Errors in the frontal portion of breakthrough data have also been reported by Taylor (1953) and Jury et al. (1991).

General discussion and conclusions

It is important to note that, as long as the transport of solute can be described by the simple CDE, as in this study, this transfer-function method permits unambiguous identification of parameters from partial breakthrough data. This is because the CDE has random, symmetric mixing occurring in the centre of fluid motion, so that the early stage contains as much information as the latter. When other processes, such as rate-limited exchange between fast and slowly moving water in bimodal flow or between mobile and immobile water occur in a soil, the scenarios are different. These processes have nested time scales and much of the information about these rate processes is housed in the tail of the BTC. Although this transfer-function method has yet to be tested under such conditions, we suggest that the method, because of its robust fitting capability, might provide good fits with the earlier sections of BTCs that are influenced by rate-limited processes. The method should also be useful for reactive soil-solute combinations that have linear sorption/desorption isotherms, because of its ability to fit 3 transport parameters (R, [tau], and N) simultaneously (Mojid et al. 2004).

In this work, we followed the procedure described by Wakao and Kaguei (1982), who did not distinguish between resident and flux concentrations or specify the method of measuring the input and response signals, C(t). TDR, however, measures resident concentrations, whereas the transfer function in Eqn 3 is appropriate to flux concentrations. We acknowledge that this is a limitation in analysing our experiments, although the principle of the transfer-function method remains valid. However, any errors incurred as a result of this will be relatively minor at the values of P in our experiments (see, for example, lines A1 and A2 in fig. 44-1, and table 44-3, of van Genuchten and Wierenga 1986) and almost insignificant compared with experimental errors and the variability among the concentrations and replications for a particular soil illustrated in Fig. 5.

The transport of solute over an area can be predicted by means of simulation models (Vereecken et al. 1991; Vanclooster et al. 1994) into which are put local solute-transport parameters and the correct specification of their spatial variability. The success of such models depends on the accuracy of the input parameters, and so it is important to determine these accurately. However, there may be a tradeoff: quicker, less precise measurements on many samples may better represent the solute-transport parameters of the soil in the area than very precise determinations on a few samples. The use of partial BTCs would expedite the specification of the variability of these parameters, especially under unsaturated conditions. In addition, the transfer-function technique could be extended to a multi-position measurement system that allows estimates of the solute-transport parameters of sequential soil horizons down the profile (Mojid et al. 2004) because the C(t) functions at any position may be specified as an input signal. The technique we describe in this and our previous paper (Mojid et al. 2004) should be as valuable to soil scientists and hydrologists in the future as it has been to chemical engineers in the past 20 years.

Conclusions

The transfer-function method used by Mojid et al. (2004) is not limited solely to a complete set of breakthrough data. It is capable of estimating solute-transport parameters accurately from partial breakthrough data. This is an important advantage of the method over other methods of determining the solute-transport parameters. The response concentration of the breakthrough data usually has a long tail, especially for small values of P. It can be very time-consuming to measure a complete set of breakthrough data at low water-flow rates, especially in fine-textured soils, and may become impractical to monitor a complete set of breakthrough data for a wide range of unsaturated soils under such situations. The truncation of the tailing portion causes erroneous estimates of the parameters by other methods of analysis. The transfer-function method has the ability to provide the required parameters accurately from a partial set of breakthrough data. However, since initial breakthrough data are often subject to error, it is better to apply the method to estimate the solute-transport parameters from a dataset of longer duration. A useful rule of thumb is that the section of breakthrough data that contains the peak of the BTC of the response concentration should be used to get an accurate estimate of the parameters. However, the transfer-function method can fit data of even shorter duration and the parameters can be estimated from such data if the experimental conditions are ideal. The method has yet to be tested on BTCs that are influenced by rate-limited mass transfer between fast- and slow-moving solution or between mobile and immobile solution. However, Wakao and Kaguei (1982) work an example from adsorption chromatography, in which solute moves by longitudinal dispersion in the macroporosity and radial diffusion in the microporosity, to demonstrate the power of the method. Also, as demonstrated by Passioura and Rose (1971), it is possible to characterise the dispersion of solute from systems of aggregates having a bimodal pore-size distribution with Eqn 1 over a wide range of pore-water velocities and aggregate diameters.

Abbreviations used: C Concentration of solute in pore-water, M/[L.sup.3]; [C.sub.t] time-dependent concentration of solute in soil water (for input and response), M/[L.sup.3]; [C.sub.0(t)] time-dependent normalised concentration of solute, dimensionless; [C.sub.i] initial concentration of solute in soil water, M/[L.sub.3]; [C.sub.i([alpha])] time-dependent measured input concentration of solute in the soil column, M/[L.sup.3];[C.sub.r(t)] time-dependent measured response concentration of solute, M/[L.sup.3]; [C.sub.r.est] estimated response concentration of solute, M/[L.sup.3]; D solute dispersion coefficient, [L.sup.2]/T; [E.sub.a(i)] initial electrical conductivity of bulk soil, [T.sup.3][A.sup.2]/[L.sup.3]M; [E.sub.a(t)] time-dependent electrical conductivity of bulk soil (for input and response), [T.sup.3][A.sup.2]/[L.sup.3]M; f(t) impulse response of a Dirac delta input of solute to the soil column, M/[L.sup.3]; N mass-dispersion number, which is the reciprocal of the column Peclet number, dimensionless; P column Peclet number, dimensionless; [r.sup.2] coefficient of determination, dimensionless; R solute retardation factor, dimensionless; t variable time, T; T total duration of experiment, T; V average velocity of pore-water, L/T; z axial distance of the soil column (positive downward), L; Z distance between the positions of the measurements of input and response concentrations, L; [alpha] time interval between 2 consecutive measurements of the input concentration, T; [[alpha].sub.1] calibration constant (Eqns 5 and 6), M/[L.sup.3]; [[alpha].sub.2] calibration constant (Eqns 5 and 6), [M.sup.2]/[T.sup.3][A.sup.2-]; [lambda] solute dispersivity, L; [tau] mean travel time or mean residence time of solute, T.

Manuscript received 25 August 2004, accepted 14 December 2005

References

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M. A. Mojid (A), D. A. Rose (B,D), and G. C. L. Wyseure (C)

(A) Department of Irrigation and Water Management, Bangladesh Agricultural University, Mymensingh 2202, Bangladesh.

(B) School of Agriculture, Food and Rural Development, University of Newcastle, Newcastle upon Tyne, NE1 7RU, UK.

(C) Faculty of Agricultural and Applied Biological Sciences, K. U. Leuven, Kasteelpark Arenberg 21, B 3001 Leuven, Belgium.

(D) Corresponding author. Email: jan.fife@ncl.ac.uk
Table 1. Some physical properties of the soils used in flow experiments

Textural Sand Silt Clay Bulk Organic
class (g/kg) density matter
 (kg/[m.sup.3]) (g/kg)

Coarse sand 935 29 36 1550 5.5
Sandy loam 745 109 146 1478 36.9
Clay loam 450 260 290 1363 35.9
Clay 268 278 454 1203 17.3

Table 2. Physical conditions of the flow experiments

 Textural Water flux Soil-water Pore-water
 class (cm/h) content velocity
 ([cm.sup.3]/ (cm/h)
 [cm.sup.3])

Coarse sand 0.497 0.143 3.48
Sandy loam 0.219 0.321 0.68
Clay loam 0.149 0.386 0.39
Clay 0.100 0.415 0.24

Table 3. Mean and standard error (in parentheses) of solute-transport
parameters for four segments of BTCs over all concentrations and
replications for coarse sand, sandy loam, and clay loam soil

Soil type Duration [tau] [N.sup.-1]
 (h) (h) = P

Coarse sand 5 3.92 (0.07) 54.91 (3.9)
 7 4.25 (0.03) 21.63 (l.1)
 10 4.32 (0.03) 18.91 (0.7)
 15 4.34 (0.03) 18.50 (0.6)

Sandy loam 30 18.76 (0.78) 8.45 (2.9)
 50 18.85 (0.76) 7.01 (2.2)
 70 18.84 (0.76) 7.07 (2.2)
 119 19.11 (0.73) 7.01 (2.2)

Clay loam 30 29.99 (0.27) 11.31 (2.0)
 60 29.78 (0.30) 12.97 (l.4)
 90 29.95 (0.34) 12.25 (1.4)
 119 29.95 (0.34) 12.13 (l.4)

Soil type V D [lambda]
 ([cm.sup.2]/h) ([cm.sup.2]/h) (cm)

Coarse sand 3.32 (0.06) 0.81 (0.04) 0.25 (0.02)
 3.06 (0.02) 1.87 (0.10) 0.61 (0.03)
 3.01 (0.02) 2.10 (0.08) 0.69 (0.02)
 3.00 (0.02) 2.13 (0.08) 0.71 (0.02)

Sandy loam 0.70 (0.03) 2.16 (0.43) 3.27 (0.75)
 0.70 (0.03) 2.30 (0.43) 3.45 (0.73)
 0.70 (0.03) 2.25 (0.42) 3.38 (0.70)
 0.69 (0.03) 2.28 (0.42) 3.48 (0.71)

Clay loam 0.43 (0.00) 0.58 (0.07) 1.35 (0.15)
 0.44 (0.00) 0.47 (0.04) 1.08 (0.10)
 0.43 (0.01) 0.50 (0.05) 1.15 (0.11)
 0.43 (0.01) 0.51 (0.05) 1.17 (0.11)

Table 4. Solute-transport parameters determined by transfer-function
method for clay soil

Duration [tau] V D 1/N = P
(h) (h) (cm/h) ([cm.sup
 .2]/h)

100 30.00 0.43 0.056 100.00
160 30.00 0.43 0.056 100.00
260 30.00 0.43 0.019 293.50
364 30.00 0.43 0.019 291.50

Duration [lambda] [R.sup.2] RMSE
(h) (cm)

100 0.13 0.92 0.24
160 0.13 0.81 0.23
260 0.04 0.87 0.23
364 0.04 0.91 0.23
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Author:Mojid, M.A.; Rose, D.A.; Wyseure, G.C.L.
Publication:Australian Journal of Soil Research
Geographic Code:8AUST
Date:Mar 15, 2006
Words:5481
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