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Transport of bromide in the Bainsvlei soil: field experiment and deterministic/stochastic model simulation. I. continuous water application.

Introduction

The rate at which agrochemicals move with percolating water from the soil surface to the groundwater is important in the management of agricultural lands and the subsurface of the earth. Although considerable progress has been made in describing the transport of water and chemicals under laboratory conditions (Nielsen et al. 1986), the results have not been carried to the field because of the natural heterogeneity of soils. At the field scale, even the most basic solute transport process, the mean convection rate, cannot be described with reasonable accuracy (Ashraf et al. 1997).

Furthermore, field investigations indicate that the fluid velocity in the field is highly variable, but can be characterised with a log-normal distribution (Biggar and Nielsen 1976; Butters and Jury 1989). A number of models, which ignore lateral mixing and treat solute movement as though it occurred in isolated stream tubes with different transport parameters, have therefore been proposed for applications in the field. In these models the local (or stream tube) parameters are treated as random, log-normal distributed variables, with zero correlation length normal to the vertical direction of flow. The area-averaged solute concentration can then be calculated by averaging the local transport model over all possible values of the random parameters. Models of this kind, which include parallel soil columns obeying local convection--dispersion, are called stochastic stream tube models (STM) (Dagan and Bresler 1979; Amoozegard-Fard et al. 1982; Bresler and Dagan 1983; Jaynes et al. 1988). The theoretical background of STMs has recently been discussed in detail by Toride and Leij (1996a, 1996b), who also developed procedures for its application to both reactive and nonreactive solutes under chemical equilibrium and nonequilibrium conditions.

Deterministic convection--dispersion equation (CDE) models have been used for years in many fields where the transport of contaminants is important and well understood (Javandel et al. 1984). Unfortunately, the same cannot be said of STM and only few attempts have been made to relate the performance of the stochastic STM to the deterministic CDE models through field tracer experiments. STM are supposedly more reliable as they stochastically take into account the variability of transport parameters in the field. The objective of this study was to conduct a steady-state field experiment to quantify the transport of [Br.sup.-] in the Bainsvlei soil of South Africa and analyse the data with CDE and STM.

Tracers have long been used to follow the movement of soil water and solutes in the laboratory and in the field under ponding water conditions (Biggar and Nielsen 1976; Starr et al. 1978; Jaynes et al. 1988; Rice et al. 1991) and non-ponding conditions (Wild and Babiker 1976; Van de Pol et al. 1977; Butters and Jury 1989; Ellsworth et al. 1991). In this study, a rainfall simulator was used as it provides a non-ponding, fairly uniform application, and a conservative ion bromide was used as a tracer. The experimental results were analysed with the CXTFIT package of Toride et al. (1995).

Materials and methods

Field investigations

A field experiment with simulated rainfall intensity of 5.41 mm/h was conducted on a patch of soil (100 by 100 cm) at the experimental site of the Department of Soil, Climate and Crop Sciences at the University of the Free State, South Africa. The soil of the experimental site is Bainsvlei sandy loam, which covers much of the South African land mass (Soil Classification Working Group 1991). It is characterised by orthic topsoil and red apedal/soft plinthic subsoil. The results of soil particle analysis and the textural group of each layer is presented in Table 1.

The layout and instrumentation of the plot is shown in Fig. 1 together with the sampling locations used in the experiment. A neutron probe access tube was installed at the centre of the plot to a depth of 200 cm for soil water content determination. Four tensiometers were installed at depths of 30, 45, 90, and 120cm around the access tube to monitor matric potential. The steady-state condition during the experiment was monitored using the neutron probe and tensiometers.

[FIGURE 1 OMITTED]

A solution of 20 g KBr dissolved in 500 mL distilled deionised water was applied to the surface of the plot to give a [Br.sup.-] concentration of 27 g/L (equivalent to 13.5 g/[m.sup.2]). Similar concentrations of [Br.sup.-] have been used in other studies (Baker and Laflen 1982; Owens et al. 1985; Rice et al. 1986; Bicki and Guo 1991; Nachabe et al. 1999). The solution was applied in perpendicular spray patterns in 2 traverses. After application of the [Br.sup.-] solution, water was applied to the experimental plot at the specified intensity.

The rainfall simulator used for the experiment was manufactured according to the design of Claassens and Van Der Watt (1993). The uniformity of application was checked repeatedly before the start of each experiment and the coefficient of uniformity was found to be > 90% in all cases.

The (100 by 100 cm) test plot was divided into 2 equal halves, which were further divided into smaller subunits (17 by 17 cm) (not shown in Fig. 1) for sampling purposes. The central subunits were not sampled, in order to avoid interference with the tensiometers and neutron probe measurements. Soil samples were taken from 2 subunits, one from each half of the plot, during each sampling time, and the samples taken from similar depth were mixed to give a composite sample. To avoid soil disturbance and compaction, the plots were accessed via a wooden beam placed on the metal frame surrounding the plot.

Of the various methods to acquire data in solute transport studies, the removal of soil cores for subsequent extraction and analysis of solutes, as used in this study, is perhaps the most intuitive approach to evaluate the spatial distribution of solutes within the soil profile (Rhoades 1982). The soil samples were taken at predetermined randomised positions as shown in Fig. 1, at intervals of 20 cm to a maximum depth of 160 cm with an auger type coring tube 20 cm long and 4.2 cm in diameter. After each sampling, the core sampler was cleaned with tap water and rinsed in distilled water before taking the sample at the next depth. After retrieving all of the cores, the holes were backfilled with soil from outside of the plot. A subsample of approximately 100 g of soil was taken from each core sample and placed in paper bags to determine the water content of the soil with the gravimetric method (dry the sample for 24 h at 105[degrees] C). The balances of the soil samples were stored in polyethylene bags for [Br.sup.-] analysis.

The soil samples for [Br.sup.-] analysis were oven-dried and crushed to pass through a 2-mm sieve. A mixture consisting of 50 g of the dried soil sample and 50 mL of distilled deionised water was then prepared for each soil sample and shaken with a laboratory shaker and then filtrated. The filtrates were all stored in a refrigerator at 4[degrees]C before analysis with an ion chromatograph (Dionex 22001 spectrograph) at the Institute for Groundwater Studies at the University of the Free State, South Africa. The [Br.sup.-] concentration of the field soil solution was then calculated from the filtrate concentration, taking into account the mass of soil extracted and the volume of water used for the extraction (1 : 1 ratio) and the gravimetric water content of the soil sample.

Equations and data analysis

The fundamental solute transport process for the non-reactive bromide in the steady (continuous water application) experiment is described by the l-dimensional CDE:

(1) [defferential]c/differential]t = D [[[differential].sup.2]c/[differential][z.sup.2]] - v [[differential]c/[differential]z]

where c is the concentration of the solution, D the dispersion coefficient, t the time, and z is the distance.

There are no known solutions of the general hydrodynamic equation given in Eqn 1, but there are a number of analytical solutions. These solutions will be referred to here as the classical CDE models. Although CDE models perform well for laboratory soil columns and homogenous field soils, the models are not always able to describe transport processes in heterogeneous field soils accurately (Jury and Fluhler 1992). Therefore, an STM is used for better accuracy. The deterministic CDE hypothesis assumes that the parameters D, v, (or [theta], [lambda]) each have one value of the whole experimental dataset. The stochastic STM hypothesis assumes that the parameters D, v, (or [theta], [lambda]) have a joint probability distribution, so that each streamtube has its own set of parameter values. The initial condition used in the experiment was that the background concentration of bromide along the soil profile is the same. The bottom boundary was taken to be very deep and the upper boundary taken as one time application of known concentration of bromide.

It is imperative that every effort should be made to ensure that mass is conserved in all studies pertaining to the movement of soil water and solutes in laboratory and field studies. The method used in the present study for this purpose was to compare the mass of [Br.sup.-] recovered from the core samples with the mass of the applied solute. The recovered mass of [Br.sup.-] associated with a core sample, [m.sub.i], was calculated as:

(2) [m.sub.i,j] = c([z.sub.i], [t.sub.j]) [theta]([z.sub.i], [t.sub.i]) A[DELTA][z.sub.i]

where c([z.sub.i], [t.sub.j]) is the [Br.sup.-] concentration at depth [z.sub.i] and sampling time [t.sub.j], [theta]([z.sub.i], [t.sub.j]) the volumetric soil water content, A the cross-sectional area of the core sampler, and [DELTA]z the thickness of the sample. The total mass, [m.sub.k], of [Br.sup.-] recovered from the soil profile over a depth Z at a given horizontal position ([x.sub.k], [y.sub.k]) is then simply the sum of the [m.sub.ij] at ([x.sub.k], [y.sub.k]):

(3) [m.sub.k] = [summation over (i)] [m.sub.ij]([z.sub.i] [less than or equal to] Z)

The mass of [Br.sup.-] recovered as a percentage of total applied mass of [Br.sup.-] was calculated as r(%) = 100 x [m.sub.k]/M, where M is the total mass of [Br.sup.-] applied at the soil surface (13.5 g/[m.sup.2] or 18.7 mg for the sampled area of 4.2 cm diameter).

The average downward movement of a solute front, if piston flow happens, can be estimated by the pore-water velocity:

(4) [v.sub.w] = q/[theta]

where q is the Darcian flux and [theta] is the average volumetric water content from the soil surface to the maximum depth at a given sampling time.

The [Br.sup.-] velocity [v.sub.s] was determined (a) as the ratio between the depth of observation and the arrival time of the concentration peak (Butters and Jury 1989), and (b) as the location of the centre of mass of the solute plume (Ellsworth et al. 1991).

The velocity of the concentration peak of the solute pulse was calculated as the depth increase by the concentration peak between 2 consecutive sampling events divided by the time interval between the 2 sampling events as:

(5) [v.sub.s] = [(z.sub.p).sub.i] - [(z.sub.p).sub.i-1]/[t.sub.i]-[t.sub.i-1] = [DELTA][z.sub.p]/[DELTA]t

where [DELTA][z.sub.p] is the change in depth of concentration peak of the solute pulse and [DELTA]t is the corresponding change in time between the 2 peaks.

The depth to the centre of mass of the [Br.sup.-] profiles was calculated as:

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [bar.[z.sub.i]] is the depth to the centre of mass at the sampling time [t.sub.i].

The velocity of the centre of mass of the solute between two samplings was calculated as:

(7) [v.sub.s] = [bar.[z.sub.i]] - [bar.[z.sub.i-1]]/[t.sub.i] - [t.sub.i-1] = [DELTA][bar.z]/[DELTA]t

where [bar.[z.sub.i]] and [bar.[z.sub.i-1]] are the depth to the centre of mass of [Br.sup.-] at sampling times [t.sub.i] and [t.sub.i-1].

If a conservative tracer such as [Br.sup.-] is transported continuously throughout the soil profile at the same rate as the pore-water velocity [v.sub.w], the tracer is said to have moved as piston flow (Jury et al. 1991; Nachabe and Morel-Seytoux 1995). Otherwise, it is said that the solute experienced preferential flow, indicated by a shorter time of arrival at a given depth than indicated by the pore-water velocity.

The longitudinal dispersion coefficient is related to pore-water velocity as:

(8) D = [lambda]|v|

where [lambda], is longitudinal dispersivity. Several studies have shown that dispersivity increases with depth of soil or travel time (Sposito et al. 1986). However, some studies have also shown that the correlation between dispersion coefficient and velocity can be highly variable (Van Ommen et al. 1989).

Parameter estimation

The CDE with deterministic coefficients and the STM with stochastic coefficients were used to determine solute transport parameters. The deterministic 1-dimensional CDE uses a constant average pore-water velocity and a constant dispersion coefficient to describe area-averaged solute movement. The STM makes the following assumptions regarding the transport processes: (i) the hydraulic properties controlling solute transport vary horizontally across the field but not with depth; thus, (ii) we can divide the field into a number of vertical columns called 'stream tubes' within which the hydraulic properties are constant; (iii) each stream tube is independent of its nearest neighbours and does not interact with them (1-dimensional flow); and (iv) the local-scale transport or transport in each stream tube is described deterministically assuming convective--dispersive model (Jury and Roth 1990; Dagan 1993).

The parameters of the analytical CDE and STM were estimated with the nonlinear parametric estimation package (CXTFIT) of Toride et al. (1995) by fitting analytical solutions for the models to observed field [Br.sup.-] concentrations.

Results and discussion

The soil water distribution

The water contents and matric heads along the soil profile indicated that the water distribution in the soil profile was at a steady-state during the experiment. The average standard deviation of only 0.25% in the volumetric water content was observed at the different soil depths.

Bromide recovery

The sum of the recovered [Br.sup.-] masses for the soil profile during each sampling event was determined using Eqn 3 and expressed as a percentage of the applied [Br.sup.-] mass. The bromide recovery at different times of sampling was found to be as follows: 2.0 h, 99.0%; 4.8 h, 100.0%; 9.8 h, 114.4%; 17.8 h, 131.0%; 27.5 h, 91.4%; 37.0 h, 65.5%; 46.5 h, 45.6%; 56.0 h, 55.6%; 66.0 h, 41.6%; 77.0 h, 39.1%; and 96.0 h, 12.3%. Considering only those times (the first 6 sampling events) for which the concentration had returned to near the background concentration at the maximum sampling depth, the average mass recovery was 100%. Starting from the seventh sampling, the average recovery went on decreasing as the bromide moved beyond the maximum sampling depth. Higher or lower than full recovery is common in field-scale solute transport experiments (Rice et al. 1986; Butters and Jury 1989; Izadi et al. 1993; Ashraf et al. 1997; Nachabe et al. 1999) and might be caused by incomplete extraction and lateral divergence outside of the subject area due to water content and concentration gradients.

Figure 2 shows graphs of the observed [Br.sup.-] concentration profiles at a number of sampling periods after the start of the experiment. Notice how the peak concentration decreased and moved deeper into the soil with time, ultimately beyond the maximum sampling depth, while the concentration profile became flatter and wider as time increased.

[FIGURE 2 OMITTED]

The peak concentration of the [Br.sup.-] plume appears to have been captured in all but the final 2 sampling times. At the last 2 sampling times, 77.0 and 96.0 h (the latter not shown in Fig. 2), the peak concentration had moved beyond the maximum sampling depth. This means that the total amount of simulated rain (519.4 mm) applied within the 4 days of the experiment had displaced the tracer peak beyond the maximum depth of sampling.

Parameter estimation

The [Br.sup.-] concentration data was fitted to the CDE and STM using CXTFIT (Fig. 3). In the relative concentration (C/Co), C is the [Br.sup.-] concentration of any soil sample corresponding to a given depth or time after the [Br.sup.-] pulse was applied and Co is the concentration of the pulse applied at the beginning of the experiment. Both models fit the observed data excellently considering an in situ experiment where many factors interact and affect the transport process. Since the fitted curves were very similar, only the CDE fit is presented in Fig. 3.

[FIGURE 3 OMITTED]

The transport parameters determined from the fits are presented in Table 2. As indicated by the coefficients of determination ([r.sup.2]), the fit was better for the earlier sampling times than the later times, probably because of the mass balance effect. From Table 2, it can be seen that both models gave very similar transport velocities with an average of 2.24cm/h for the CDE and 2.20cm/h for the STM. The average dispersion coefficient values were 3.48 [cm.sup.2]/h for the CDE and 3.29 [cm.sup.2]/h for the STM.

Dispersivity changed with distance (Table 2). This might be due to the vertical and horizontal soil heterogeneity, since the soil cores were taken from different positions at different times, which can result in different transport parameters. Several studies have shown that dispersivity increases with travel distance/time in the saturated zone (Sposito et al. 1986). In the unsaturated zone, it is less clear whether dispersivity increases with travel distance/time (Persson and Berndtsson 1999).

The standard deviation for the velocities shown in Table 2 were relatively small when compared, for example, with the values of [[sigma]s.ub.v] = 1.25 obtained by Biggar and Nielsen (1976) for ponded infiltration condition. This behaviour and the small dispersion coefficients can be taken as an indication that dispersion is limited during this experiment and that the transport could be described adequately with the convection--dispersion model.

Water and bromide velocity

One of the objectives of this study was to investigate the nature of solute transport in the Bainsvlei sandy loam soil, i.e. whether the mean convective movement of a solute can be described by piston flow. If the transport of [Br.sup.-] in the soil solution is like a piston displacing fluid, then the pore water velocity, [v.sub.w], will be the velocity of movement of [Br.sup.-]. This piston behaviour is well explained by, for example, Jury et al. (1991) and Nachabe and Morel-Seytoux (1995). If there is preferential flow, the infiltrating water and solute may bypass (not displace) the initial soil water. The possibilities of bypass flow was determined by comparing bromide velocities with pore-water velocity.

Pore-water velocity

If piston flow is assumed at this site, expected solute velocities could be estimated by dividing the Darcian flux by the average soil water content. The Darcian flux was calculated as the net water applied divided by the duration of application. The amount of water applied during the experimental period of 4 days (96h) was 519.4mm, which gives q = 5.41 mm/h. The average volumetric water content of the soil profile was 0.259. The pore-water velocity, calculated using Eqn 4, was 2.08 cm/h.

The pore-water velocity was also determined using the least-square inversion package CXTFIT (Toride et al. 1995). Using this approach, the average pore-water velocities for the CDE and STM were 2.24 cm/h and 2.20 cm/h, respectively (Table 2). Therefore, the average pore-water velocity [v.sub.w] (from water balance, CDE, and STM) was 2.17 cm/h.

Velocity of bromide concentration peak

The velocity calculated from the arrival times of the concentration peaks of the [Br.sup.-] pulse at various depths in the soil profile, Eqn 5, is presented in Table 3. It is clear that the mean measured bromide velocity of 2.05 cm/h is very similar to the theoretical pore-water velocity of 2.08 cm/h.

Bromide centre of mass velocity

The velocity of the [Br.sup.-] centre of mass through the soil profile, calculated using Eqn 7, is presented in Table 4 where [bar.Z] is the depth of the centre of mass of the [Br.sup.-] concentration profile. The average centre of mass velocity of the concentration profile is 2.02 cm/h, which is again very similar to the theoretical pore-water velocity calculated using Eqn 4.

Based on the method in which solute velocity is determined, different and sometimes contradictory results are obtained regarding the possibility of describing field solute transport by simple piston flow models (Biggar and Nielsen 1076; Starr et al. 1978; Rice et al. 1991; Butters and Jury 1989; Ellsworth et al. 1991). Biggar and Nielsen (1976) found good agreement between the piton flow velocity and the solute velocity ([v.sub.s]/[v.sub.w] = 1.0) under steady ponding condition in a clay loam soil. Rice et al. (1991) found that under intermittent ponding conditions, the solute velocity was almost 5 times larger than the piston flow velocity in a sandy loam soil. Starr et al. (1978) conducted a study in a layered sandy loam soil and found that the solute velocity based on arrival time of the peak concentration was only about half as large as the piston flow prediction under ponded conditions. Similar conditions exist under non-ponding situations. Butters and Jury (1989) studied leaching of [Br.sup.-] in a loamy sand soil using bidaily sprinkler irrigation. They found the solute velocity (calculated as the ratio of the depth of observation to the mean solute pulse arrival time) to be substantially less than the piston flow velocity in the top 1.8 m, below which the estimates agreed. The study of Ellsworth et al. (1991) involved leaching of solute using trickle irrigation in a loamy sand soil, and the solute velocity (estimated by the location of the centre of mass of the solute plume) agreed almost perfectly with the local piston flow estimates. Wild and Babiker (1976) found peak velocity of the resident concentration pulse much smaller than the centre of mass velocity. These research studies illustrate the complexity of the field regime, which can obscure the identification of even the most basic of solute transport processes, the mean convection rate.

In this study, the ratios of [v.sub.s] to [v.sub.w] were 0.98 and 0.93 when the [Br.sup.-] velocity [v.sub.s] was determined from the movement of the [Br.sup.-] concentration peak and centre of mass, respectively. These ratios (average 0.96) indicate that there was no preferential transport of [Br.sup.-] in this soil during this steady-state experiment. The soil was relatively homogeneous and of a low clay content. It can be classified as weakly structured soil and there is less possibility of preferential flow in such soils.

Conclusions

The solute velocities determined from the concentration peak movement and the movement of the solute centre of mass were in good agreement with the average pore-water velocity determined from soil water balance and fitting concentration data to theoretical models, indicating that piston flow can describe the transport process. The Bainsvlei soil is a weakly structured fine sandy loam and preferential flow paths are less likely to develop under this steady-state condition.

The deterministic convection--dispersion model and the STM performed almost equally well in estimating the transport parameters and simulating the solute concentrations as functions of time and space. No significant differences were observed between the results obtained by these methods.

The uniform manner in which the water and solute was applied and subsequently leached, the consistency of sampling technique, and the near 100% average recovery of solutes contributed towards compiling a comprehensive dataset for model development and verification.
Table 1. Particle size distribution and
bulk density of the soil profile

 Sand (%)

Depth
(cm) Coarse Medium Fine Total

 0-20 0.4 6.8 63.8 91
 20-40 0.4 7.7 78.9 87
 40-60 0.3 5.5 70.2 74
 60-80 0.4 5.5 72.1 76
 80-100 0.2 4.8 73.0 76
100-120 0.3 4.8 73.9 78
120-140 0.3 5.4 71.3 76
140-160 0.2 2.8 73.0 76

Depth Silt Clay Soil Bulk density
(cm) (%) (%) texture (Mg/[m.sup.3]) (A)

 0-20 4 5 Sand 1.64 [+ or -] 0.05
 20-40 2 11 Loamy sand 1.72 [+ or -] 0.07
 40-60 6 20 Sandy loam 1.62 [+ or -] 0.04
 60-80 6 18 Sandy loam 1.58 [+ or -] 0.05
 80-100 4 20 Sandy loam 1.64 [+ or -] 0.06
100-120 4 18 Sandy loam 1.67 [+ or -] 0.08
120-140 4 20 Sandy loam 1.68 [+ or -] 0.08
140-160 4 20 Sandy loam 1.71 [+ or -] 0.04

(A) Mean of 8 values [+ or -] standard deviation.

Table 2. Deterministic (v, D, [lambda]) and stochastic
((v), (D), [lambda]) transport parameters determined
from the concentration profiles

 Stream
 tube
 Convection--dispersion model model

Time (h) D
and drainage v ([cm.sup.2] [lambda] (v)
(mm) (cm/h) /h]) (cm) [r.sup.2] (cm/h)

 4.8 (26.0) 2.75 3.15 1.15 1.000 2.10
 9.8 (53.0) 2.94 2.31 0.79 0.999 2.94
17.8 (96.3) 2.95 1.12 0.38 0.982 3.00
27.5 (148.8) 2.40 3.01 1.25 0.950 2.40
37.0 (200.2) 2.15 2.60 1.21 0.995 2.16
46.5 (251.6) 1.70 7.96 4.68 0.804 1.74
56.0 (303.0) 1.89 4.61 2.44 0.971 1.89
66.0 (357.1) 1.96 2.79 1.42 0.778 2.00
77.0 (416.6) 1.85 5.27 2.85 0.989 1.88
96.0 (519.4) 1.84 1.96 1.07 0.961 1.84

 Stream tube model

Time (h) D
and drainage ([cm.sup.2] [lambda] [[sigma].
(mm) /h]) (cm) sub.v] [r.sup.2]

 4.8 (26.0) 2.31 1.10 0.27 1.000
 9.8 (53.0) 2.40 0.82 0.01 0.998
17.8 (96.3) 1.02 0.34 0.05 0.982
27.5 (148.8) 2.87 1.20 0.15 0.950
37.0 (200.2) 2.03 0.94 0.08 0.996
46.5 (251.6) 7.83 4.50 0.12 0.808
56.0 (303.0) 4.74 2.51 0.01 0.971
66.0 (357.1) 2.67 1.34 0.04 0.778
77.0 (416.6) 5.13 2.72 0.05 0.989
96.0 (519.4) 1.94 1.05 0.02 0.961

Table 3. Bromide concentration peak velocity calculated using Eqn 5

 Difference

 Time, [z.sub.i] - [t.sub.i] - Velocity,
Depth, z t [z.sub.i-1] [t.sub.i-1] [v.sub.s] [v.sub.s]/
 (cm) (h) (cm) (h) (cm/h) [v.sub.w]

 10 2.0
 30 9.8 20 7.8 2.56 1.2
 50 17.8 20 8.0 2.50 1.2
 70 27.5 20 9.7 2.06 1.0
 90 37.0 20 9.5 2.11 1.0
110 56.0 20 19.0 1.05 0.5
130 66.0 20 10.0 2.00 0.9
Average 2.05 0.97

Table 4. Bromide centre-of-mass velocity calculated using Eqn 7

Time t (h) 4.8 9.8 17.8 27.5
[bar.Z] 11.5 31.0 50.7 67.9
[DELTA][bar.Z] = 19.5 19.7 17.2
 [bar.Z]([t.sub.i]) -
 [bar.Z]([t.sub.i-1])
[DELTA]t = [t.sub.i] - 5.0 8.0 9.7
 [t.sub.i-1]
[v.sub.s] = [DELTA] 3.90 2.46 1.77
 [bar.Z]/[DELTA]t

Time t (h) 37.0 46.5 56.0 66.0
[bar.Z] 82.8 88.8 106.0 126.2
[DELTA][bar.Z] = 14.9 6.1 17.2 20.1
 [bar.Z]([t.sub.i]) -
 [bar.Z]([t.sub.i-1])
[DELTA]t = [t.sub.i] - 9.5 9.5 9.5 10.0
 [t.sub.i-1]
[v.sub.s] = [DELTA] 1.57 0.64 1.81 2.01
 [bar.Z]/[DELTA]t


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Manuscript received 29 January 2003, accepted 20 September 2004

Ketema Tilahun (A,C), J. F. Botha (A), and A. T. P. Bennie (B)

(A) Institute for Groundwater Studies, The University of the Free State, PO Box 339, Bloemfontein 9300, Republic of South Africa.

(B) Department of Soil, Crop, and Climate Sciences, The University of the Free State, PO Box 339, Bloemfontein 9300, Republic of South Africa.

(C) Corresponding author. Present address: Alemaya University, PO Box 138, Dire Dawa, Ethiopia. Email: ketematilahun@yahoo.com
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Author:Tilahun, Ketema; Botha, J.F.; Bennie, A.T.P.
Publication:Australian Journal of Soil Research
Geographic Code:8AUST
Date:Jan 1, 2005
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