INVERSE ESTIMATION OF SOIL HYDRAULIC AND SOLUTE
TRANSPORT PARAMETERS FROM TRANSIENT FIELD
EXPERIMENTS: HETEROGENEOUS SOIL
F. Abbasi, D. Jacques, J. Simunek, J. Feyen, M. Th. van Genuchten
ABSTRACT. While inverse parameter estimation techniques for determining key parameters affecting water flow and solutetransport are becoming increasingly common in saturated and unsaturated zone studies, their application to practicalproblems, such as irrigation, have received relatively little attention. In this article, we used the Levenberg–Marquardt
optimization algorithm in combination with the HYDRUS–2D numerical code to estimate soil hydraulic and solute transportparameters of several soil horizons below experimental furrows. Three experiments were carried out, each of the same
duration but with different amounts of water and solutes resulting from 6, 10, and 14 cm water depths in the furrows. Two moreexperiments were performed with the same amounts of applied water and solute and, consequently, for different durations,on furrows with depths of 6 and 10 cm of water. We first used a scaling method to characterize spatial variability in the soil
hydraulic properties, and then simultaneously estimated the saturated hydraulic conductivity (Ks ) and the longitudinal
dispersivity (DL ) for the different horizons. Model predictions showed only minor improvements over those previously
obtained assuming homogeneous soil profiles. In an effort to improve the predictions, we also carried out a two–step,
sequential optimization in which we first estimated the soil hydraulic parameters followed by estimation of the solutetransport parameters. This approach allowed us to include additional parameters in the optimization process. A sensitivity
analysis was performed to determine the most sensitive hydraulic and solute transport parameters. Soil water contents were
found to be most sensitive to the n parameter in van Genuchten’s soil hydraulic model, followed by the saturated water content(qs ), while solute concentrations were most affected by qs and DL . For these reasons, we estimated qs and n for the various
soil horizons of the sequential optimization process during the first step, and only DL during the second step. Sequential
estimation somewhat improved predictions of the cumulative infiltration rates during the first irrigation event. It alsosignificantly improved descriptions of the soil water content, particularly of the upper horizons, as compared to those obtainedusing simultaneous estimation, whereas deep percolation rates of water did not improve. Solute concentrations in the soil
profiles were predicted equally well with both optimization approaches.
Keywords. Furrow irrigation, Heterogeneous soil, Inverse solution, Scaling, Solute transport, Water flow.

Understanding solute transport processes between
the soil surface and the groundwater table isessential for limiting or effectively managing soil
and groundwater pollution. Unfortunately, solutetransport, particularly at the field scale under transientconditions, can be very complex because of the presence ofseveral mutually interactive soil physical, chemical, andbiological processes that may vary substantially over space
and time. Previous studies have shown that water flow andsolute transport processes are influenced by soil type
(Vanderborght et al., 1997, 2001), flow rate (Vanderborght et

Article was submitted for review in July 2002; approved for publicationby the Soil & Water Division of ASAE in May 2003.
The authors are Fariborz Abbasi, Soil Physicist, and Jan Feyen,
Professor, Institute for Land and Water Management, Katholieke
Universiteit Leuven, Belgium; Deiderik Jacques, Soil Physicist, Waste andDisposal Department, SCK–CEN, Mol, Belgium; and Jirka Simunek, Soil
Physicist, and Martinus Th. van Genuchten, Research Leader,
USDA–ARS, George E. Brown Jr. Salinity Laboratory, Riverside,
California. Corresponding author: Jan Feyen, Institute for Land andWater Management, Vital Decosterstraat 102, 3000–Leuven, Belgium;
phone: +32–16–329756; fax: +32–16–329760; e–mail: jan.feyen@agr.
kuleuven.ac.be.

al., 1997; Forrer et al., 1999; Wildenschild et al., 2001), flow
regime (Bowman and Rice, 1986; Jaynes et al., 1988),
irrigation method (Ghodrati and Jury, 1990; Troiano et al.,
1993; Flury et al., 1994), surface flow depth (Abbasi et al.,2003a), preferential flow (Flury et al., 1994; vanWeesenbeeck and Kachanoski, 1994), soil heterogeneity(Roth et al., 1991; Jacques, 2000), initial conditions (Whiteet al., 1986; Steenhuis and Muck, 1988), and/or boundary
conditions (Russo et al., 1994a, 1994b).
Some field studies suggest that solute transport inheterogeneous soils can be described with the classicalconvection–dispersion equation (CDE) (e.g. Roth et al.,
1991; Butters and Jury, 1989), while others show considerable deviations between model predictions and field data
(e.g. Snow et al., 1994; de Vos, 1997; Jacques et al., 1998).
Roth at al. (1991) found that the CDE model described theirtracer data well for transport distances less than 1 m, whereas
the findings of Butters and Jury (1989) showed goodcorrespondence only after a travel distance of 4 m. Several
commonly observed features of solute transport, such as
multiple peak breakthrough curves and fingered flow, cannot
be described with the CDE model. Ventrella et al. (2000) andJacques et al. (2002) found that the mobile–immobile (MIM)
approach described observed breakthrough curves (BTC)

Transactions of the ASAE
Vol. 46(4): 1097–1111

2003 American Society of Agricultural Engineers ISSN 0001–2351

1097

better than the CDE model for chloride transport inheterogeneous soils under natural boundary conditions. By
contrast, Jacques et al. (1998, 2002) and Snow et al. (1994),
among others, found that the CDE and MIM models bothunderestimated the rate of transport (early breakthrough) of
their observed data.
Identification and accurate simulation of field–scalewater flow and solute transport processes, especially whensubject to natural boundary conditions, is a complex task
since many input parameters are difficult to measure at thedesired scale, while their temporal and spatial variability
often cannot be easily described. However, combining
numerical solutions of the governing flow and transport
equations with inverse optimization algorithms and detailed
measurement of different variables is a promising approachfor process and parameter identification (Jacques et al.,2002). A common way to estimate flow and transportparameters is by using inverse optimization techniques
whereby differences between measured and model–predicted values are minimized. Inverse methods are increasingly being used to estimate soil hydraulic parameters (e.g.,
Gribb, 1996; Simunek and van Genuchten, 1996, 1997;
Kodesova et al., 1998, 1999; Simunek et al., 2001; among
others), or in combination with solute transport or root water
uptake parameters (Ventrella et al., 2000; Inoue et al., 2000;
Wildenschild et al., 2001; Vrugt et al., 2001; Jacques et al.,
2002; Simunek et al., 2002). In spite of considerable efforts
in estimating soil hydraulic and transport properties bymeans of inverse modeling, their application to practicalproblems, particularly irrigation, has received a little attention.
Abbasi et al. (2003b) recently applied inverse methods toseveral 2–D furrow irrigation experiments assuming homogeneous soil profiles, and simultaneously estimated thesaturated hydraulic conductivity (Ks ) and several solutetransport parameters. For solute transport, they used both theequilibrium (CDE) and nonequilibrium (MIM) transportmodels. In this study, we revisit the analysis by Abbasi et al.
(2003b) assuming a layered profile and also consider
sequential estimation. Since the MIM model did not lead tosignificant improvements in the model predictions for the
homogeneous case (Abbasi et al., 2003b), we limit ourselvesin this study to only the equilibrium transport model. Hence,the purpose of this article is to use simultaneous andsequential optimization to estimate the soil hydraulic andsolute transport parameters for different soil horizons fromseveral furrow flow and transport experiments. A scaling
method will be used to characterize spatial variability in thesoil hydraulic properties. Simulations and inverse analyses
will be carried out using the HYDRUS–2D variably saturated
flow/transport code of Simunek et al. (1999), which includes
the Levenberg–Marquardt parameter optimization algorithm. Model predictions will be compared with field–measured cumulative infiltration rates, soil water contents,
bromide concentrations, and both water and solute deep
percolation rates.

MATERIALS AND METHODSFIELD EXPERIMENTS
Field experiments were conducted at the Maricopa
Agricultural Center (MAC) in Phoenix, Arizona, on a bare

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Casa Grande sandy loam soil on short blocked–end furrows.
Five soil horizons were identified in the experimental area:
a 33 cm thick surface Ap horizon, followed by a Btkn1
horizon between 33 and 58 cm having a calcium carbonate
content that increased with depth, a Btkn2 horizon between
58 and 71 cm, a Btkn3 horizon between 71 and 125 cm, anda 2Bkn horizon between 125 and 152 cm.
Two series of experiments were carried out. We first
performed same–duration (SD) experiments that involved
three flow depths (6, 10, and 14 cm), each with two
irrigations, 10 days apart. The first irrigation took place in
two phases. During the first phase, water was applied for
60 min (the same for all three experiments) to wet the soil
profile. During the second phase, water amended with CaBr2
was applied for 30 min, again the same for the three plots. The
second irrigation lasted 90 min using the same depths ofunamended water as for the first irrigation.
The second series of experiments involved similar
amounts of applied water and solutes (SWS). The amount ofapplied water was the same as that infiltrated for the 14 cm
depth treatment of the SD experiments. Water levels of 6 and10 cm were used in this case. The first irrigation was again
carried out in two steps, similarly as for the SD scenarios,with unamended water being applied first followed bybromide–amended water. However, the irrigation times were
adjusted for each of the water levels such that predefined
amounts of water infiltrated (i.e., the amount given during theprevious 14 cm SD experiment). The second irrigation used
unamended water applied with the same water levels as forthe first irrigation, but with times adjusted so that again thesame amount of water infiltrated during the second irrigation
as for the 14 cm depth treatment in the first set of SD
experiments.
A set of five neutron probe access tubes (each 3.3 m indepth) was installed to measure soil water contents at
different locations perpendicular to the axis of the monitored
furrows. Readings were taken at depths of 20, 40, 60, 80, 100,
140, 180, 220, and 260 cm before each irrigation to provide
initial conditions, immediately after each irrigation, thenhourly up to 6 hours after each irrigation, and subsequently
each 3 hours up to 24 hours. Measurements were later taken
3 to 4 times per day up to 3 days after irrigation, and then 1
to 2 times per day up to the next irrigation. Water contents ofthe surface layer (0 to 30 cm) were also measured using asite–calibrated time domain reflectometry (TDR) probe. The
TDR and neutron probe readings were taken at the sametimes as indicated above.
Soil samples for analyzing bromide concentrations weretaken manually, at depths (up to 180 cm) and locations
corresponding to the neutron probe access tube measurements, four times during the experiments, i.e., prior to theexperiments as initial values, 5 days after the first irrigation,and 6 and 20 days after the second irrigation. Soil extractions(1:1 weight:volume) were made and analyzed for bromide
with a Lachat QuikChem flow injection analyzer using
standard colorimetric procedures.
The SD and SWS experiments started on 30 January and
26 February 2001, respectively, and each lasted 30 days.
Further information about the experiments can be found in
Abbasi et al. (2003a, 2003b).

TRANSACTIONS OF THE ASAEGOVERNING WATER FLOW AND SOLUTE TRANSPORTEQUATIONSThe Richards equation (Richards, 1931) describing two–
dimensional isothermal Darcian water flow in a variably
saturated rigid porous medium and the physical equilibrium
convection–dispersion equation (CDE) were used as thegoverning equations for water flow and solute transport,
respectively. The CDE equation was previously found to besuitable for this soil (Abbasi et al., 2003b). No significant
improvements in predictions of calculated concentrations
were obtained when the physical nonequilibrium MIM
model was used. Calculations based on the governing flow
and transport equations were carried out using the HYDRUS–2D code (Simunek et al., 1999), and the minimization
problem for parameter optimization was carried out using theLevenberg–Marquardt nonlinear weighted least–squares approach (Marquardt, 1963) as implemented in HYDRUS–2D.
Measured bromide concentrations and soil water contents
before the experiments were used as initial conditions within
the soil profile below the furrows. A constant pressure head(surface ponding) was specified as the upper boundary
condition in the furrow during irrigation, while an atmospheric boundary condition was used after irrigation during
the redistribution phase. A Cauchy condition was used for theupper boundary condition for solute transport, and free–drainage conditions for both water and solute were used at thelower boundary of the domain. No–flux boundary conditions
were applied to both sides of the flow domain. The furrows
were 100 cm wide, while the soil profile was assumed to be
100 cm deep in our simulations.
CHARACTERIZATION OF THE SOIL HYDRAULIC PROPERTIES
Spatial variability in the soil hydraulic properties remains
one of the major problems in field–scale analyses of waterflow and solute transport. One way of characterizing thisspatial variability is by scaling locally measured soil water
retention and hydraulic conductivity data to reference curves
using scaling factors (Hopmans, 1989; Vogel et al., 1991;
Eching et al., 1994). This concept is known as functional
similarity (Simmons et al., 1979). In this article, laboratory–measured soil water retention curves (no hydraulic conductivity measurements were made) for the different horizons
were scaled using linear scaling relationships proposed by
Vogel et al. (1991). Neglecting time variability in the soil
hydraulic properties, the scaling factors and their relationships to the reference curves are as follows (Vogel et al.,1991):
θ(x, h) = θr (x) + αθ (x)[θ*(h*) − θ*r ](1)

h = αh (x)h*

(2)

where
q
= water content (L3 L–3)h= pressure head (L)q*(h*) = reference retention curve
qr= residual soil water contents (L3 L–3) of the local
retention curves
q*r= residual soil water contents (L3 L–3) of thereference retention curvesx= vector of spatial coordinatesaq= scaling factors for water contentsah= scaling factors for pressure heads.

Vol. 46(4): 1097–1111

The reference soil water retention curve, q*(h*), is
described using the closed–form equation of van Genuchten(1980):
θ* ( h * ) = θ *r +

θ*s − θ *r
(1 + αh *

n

)m

(3)

and the unsaturated soil hydraulic conductivity function,
K(h*), using the pore–size distribution model of Mualem(1976), as follows (van Genuchten, 1980):
K(h*) = Ks Se l [1 – (1 – Se 1/m)m]2

(4)

where
Se =

θ* − θ*r ,θ*s − θ*r1
m = 1− ,
n

n>1

(5)

and
q*s
= saturated water content (L3 L–3)
Se
= relative saturation (dimensionless)m, n, l, a = empirical parameters (m, n, and l dimension–
less; a in L–1).
The parameters of the reference water retention curve andthe scaling factors aq and ah were determined by minimizing
the squared differences between the reference water contents
and scaled measured water contents, subject to the constraintthat the average of aq and ah be unity (e.g., Warrick et al.,
1977; Vogel et al., 1991; Jacques, 2000). In this study, q*s andq*r were fixed at the average of the fitted local θr and θsvalues, respectively, while l in equation 4 was assumed to be
0.5 (Mualem, 1976). Undisturbed soil samples (6 cm longand 5.4 cm diameter) were collected from different soil
horizons up to a depth of 100 cm to estimate parameters in thevan Genuchten (VG) soil hydraulic model. The scaling
analysis was carried out separately for each of the soil
horizons.
SENSITIVITY ANALYSIS
Prior to analyzing the actual field data, a sensitivity
analysis was performed to determine the most sensitive soil
hydraulic and solute transport parameters. The sensitivity ofthe measured variables to various model parameters wasevaluated using sensitivity coefficients. Sensitivity coefficients, s(z,t,bj ), were determined according to Simunek et al.
(1998) as follows:s(z, t, b j ) = Y (b + ∆be j )−Y (b)

(6)

where
s(z,t,bj ) = change in variable Y (the soil water content or
the solute concentration) corresponding to a 1%change in parameter bj
ej= jth unit vectorDb= 0.01b.
Sensitivity coefficients were multiplied by 100 in equation 6 to avoid very small numerical values.
The sensitivity analysis was performed only for plot 3
assuming experimental conditions that were between the SD
and SWS experiments, and for a duration of up to 10 days
after the first irrigation event. We assumed 60 min irrigation
with tap water, followed by a 30 min solute pulse application,
as done during the field experiments. We evaluated sensitivi–

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Table 1. Reference van Genuchten soil water retention
parameters for the different soil horizons.
Soil
Depth
θr
θs
α
n
Horizon(cm3 cm–3) (cm3 cm–3) (cm–1)
(m)
(–)

[a]

Ap
Btkn1
Btkn2
Btkn3

0–0.330.33–0.58
0.58–0.71
0.71–1.25 [a]

0.104
0.111
0.107
0.103

0.374
0.444
0.420
0.412

0.035
0.063
0.040
0.047

1.611
1.539
1.555
1.554

CV (%)

––

2.93

6.10

22.9

1.75

Soil samples were collected at a depth of 0.8.

ties of the soil water contents and solute concentrations to all
retention parameters (assuming their scaled values as given
in table 1), the saturated hydraulic conductivity (Ks ), and thelongitudinal (DL ) and transverse (DT ) dispersivities in the
CDE transport model. Estimated values from our previous
study (Abbasi et al., 2003b) were used for DL , DT, and Ks ,while the exponent l in Mualem’s expression was taken to be
0.5 during the sensitivity analysis.
SELECTION OF PARAMETERS FOR OPTIMIZATION
The main objective of this study was to estimate soil
hydraulic and solute transport parameters of the different soil
horizons using two different optimization approaches. We
tried to limit the number of unknown parameters for thefollowing reasons: (1) uncertainty is generally reduced andnon–physical parameter values are more likely avoided whena limited number of parameters is used in optimization
studies (e.g., Simunek et al., 2001); (2) data showed only
minor changes in water contents and solute concentrations
below a depth of 100 cm, particularly for the same duration(SD) experiments (Abbasi et al., 2003a); (3) the majority ofmeasurements were made in the upper 100 cm of the soil
profiles, and only a limited number of measurements weremade at deeper depths, thus providing insufficient information to optimize parameters for the lower horizons; and(4) simulation times needed to run the inverse option of
HYDRUS–2D for our problem were considerable. Optimization of a complete set of parameters for our experiments took
several days using a Pentium III PC.
For the simultaneous parameter estimation approach,twelve additional parameters must be optimized for the first
four identified soil horizons (four values each of Ks , DL , andDT ) when assuming that the retention properties are fully
characterized by the reference scaling curves. For the reasonsstated above, we limited the number of optimized parameters
to less than ten. Previous optimization results assuming
homogeneous soil profiles indicated low confidence in the
optimized DT values (broad confidence intervals). Thisparameter was also the least sensitive parameter among all
soil hydraulic and solute transport parameters in oursensitivity analysis (see results below). For these reasons, we
did not include DT in the optimization process, but rather
fixed it at values obtained for the homogeneous profiles. The
remaining two parameters (Ks and DL ) were considered to be
the most important and hence were optimized for each soil
horizon separately (eight parameters in total). Among the soil
hydraulic parameters, only Ks was optimized for the different
soil horizons since we did not have direct measurement for
Ks . Other soil hydraulic parameters during optimization were
fixed for each soil horizon at the values given in table 1.

1100

As will be discussed in the following sections, simultaneous estimation of the Ks and DL parameters for the different
horizons produced only minor improvements in the model
predictions as compared to results obtained from ourprevious analysis assuming homogeneous soil profiles. We
wanted to know if including additional soil hydraulicparameters would improve the predictions. However, as
discussed above, it is not recommended to estimate too many
parameters simultaneously due to instability of the inverse
optimization methods. To overcome this problem, we also
used sequential estimation, in which the soil hydraulicparameters are estimated first, followed by estimation of thesolute transport parameters. In our study, we used two
sequential steps. During the first step, the saturated soil watercontent (qs ) and the n parameter in the VG soil hydraulicmodel, which were found to be the most sensitive parametersin our sensitivity analysis (discussed below), were optimized
for the different soil horizons. During the second step, only
DL values were estimated. We did not include DT in thesecond step since this often caused instability in the
optimizations, with the model not converging to a uniquesolution for some experiments.
As indicated earlier, the optimization process was carriedout using the Levenberg–Marquardt optimization algorithm(Marquardt, 1963), leading to minimization of an objectivefunction, as discussed in detail by Abbasi et al. (2003b).

RESULTS AND DISCUSSIONSCALING OF RETENTION PROPERTIES
Measured (unscaled) and scaled soil water retention
curves for the various soil horizons are shown in figure 1.
Parameters of the reference curves and their coefficients ofvariations (CV) are given in table 1. The effect of scaling on
the retention curves was more noticeable for the first andsecond soil horizons since more soil samples had beencollected from these two horizons, thus revealing more
spatial variability. The qr and n values remained almost
constant with depth and showed relatively low CVs, while thea parameter exhibited much more variability, with a CV of
22.9%. Notice that qs for the first horizon was somewhat
lower than for the other horizons. The CVs for n and a before
scaling were 7% and 50%, respectively. These results agree
well with findings of Mallants et al. (1996a), who also foundmore variability in a as compared to n.
SENSITIVITY ANALYSESThe sensitivities of the soil water contents and soluteconcentrations to soil hydraulic and transport parameters are
given in figures 2a and 2b, respectively. Results are only
shown for the vertical below the bottom of the furrow down
to a depth of 100 cm since most changes generally occur inthis region (note that the furrow depth was approximately
20 cm). Overall, water contents (fig. 2a) in the soil profile
showed the highest sensitivity near the moisture front as it
gradually moved to deeper depths with time. However, somesensitivity was also observed near the soil surface during thedrier conditions starting about 8 days after water application.
The largest sensitivity coefficients were observed in thelower part of the soil profile about 10 days after the irrigation.
The most sensitive parameter was n, having a maximum
sensitivity coefficient of 1.5, followed by qs . The exponent

TRANSACTIONS OF THE ASAE

0.5

Measured data

Soil water content(cm 3 cm – 3 )0.4(cm 3 cm – 3 )Soil water content

0.5
0.3
0.2
0.1
0

(a)

1

10

100

1000

Reference curve
Scaled data

0.4
0.3
0.2
0.1

(a)

0
1

10000

10

0.3
0.2
0.1
0

(b)

1

10

100

1000

10000

Soil water content(cm 3 cm – 3 )Soil water content(cm 3 cm – 3 )0.5

Measured data

0.4

0.4
0.3
0.2
0.1

(b)

0
1

10

1000

10000

0.5
Measured data

0.4

Soil water content(cm 3 cm – 3 )Soil water content(cm 3 cm – 3 )100

Pressure head (cm)

0.5
0.3
0.2
0.1

(c)

0
1

10

100

1000

Reference curve
Scaled data

0.4
0.3
0.2
0.1
0

10000

(c)

1

10

0.5

0.5

0.3
0.2
0.1

(d)

1

10

100

1000

10000

Pressure head (cm)Soil water content(cm 3 cm – 3 )Measured data

0.4

100

1000

10000

Pressure head (cm)Pressure head (cm)Soil water content(cm 3 cm – 3 )10000

Reference curve
Scaled data

Pressure head (cm)

0

1000

Pressure head (cm)Pressure head (cm)

0.5

100

Reference curve
Scaled data

0.4
0.3
0.2
0.1

(d)

0
1

10

100

1000

10000

Pressure head (cm)Figure 1. Measured (unscaled), scaled, and reference (fitted) soil water retention curves for the different soil horizons: (a) 0 to 0.33 m, (b) 0.33 to 0.58 m,(c) 0.58 to 0.71 m, and (d) 0.71 to 1.25 m.

l in the capillary model of Mualem (1976) was the leastsensitive parameter. Contrary to the water contents, soluteconcentration sensitivities were more pronounced near thesoil surface and gradually increased with time as the soil
surface dried out (fig. 2b). The saturated water content (qs )was the most sensitive parameter, followed by the saturated
hydraulic conductivity (Ks ). The longitudinal dispersivity(DL ) appeared to be much more sensitive than the transverse
dispersivity (DT ). This was not consistent with findings by
Forrer et al. (1999), who found that DT was more sensitive
than DL . Note that white zones in the different subplots offigure 2 are representative of small but non–zero values.

Vol. 46(4): 1097–1111

PARAMETER OPTIMIZATION
Simultaneous Optimization
Simultaneously optimized Ks and DL values for thevarious soil horizons and experimental plots are summarizedin table 2. Corresponding values obtained from analyses
assuming homogeneous soil profiles (Abbasi et al., 2003b)
are also included in table 2 for comparison. The Ks values forthe first horizon for the different experiments corresponded
well with the optimized values for an equivalent homogeneous soil profile and those predicted with pedotransfer
functions for a sandy loam soil (Schaap and Leij, 1998). The
Ks values for the other horizons were slightly higher than for
the first horizon (except for plot 1) and somewhat larger thanthose reported by Schaap and Leij (1998). However, they
were consistent with those given by Carsel and Parrish (1988)

1101

θr

θs

α

n

Ks

l

Figure 2a. Soil water content sensitivities to different soil hydraulic parameters as evaluated for plot 3, plotted along a vertical from the bottom of the
furrow (20 cm below the soil surface) to a depth of 100 cm.

θ r

θs

Ks

l

α

D

n

L

D

T

Figure 2b. Solute concentration sensitivities to different soil hydraulic and solute transport parameters as evaluated for plot 3, plotted along a verticalfrom the bottom of the furrow (20 cm below the soil surface) to a depth of 100 cm.

for corresponding soil textures. We believe that, rather thanthe deeper horizons having inherently higher saturated
conductivities, the higher estimated Ks values of thesehorizons were more likely due to having fewer data fromthese depths. The Ks values of the upper horizons should be
relatively well defined since they were estimated from the1102

measured infiltration rates and measured soil water contents.
However, the infiltration rate at the soil surface should not
help much in estimating Ks for the deeper layers, for which
the measured water content is the main information. Havingsoil water pressure head measurements probably would havehelped to improve the estimation of Ks for the lower horizons.

TRANSACTIONS OF THE ASAE

Table 2. Summary of the simultaneously optimized saturated hydraulicconductivity (Ks ) and longitudinal dispersivity (DL ) parameters for thedifferent soil horizons and experimental plots. Corresponding
parameters for analyses assuming homogeneous soil profiles are given
in parentheses. Also included are values of the minimized objectivefunction (SSQ) and the R2 values of the optimizations.
Depth
Ks
DL
SSQ
R2
(m)
(cm min–1)
(cm)
(–)
(–)
Plot 1 (6 cm)[a]0–0.330.33–0.58
0.58–0.71
0.71–1.25(0.0389)
0.0311
0.04840.03730.0265(18.30)
34.61
0.03
3.65
0.02

1.56 (1.69)

0.989 (0.988)

Plot 2 (10 cm)
0–0.330.33–0.58
0.58–0.71
0.71–1.25(0.0392)
0.02900.0711
0.06350.0547(17.0)
17.39
0.40
37.76
49.20

2.29 (4.22)

2.29 (4.22)

Plot 3 (14 cm)
0–0.330.33–0.58
0.58–0.71
0.71–1.25(0.0497)
0.03800.07360.07090.0560(12.50)
15.11
0.93
3.25
0.50

5.11 (6.32)

0.984 (0.984)

Plot 4 (6 cm)
0–0.330.33–0.58
0.58–0.71
0.71–1.25(0.0436)
0.02310.08890.06120.0754(8.30)
13.90
7.67
32.73
0.35

0.984 (0.984) 0.996 (0.995)

Plot 5 (10 cm)
0–0.330.33–0.58
0.58–0.71
0.71–1.25(0.0996)0.03970.08350.06350.0392(2.60)
5.73
9.57
24.69
13.91

3.10 (3.50)

CV (%)[a]0.990 (0.988)

36.1 (47.5) 102.5 (55.1)

Water level in the furrow.

Much of the differences in the optimized Ks values for the
homogeneous and heterogeneous (layered) soil profiles were
due to differences in the VG parameters (a and n inparticular) caused by scaling of the soil water retention data.
Despite the relatively large differences in estimated Ks
values, SSQ values of the objective function (table 2) did not
significantly decrease (except for plot 4 and somewhat forplot 2). This was mostly because of the low sensitivity of thesoil water contents to Ks (fig. 2a).
The optimized DL values (table 2) varied with depth, most
likely because of variability in other soil properties. The DL
values differed significantly from one horizon/plot toanother, ranging from 0.02 to 49.20 cm. This somewhat
erratic behavior is similar to results obtained from ouranalyses assuming homogeneous profiles (Abbasi et al.,2003b), for other experimental columns (e.g., Mallants et al.,
1994, 1996b), and for many other field–scale studies (e.g.,
Forrer et al., 1999; Jacques, 2000). The erratic behavior of DL
in field studies appears to be related to spatial variability inthe soil water content and the presence presumably of more
irregular flow patterns. Unlike results for the homogeneous
profiles, the estimated DL values did not show a noticeable
dependency on either the water level or the water/solute
application time (except for the first soil horizon). Thissuggests that variability in the soil physical and chemical
properties played a more important role than the imposed
flow conditions. The DL values for the first horizon showed

Vol. 46(4): 1097–1111

Table 3. Summary of the optimized soil hydraulic (n and qs ) and solutetransport (DL ) parameters for the different soil horizons andexperimental plots obtained using the two–step optimization method.
Values of SSQ and R2 after estimating DL (the second
step of the analyses) are given in parentheses.
Depth
n
θs
DL
SSQ
R2
(m)
(–) (cm3 cm–3) (cm)(–)(–)
Plot 1 (6 cm)[a]0–0.330.33–0.58
0.58–0.71
0.71–1.25

1.56
1.56
1.75
1.92

0.300
0.320
0.383
0.430

34.52
0.03
3.95
2.26

Plot 2 (10 cm)
0–0.330.33–0.58
0.58–0.71
0.71–1.25

1.72
1.73
1.96
1.56

0.300
0.430
0.300
0.300

15.54
0.06
34.37
26.60

Plot 3 (14 cm)
0–0.330.33–0.58
0.58–0.71
0.71–1.25

1.80
1.66
1.94
2.35

0.368
0.300
0.300
0.340

15.49
0.32
0.55
62.14

Plot 4 (6 cm)
0–0.330.33–0.58
0.58–0.71
0.71–1.25

1.50
1.57
1.51
1.53

0.330
0.300
0.300
0.306

10.14
1.95
6.12
0.15

Plot 5 (10 cm)
0–0.330.33–0.58
0.58–0.71
0.71–1.25

1.56
1.56
1.56
1.56

0.304
0.341
0.352
0.368

2.62
35.10
100.0062.20

CV (%)

12.6

12.8

131.7

[a]

1.12 (0.444) 0.988 (0.610)

0.645 (0.511) 0.987 (0.579)

0.927 (0.583) 0.984 (0.47)

0.85 (0.466) 0.996 (0.566)

0.847 (0.191) 0.991 (0.835)

Water level in the furrow.

some dependency on the water level and the water/solute
application time. The SD experiments produced larger DL
values as compared with the SWS experiments, which agrees
with results for the equivalent homogeneous soil profiles.
Two–Step OptimizationTwo–step, sequential estimations of the soil hydraulic andsolute transport parameters were carried out in an attempt tofurther improve our model predictions. During the first step,
the most sensitive parameters (qs and n) were optimized forthe different soil horizons, while during the second step, only
DL values were estimated. Two–step optimization results are
summarized in table 3. Note that SSQ and R2 for theestimated DL values during the second step are given inparentheses. The two–step method produced much lower
SSQ values (table 3) compared to simultaneous estimation of
the Ks and DL parameters (table 2). This is an immediate
consequence of using classical sequential estimation for thesoil hydraulic parameters followed by estimation of thesolute transport parameters. However, regressions between
measured and predicted solute concentrations were still
relatively low, generally between 0.47 and 0.84 for thedifferent plots (table 3). Optimized values for n varied
between 1.50 and 2.35, with a CV of 12.6%, while qs ranged
from 0.30 to 0.43 cm3 cm–3, with a CV of 12.8%. Bothparameters exhibited relatively low variability with depth
and among the plots (except n in plot 3 and qs in plot 1). The

1103

obtained qs values overall were lower than those measured in
the laboratory (table 1). This is consistent with the fact thatfield–measured qs values are generally much lower than theporosity because of entrapped air (e.g., Klute, 1986).
Contrary to the qs and n parameters, DL varied considerably with depth and between plots, ranging from 0.03 to
100.0 cm, with an overall CV of 131.7%. However, the values
were fairly similar, particularly for the first soil horizon, tothose estimated simultaneously with Ks and to those obtained
assuming homogeneous soil profiles (table 2). Estimated DL
values were also comparable with those previously obtainedat the same field site by Bowman and Rice (1986) underintermittent basin irrigation conditions (semi–weekly water
application). Their dispersivity values, fitted to a one–dimensional CDE, ranged between 20.8 and 141.0 cm. The higher
CV of the estimated DL values, as compared to that obtained
during simultaneous estimation of Ks and DL , was likely
related to differences in the estimated n and qs parameters
between the two optimization methods. Note that during thetwo–step optimization process, Ks values were set equal tothose previously estimated assuming homogeneous soil
profiles (table 2). Optimization using the Ks values obtainedusing the simultaneous method produced somewhat higher
SSQ values (results not shown here).
MODEL PREDICTIONS
Measured infiltrated data, water contents, and soluteconcentrations from all five experimental plots are comparedwith predicted values in figure 3. The calculated values wereobtained with the final optimized parameters estimated using
the simultaneous and the two–step optimization methods.
Two–step optimization generally produced higher R2 values,
especially for the soil water contents. However, the agreement was still relatively poor, probably because of temporal
Infiltration(L/m)

Cumulative Infiltration Rates
Measured and calibrated cumulative infiltration rates forthe different optimization scenarios and experiments involving two irrigations are compared in figure 4. Overall, model
predictions using the two–step optimization method weresatisfactory for the first irrigation (fig. 4a), whereas thesimultaneous method considerably underestimated infiltrated water in several cases (plots 4 and 5 in particular).
Infiltration rates during the second irrigation were slightly
overestimated (fig. 4b), which reflects temporal variability inthe soil hydraulic properties from one irrigation to the next.
We believe that this was caused by some deterioration of thephysical structure of the soil surface during initial wetting,and perhaps by some soil erosion (detachment) and deposition during the irrigations. Note that the simultaneous
optimization method resulted in better predictions during thesecond irrigation (fig. 4b). Model predictions for the
homogeneous soil profiles and the two–step methods weremore or less the same, partly because the same Ks values wereused in the two optimizations.
Water Contents and Bromide Concentrations
Comparisons between measured and calibrated soil watercontents and bromide concentrations at various depths,
experimental plots, and times (6 hours after the first irrigationand 10 days after the second irrigation for the soil watercontents, representing wet and dry conditions, respectively;
and 5 days after the first irrigation and 20 days after thesecond irrigation for solute concentrations) are presented infigures 5 and 6, respectively. Experimental plots 1, 3, and 5
Concentration(g/L)

Soil water content
3(cm cm–3 )
6

80
40
0
0

40 80 120 160 200
Observed

0.30

5

1:1 Line
R2 = 0.297
Predicted

1:1 Line
R2 = 0.935120

Predicted

0.40

160
Predicted

Simultaneous

200

and spatial variability in the soil hydraulic and solutetransport properties.

0.20

1:1 Line
R 2 = 0.563

4
3
2
1

0.10
0.10 0.20 0.30 0.40
Observed

0

0.40

6

0

1 2 3

4 5 6

Observed

1:1 Line
R2 = 0.923

120
80
40
0
0

40 80 120 160 200
Observed

0.30

1:1 Line
R2 = 0.586
Predicted

Predicted

Two–Step

160

Predicted

200

0.20
0.10
0.10

5
4

1:1 Line
R2 = 0.598

3
2
1
0

0.20 0.30 0.40
Observed

0

1 2 3

4 5 6

Observed

Figure 3. Measured versus predicted (using simultaneous and two–step parameter estimation) cumulative infiltration rates, soil water contents, andsolute concentrations for five experimental plots.

1104

TRANSACTIONS OF THE ASAE

50
25
0
0

50
Time (min)

100

50
25
0
0

50

100

Infiltrated water(L/m)(b) 2nd Irrigation

Infiltrated water(L/m)

Infiltrated water(L/m)
Infiltrated water(L/m)

Plot 2Plot 1(a) 1st Irrigation
50
25
0
0

50
Time (min)

100

0

50
Time (min)

100

0

50
Time (min)

100

0

100
Time (min)

0

50
100
Time (min)

50
25
0

100
50
0
0

50

100

Infiltrated water(L/m)

Infiltrated water(L/m)

Plot 3Time (min)

100
50
0

50
0
0

200
Time (min)

400

100
50
0
0

50

100

150

Infiltrated water
(L/m)

100

Infiltrated water(L/m)

Infiltrated water(L/m)
Infiltrated water(L/m)

Plot 5

Plot 4

Time (min)

100
50
0
200

100
50
0

Time (min)150

Measured
Heterogeneous–Simultaneous
Heterogeneous–Two–Step
Homogeneous
Figure 4. Measured and calibrated cumulative infiltration rates (using different calibration methods) for the different plots during two irrigation
events: (a) first irrigation, and (b) second irrigations.

were selected for the comparisons since they were the mostrepresentative of the SD experiments, the transient experiment between the SD and SWS experiments, and the SWS
experiments, respectively. Results are given by means of
one–dimensional curves for better visual comparison between measured data and model predictions. Agreement
between the observed and simulated results was relatively
poor during wet conditions 6 hours after the first irrigation

Vol. 46(4): 1097–1111(fig. 5a). All invoked optimization scenarios overestimatedthe soil water contents in the soil surface horizons (20 and
40 cm) but provided better predictions at deeper depths. The
two–step method considerably improved the predictions forthe surface horizons of plots 1 and 5. Agreement during thedrier conditions 10 days after the second irrigation (fig. 5b)
was somewhat better, but not excellent. The two–step method
again produced better predictions, particularly for the surface

1105

Plot 1Plot 340

at 80 cm depth

at 60 cm depth

at 40 cm depth

at 20 cm depth

40

Soil Water Content (cm 3 cm – 3 )Plot 5
40

30
20

30

30

20

20

10

10
0

25

50

75

10
0

100

25

50

75

100

40

40

40

30

30

30

20

20

20

10

10
0

25

50

75

25

50

75

100

40

40

40

30

30

30

20

20

20

10

10
0

25

50

75

25

50

75

100

40

40

40

30

30

30

20

20

20

10

10
0

25

50

75

100

Measured
Heterogeneous–Simultaneous
Heterogeneous–Two–Step
Homogeneous

50

75

100

0

25

50

75

100

0

25

50

75

100

0

25

50

75

100

10
0

100

25

10
0

100

0

10
0

25

50

75

100

Furrow Width (cm)Figure 5a. Measured and calibrated (assuming homogeneous soil profiles, and assuming heterogeneous soil profiles using parameters estimated withsimultaneous and two–step calibration) soil water contents for various depths and experimental plots, 6 hours after the first irrigation.

horizon (e.g., 20 cm). Agreement between measured andpredicted values obtained using homogeneous soil profiles
was less than that obtained using the simultaneous andtwo–step optimization methods for the heterogeneous profiles.
In spite of the considerable variability and uncertainty insolute transport parameters commonly derived from thefield–scale studies, reasonably good agreement was obtainedin our study between measured and predicted bromide
concentrations (fig. 6). The different optimization methods
did not produce good descriptions of the relatively highconcentrations observed at a few locations in the soil surface
horizons (e.g., 20 cm depth of plot 1 in fig. 6a, and plot 3 in
fig. 6b). We believe that this was partly due to overestimationof the soil water contents (fig. 5), thus keeping approximately
the same total amounts of solute in the surface layers.
Concentrations below depths of 60 cm were not included infigure 6a since the measured and predicted values were both1106

very low. Generally, no obvious differences were foundbetween predictions using the assumptions of profile homogeneity or heterogeneity (for both simultaneous and two–step
calibration), except in a few cases where none of themdescribed the solute distributions well (e.g., 60 cm depths ofplots 1 and 5 in fig. 6b). Two–step sequential optimization
was found to have only a small advantage over thesimultaneous approach for the solute concentration as
opposed to the water content. In some cases, the two–stepmethod (e.g., the 20 cm and 80 cm depths of plot 5 in fig. 6a)
produced better results, while in other cases, the simultaneous approach seems to work better (e.g., the 60 cm depthsof plots 3 and 5 in fig. 6a).
Deep Percolation Rates
Measured and predicted rates of deep percolation forwater (WDP) and solute (SDP) during the 30–day experi–mental period are presented in table 4. WDP and SDP are

TRANSACTIONS OF THE ASAE

Plot 1Plot 3at 40 cm depth
at 60 cm depth
at 80 cm depth

Soil Water Content (cm3 cm– 3 )at 20 cm depth

40
30
20

Plot 5

40

40

30

30

20

20

10

10
0

25

50

75

10
0

100

25

50

75

100

0

25

50

75

100

0

25

50

75

100

0

25

50

75

100

0

25

50

75

100

40

40

40

30

30

30

20

20

20

10

10
0

25

50

75

10
0

100

25

50

75

100

40

40

40

30

30

30

20

20

20

10

10
0

25

50

75

10
0

100

25

50

75

100

40

40

40

30

30

30

20

20

20

10

10
0

25

50

75

100

Measured
Heterogeneous–Simultaneous
Heterogeneous–Two–Step
Homogeneous

10
0

25

50

75

100

Furrow Width (cm)Figure 5b. Measured and calibrated (assuming homogeneous soil profiles, and assuming heterogeneous soil profiles using parameters estimated withsimultaneous and two–step calibration) soil water contents for various depths and experimental plots, 10 days after the second irrigation.

defined as the amounts of water and solute, respectively,
percolating below a depth of 100 cm. In order to calculate
WDP and SDP, soil profiles below the 100 cm depth down to
180 cm were divided into a network of rectangular elements,
each being approximately 25 cm by 30 cm. Volumes of wateror masses of bromide were determined for each rectangle andthen summed to obtain the total amounts of water/bromide.
We used daily measured soil water contents of the different
depths taken with a neutron probe to estimate the amounts ofwater, and the measured gravimetric soil water contents, bulk
densities, and bromide concentrations of different depths toestimate bromide masses in the soil profiles. Predicted WDPvalues assuming homogeneous soil profiles were underestimated for the SD plots, and overestimated for plot 5, but
provided a reasonable value for plot 4 (table 4). The
simultaneous approach generally produced better WDP
predictions as compared to the two–step optimization
method. However, estimated values were still somewhat

Vol. 46(4): 1097–1111

underestimated. The two–step method considerably overestimated WDP for all experiments, except for plot 1. The
relatively large estimated Ks values of the deeper horizonsobtained with the simultaneous optimization method (table2) did not lead to overestimated WDP values, mostly becauseof low sensitivity of the soil water contents to the Ks values.
The homogeneity assumption and the two–step method gave
relatively better predictions of the SDP values (except forplot 5), although SDP was generally underestimated using
simultaneous optimization. This in turn was largely due tounderestimation of WDP, since convective solute transport isdirectly related to water flow.

CONCLUSIONS

Several soil hydraulic parameters and longitudinal dispersivities were inversely estimated from five different transient
experiments on blocked–end furrows assuming the presence

1107

Plot 1Plot 3

Plot 512

at 40 cm depth
at 60 cm depth

Bromide Concentration (g/L)

at 20 cm depth

9
4

6

2

3

9
6
3

0

0
0

25

50

75

0
0

100

25

50

75

4

4

4

2

2

2

0
0

25

50

75

100

0
0

25

50

75

100

2

4

0.25

1

2

0
0

25

50

75

100

Measured
Homogeneous
Heterogeneous–Simultaneous
Heterogeneous–Two–Step

0

25

50

75

25

50

75

100

0

25

50

75

100

0

25

50

75

100

0

0.5

0

0

100

100

0

Furrow Width (cm)Figure 6a. Measured and calibrated (assuming homogeneous soil profiles, and assuming heterogeneous soil profiles using parameters estimated withsimultaneous and two–step calibration) bromide concentrations for various depths and experimental plots, 5 days after the first irrigation.

of four soil horizons. First, the saturated hydraulic conductivity and the longitudinal dispersivities were simultaneously
estimated for the different soil horizons and compared withthose previously obtained from the same experiments using
the assumption of profile homogeneity. Results showed only
minor improvements in the model predictions. To improve
the predictions, the most sensitive unknown soil hydraulicand solute transport parameters were estimated in two steps
by means of sequential estimation of the soil hydraulicparameters followed by the estimation of the transportparameters. This two–step method somewhat improved thepredicted cumulative infiltration rate during the first irrigation event, and more significantly the soil water contents,
particularly of the surface horizons, while predictions of the
deep percolation rates of water did not improve. The different
parameter estimation techniques predicted the solute concentrations in the soil profiles reasonably well, considering
the temporal and spatial variability in the soil hydraulic andsolute transport properties, without obvious differences
between the various optimization approaches.
In spite of implementing a scaling procedure for theunsaturated soil hydraulic properties, variability in the
optimized longitudinal dispersivity values for the layeredsoil profiles was found to be higher than for the homogeneous
ones. This increased variability was associated with variations in the scaled soil water retention curves of the different
soil horizons.

1108

The longitudinal dispersivity of the upper soil horizons
showed some dependency on the water level in the furrows
and the water/solute application time; similar dependencies
were not apparent for the other horizons. An important
advantage of scaling was to decrease the computational times
by a factor of about 5. Considering temporal variability in thesoil hydraulic parameters between the first and second
irrigations, presumably because of physical deterioration of
the furrow surfaces, could have further improved the model
predictions.
In this study, we did not consider hysteresis in the local soil
water retention functions. Including hysteresis likely would
produce slightly better predictions. However, Kaluarachchi
and Parker (1987) and Lenhard et al. (1991) showed thatneglecting hysteresis had only relatively little effect on the
results in their studies. We felt that our measured data wereinsufficient to provide an accurate description of hysteresis.
Predictions probably could also be improved by usingdual–porosity and/or dual–permeability models for waterflow and/or solute transport (Simunek et al., 2001). However,models of this type require many more input parameters, andconsequently many more measurements, that generally are
not available or are difficult to obtain for field–scale
experiments.

TRANSACTIONS OF THE ASAE

Plot 1at 20 cm depth

420

at 40 cm depth

Plot 5

6

6

3

3

0

0
25

50

75

0
0

100

25

50

75

0

25

50

75

100

0

25

50

75

100

0

25

50

75

100

0

25

50

75

100

100

4

4

4

2

2

2

0

0
0

25

50

75

0
0

100

25

50

75

100

4

0.5

4

at 60 cm depth

3
2

0.25

2

1
0

0
0

at 80 cm depth

Bromide Concentration (g/L)

Plot 3

25

50

75

0
0

100

25

50

75

100

0.5

4

4

0.25

2

2

0

0
0

25

50

75

100

Measured
Homogeneous
Heterogeneous–Simultaneous
Heterogeneous–Two–Step

0
0

25

50

75

100

Furrow Width (cm)Figure 6b. Measured and calibrated (assuming homogeneous soil profiles, and assuming heterogeneous soil profiles using parameters estimated withsimultaneous and two–step calibration) bromide concentrations for various depths and experimental plots, 20 days after the second irrigation.
Table 4. Measured and predicted deep percolation rates for water and soluteat the different experimental plots using various optimization approaches.
Water (L/m)
Solute (g/m)
Predicted

Predicted

Measured

Homogeneous
Soil Profile

Simultaneous
MethodTwo–Step
MethodSD[a]Plot 1 (6 cm)[b]Plot 2 (10 cm)
Plot 3 (14 cm)

11.4
16.5
29.2

0.0
0.0
5.9

11.9
0.4
11.5

SWS[c]
Plot 4 (6 cm)
Plot 5 (10 cm)

37.5
39.5

30.4
118.4

16.3
27.3

Experiment

Measured

Homogeneous
Soil Profile

Simultaneous
MethodTwo–Step
Method

8.5
29.7
88.7

1.1
2.9
3.5

0.0
0.0
1.8

0.0
0.0
0.0

0.0
1.7
26.5

88.8
166.4

27.2
15.9

13.0
41.0

0.3
3.0

18.5
169.1

[a] SD = same duration experiments.
[b] Water level in furrow.
[c] SWS = same amount of applied water and bromide experiments.

Vol. 46(4): 1097–1111

1109

ACKNOWLEDGEMENTS
This study was partially supported by SAHRA (Sustainability of Semi–Arid Hydrology and Riparian Areas) under
the STC Program of the National Science Foundation,
Agreement No. EAR–9876800.

REFERENCES

Abbasi, F., F. J. Adamsen, D. J. Hunsaker, J. Feyen, P. Shouse, and
M. Th. van Genuchten. 2003a. Effects of flow depth on waterflow and solute transport in furrow irrigation: Field data
analysis. J. Irrig. and Drainage Eng. 129(4): 237–246.
Abbasi, F., J. Simunek, J. Feyen M. Th. van Genuchten, and P. J.
Shouse. 2003b. Simultaneous inverse estimation of soil
hydraulic and solute transport parameters from transient field
experiments: Homogeneous soil. Trans. ASAE 46(4): 1085–
1095.
Bowman, R. S., and R. C. Rice. 1986. Transport of conservative
tracers in the field under intermittent flood irrigation. Water
Resources Research 22(11): 1531–1536.
Butters, G. L., and W. A. Jury. 1989. Field–scale transport ofbromide in an unsaturated soil: 2. Dispersion modeling. Water
Resources Research 25(7): 1583–1589.
Carsel, R. F., and R. S. Parrish. 1988. Developing joint probability
distributions of soil water retention characteristics. Water
Resources Research 24(5): 755–769.
de Vos, J. A. 1997. Water flow and nutrient transport in a layeredsilt loam soil. PhD diss. Wageningen, The Netherlands:
Wageningen Agricultural University.
Eching, S. O., J. W. Hopmans, and W. W. Wallender. 1994.
Estimation of in situ unsaturated soil hydraulic functions fromscaled cumulative drainage data. Water Resources Research30(8): 2387–2394.
Flury, M., H. Fluhler, W. A. Jury, and J. Leuenberger. 1994.
Susceptibility of soils to preferential flow of water: A field study.
Water Resources Research 30(7): 1945–1954.
Forrer, I., R. Kasteel, M. Flury, and H. Fluhler. 1999. Longitudinaland lateral dispersion in an unsaturated field soil. Water
Resources Research 35(10): 3049–3060.
Ghodrati, M., and W. A. Jury. 1990. A field study using dyes tocharacterize preferential flow of water. SSSA J. 54(6):1558–1563.
Gribb, M. M. 1996. Parameter estimation for determining hydraulicproperties of a fine sand from transient flow measurements.
Water Resources Research 32(7): 1965–1974.
Hopmans, J. W., 1989. Stochastic description of field–measured
infiltration data. Trans. ASAE 32(6): 1987–1993.
Inoue, M., J. Simunek, S. Shiozawa, and J. W. Hopmans. 2000.
Simultaneous estimation of soil hydraulic and solute transportparameters from transient infiltration experiments. Advances in
Water Resources 23(7): 677–688.
Jacques, D. 2000. Analysis of water flow and solute transport at thefield scale. PhD diss. No. 454. Leuven, Belgium: Katholieke
Universiteit Leuven, Fac. Landbouwkundige en Toegepaste
Biologische Wetenschappen.
Jacques, D., D. J. Kim, J. Diels, J. Vanderborght, H. Vereecken, and
J. Feyen. 1998. Analysis of steady–state chloride transport
through two heterogeneous field soils. Water Resources
Research 34(10): 2539–2550.
Jacques, D., J. Simunek, A. Timmerman, and J. Feyen. 2002.
Calibration of Richards’ and convection–dispersion equations tofield–scale water flow and solute transport under rainfall
conditions. J. Hydrology 259: 15–31.
Jaynes, D. B., R. S. Bowman, and R. C. Rice. 1988. Transport of aconservative tracer in the field under continuous flood irrigation.
SSSA J. 52(3): 618–624.

1110

Kaluarachchi, J. J., and J. C. Parker. 1987. Effects of hysteresis with
air entrapment on water flow in the unsaturated zone. Water
Resources Research 23(10): 1967–1976.
Klute, A. 1986. Water retention: Laboratory methods. A. Klute, ed.
Methods of Soil Analysis: Part 1. Physical and Mineralogical
Methods, 2nd ed. Agronomy 9(1): 635–662. Madison, Wisc.:
American Society of Agronomy.
Kodesova, R., M. M. Gribb, and J. Simunek. 1998. Estimating soil
hydraulic properties from transient cone permeameter data. Soil
Science 163(6): 436–453.
Kodesova, R., S. E. Ordway, M. M. Gribb, and J. Simunek. 1999.
Estimating of soil hydraulic properties with cone permeameter:
Field studies. Soil Science 164(8): 527–541.
Lenhard, R. J., J. C. Parker, and J. Kaluarachchi. 1991. Comparing
simulated and experimental hystertic two–phase transient fluid
flow phenomena. Water Resources Research 27(8): 2113–2124.
Mallants, D., M. Vanclooster, M. Meddahi, and J. Feyen. 1994.
Estimating solute transport in undisturbed soil columns usingtime–domain reflectometry. J. Contaminant Hydrology 17(2):91–109.
Mallants, D., B. P. Mohanty, D. Jacques, and J. Feyen. 1996a.
Spatial variability of hydraulic properties in a multi–layered soil
profile. Soil Science 161(3): 167–181.
Mallants, D., M. Vanclooster, and J. Feyen. 1996b. Transect study
on solute transport in a macroporous soil. Hydrological
Processes 10(1): 55–70.
Marquardt, D. W. 1963. An algorithm for least squares estimation of
non–linear parameters. J. Ind. Appl. Math. 11: 431–441.
Mualem, Y. 1976. A new model for predicting the hydraulicconductivity of unsaturated porous media. Water Resources
Research 12(3): 513–522.
Richards, L. A. 1931. Capillary conduction of liquids throughporous medium. Physics 1: 318–333.
Roth, K., W. A. Jury, H. Fluhler, and W. Attinger. 1991. Transportof chloride through an unsaturated field soil. Water Resources
Research 27(10): 2533–2541.
Russo, D., J. Zaidel, and A. Laufer. 1994a. Stochastic analysis ofsolute transport in partially saturated heterogeneous soil: I.
Numerical experiments. Water Resources Research 30(3):769–779.
Russo, D., J. Zaidel, and A. Laufer. 1994b. Stochastic analysis ofsolute transport in partially saturated heterogeneous soil: II.
Prediction of solute spreading and breakthrough. Water
Resources Research 30(3): 781–790.
Schaap, M. G., and F. J. Leij. 1998. Database related accuracy anduncertainty of pedotransfer functions. Soil Science 163(10):765–779.
Simmons, C. S., D. R. Nielsen, and J. W. Biggar. 1979. Scaling offield–measured soil water properties. Hilgardia 47: 77–154.
Simunek, J., and M. Th. van Genuchten. 1996. Estimating
unsaturated soil hydraulic properties from tension disc
infiltrometer data by numerical inversion. Water Resources
Research 32(9): 2683–2696.
Simunek, J., and M. Th. van Genuchten. 1997. Estimating
unsaturated soil hydraulic properties from multiple tension disc
infiltrometer data. Soil Science 162(6): 383–398.
Simunek, J., O. Wendroth, and M. Th. van Genuchten. 1998.
Parameter estimation analysis of the evaporation method for
determining soil hydraulic properties. SSSA J. 62(4): 894–905.
Simunek J., M. Sejna, and M. Th. van Genuchten. 1999. The
HYDRUS–2D software package for simulating thetwo–dimensional movement of water, heat, and multiple solutes
in variably saturated media, Version 2.0. IGWMC–TPS–70.
Golden Colo.: Colorado School of Mines, International Ground
Water Modeling Center.
Simunek, J., O. Wendroth, N. Wypler, and M. Th. van Genuchten.
2001. Non–equilibrium water flow characterized by means of
upward infiltration experiments. European J. Soil Science 52(1):
13–24.

TRANSACTIONS OF THE ASAE

Simunek, J., D. Jacques, J. W. Hopmans, M. Inoue, M. Flury, and
M. Th. van Genuchten. 2002. Solute transport during
variably–saturated flow – Inverse methods. Chapter 6.6 in
Methods of Soil Analysis: Part 1. Physical Methods,
1435–1449. 3rd ed. J. H. Dane and G. C. Topp, eds. Madison,
Wisc.: SSSA.
Snow, V. O., B. E. Clothier, D. R. Scotter, and R. E. White. 1994.
Solute transport in a layered field soil: Experiments andmodeling using the convection–dispersion approach. J.
Contaminant Hydrology 16(4): 339–358.
Steenhuis, T. S., and R. E. Muck. 1988. Preferred movement of
nonadsorbed chemicals on wet, shallow, sloping soils. J.
Environ. Quality 17(3): 376–384.
Troiano, J., C. Garretson, C. Krauter, J. Brownell, and J. Hutson.
1993. Influence of amount and method of irrigation water
application on leaching of atrazine. J. Environ. Quality 22(2):290–298.
Vanderborght, J., C. Gonzalez, M. Vanclooster, D. Mallants, and J.
Feyen. 1997. Effects of soil type and water flux on solutetransport. SSSA J. 61(2): 372–389.
Vanderborght, J., M. Vanclooster, A. Timmerman, P. Seuntjens, D.
Mallants, D.–J. Kim, D. Jacques, L. Hubrechts, C. Gonzalez, J.
Feyen, J. Diels, and J. Deckers. 2001. Overview of inert tracer
experiments in key Belgian soil types: Relation between
transport and soil morphological and hydraulic properties. Water
Resources Research 37(12): 2873–2888.

Vol. 46(4): 1097–1111

van Genuchten, M. Th. 1980. A closed–form equation forpredicting the hydraulic conductivity of unsaturated soils. SSSA
J. 44(5): 892–898.
van Weesenbeeck, I. J., and R. G. Kachanoski. 1994. Effect ofvariable horizon thickness on solute transport. SSSA J. 58(5):1307–1316.
Ventrella, D., B. P. Mohanty, J. Simunek, N. Losavio, and M. Th.
van Genuchten. 2000. Water and chloride transport in a
fine–textured soil: Field experiments and modeling. Soil Science
165(8): 624–631.
Vogel, T., M. Cislerova, and J. W. Hopmans. 1991. Porous media
with linearly variable hydraulic properties. Water Resources
Research 27(10): 2735–2741.
Vrugt, J. A., M. T. van Wijk, J. W. Hopmans, and J. Simunek. 2001.
One–, two–, and three–dimensional root water uptake functions
for transient modeling. Water Resources Research 37(10):2457–2470.
Warrick, A. W., G. J. Mullen, and D. R. Nielsen. 1977. Scaling
field–measured soil hydraulic properties using a similar media
concept. Water Resources Research 13(2): 355–362.
White, R. E., J. S. Dyson, Z. Gerstl, and B. Yaron. 1986. Leachingof herbicides through undisturbed cores of a structured clay soil.
SSSA J. 50(2): 277–283.
Wildenschild, D., J. W. Hopmans, and J. Simunek. 2001. Flow ratedependence of soil hydraulic characteristics. SSSA J. 65(1):35–48.

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