Advances in Water Resources 23 (2000) 677±688
www.elsevier.com/locate/advwatres

Simultaneous estimation of soil hydraulic and solute transportparameters from transient in®ltration experiments
 unek b,*, S. Shiozawa c, J.W. Hopmans d
M. Inoue a, J. Sima
Arid Land Research Center, Tottori University, Hamasaka 1390, Tottori 680, Japan
US Salinity Laboratory, USDA-ARS, 450 W. Big Springs Dr., Riverside, CA 92507, USAc
Institute of Agricultural and Forest Engineering, University of Tsukuba, Tsukuba, Japand
Hydrology Program, Department LAWR, 123 Veihmeyer Hall, University of California, Davis, CA 95616, USAb

Received 6 October 1999; received in revised form 10 February 2000; accepted 11 February 2000

Abstract
Estimation of soil hydraulic and solute transport parameters is important to provide input parameters for numerical models
simulating transient water ¯ow and solute transport in the vadose zone. The Levenberg±Marquardt optimization algorithm incombination with the HYDRUS-1D numerical code was used to inversely estimate unsaturated soil-hydraulic and solute transportparameters from transient matric pressure head, apparent electrical conductivity, and euent ¯ux measurements. A 30 cm long soilcolumn with an internal diameter of 5 cm was used for in®ltration experiments in a coarse-textured soil. In®ltration experiments
were carried out with both increasing and decreasing solute concentrations following a sudden increase in the in®ltration rate.
Matric pressure heads and solute concentrations were measured using automated mini-tensiometers and four-electrode sensors,
respectively. The simultaneous estimation results were compared with independently measured soil water retention, unsaturated
hydraulic conductivity, and solute dispersion data obtained from steady-state water ¯ow experiments. The optimized values corresponded well with those measured independently within the range of experimental data. The information contained in the apparent electrical conductivity (which integrates information about both water ¯ow and solute transport) proved to be very useful forthe simultaneous estimation of soil hydraulic and solute transport parameters. Ó 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Four-electrode sensor; Bulk soil electrical conductivity; Hydraulic conductivity; Soil water retention curve; Dispersivity

1. Introduction
In arid and semiarid regions that are characterized byhigh air temperatures and low precipitation rates, salts
accumulation at or near the soil surface is common. Soilsalinization in irrigated agriculture may be accelerated
by the presence of high groundwater table when, forexample, deep drainage is reduced because of low subsoil permeability. The combined eects of waterlogging
and salinization may cause a signi®cant decrease of agricultural productivity of irrigated lands [17]. When areliable drainage system is present, salts can be removed
from the root zone by leaching using excess irrigationwater. Such practices can be conveniently described*

Corresponding author. Tel.: +1-909-369-4865; fax: +1-909-3424964.
E-mail addresses: mainoue@center.tottori-u.ac.jp (M. Inoue), jsimunek@ussl.ars.usda.gov (J. SÏimuÊnek), shiozawa@sakura.cc.tsukuba.
ac.jp (S. Shiozawa), jwhopmans@ucdavis.edu (J.W. Hopmans).

using models that simulate simultaneously water ¯ow
and solute transport processes [22].
Computer models based on numerical solutions of
the ¯ow and solute transport equations are increasingly
being used for a wide range of applications in soil andwater management. Model predictions depend largelyon the accuracy of available model input parameters.
Soil hydraulic parameters, characterizing the water retention and permeability properties, and transport andchemical parameters aecting the rate of spreading ofchemicals and their distribution between solid and liquid
phases are the most important input variables for suchmodels.
The use of parameter estimation techniques for determining soil hydraulic properties is well established[2,8]. The approach has been widely used for various
laboratory and ®eld experiments. Among others, laboratory experiments include one-step [7,30] and multistep [1,31] out¯ow experiments, upward ¯ux or head
controlled in®ltration [3], the evaporation method0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 1 1 - 7

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M. Inoue et al. / Advances in Water Resources 23 (2000) 677±688[20,24], and in®ltration followed by redistribution [25].
In separate lines of research, solute transport parameters
are often obtained from column experiments assuming
steady-state water ¯ow [15], and using parameter estimation codes such as CFITIM [33] or CXTFIT [29] for®tting analytical solutions of the transport equation toexperimental breakthrough curves. Solute transportparameters for conditions for which no analytical solutions exist, such as for nonlinear adsorption, can beobtained using numerical solutions [10,25]. The above
parameter estimation eorts for water ¯ow and solutetransport have remained relatively disjoint. Although
there are numerous studies that combined estimation of¯ow and transport parameters for groundwater ¯owproblems [12,27,34], only a very few studies have usedcombined transient variably-saturated water ¯ow andsolute transport experiments for simultaneous estimation of soil hydraulic and solute transport parameters[13].
Dierent strategies in combined estimation of water¯ow and solute transport parameters can be followed.
Only water ¯ow information (matric pressure heads
and/or ¯uxes) can be used ®rst to estimate soil hydraulicparameters, followed with estimation of transportparameters using only transport information (concentrations). Combined water ¯ow and transport information can be used to estimate sequentially soil hydraulicand solute transport parameters. Finally, combined
water ¯ow and transport information can be used tosimultaneously estimate both soil hydraulic and solutetransport parameters. The last approach is the mostbene®cial since it uses crossover eects between statevariables and parameters [27] and it takes advantage of
the whole information, because concentrations are afunction of water ¯ow [12]. Misra and Parker [13]showed that simultaneous estimation of hydraulic andtransport properties yields smaller estimation errors formodel parameters than sequential inversion of hydraulicproperties from water content and matric pressure headdata followed by inversion of transport properties fromconcentration data.
The main motive for the simultaneous estimation ofwater ¯ow and solute transport parameters in groundwater studies is to use the most information available
and to decrease parameter uncertainty. In soil studies,
this is accompanied by the motive to avoid carrying out
repeated experiments on the same sample. That is, repeated experiments on the same or identically-packed
soil columns most likely will aect the magnitude of ¯ow
and transport parameters. Moreover, the presented
transient ¯ow and transport experiments are more realistic than those requiring steady state. The combined
use of transient ¯ow and transport data for estimationof the soil hydraulic and solute transport parameters canalso result in substantial time-savings as compared tosteady-state methods.

Excellent tools have been developed over the years toanalyze transient ¯ow experiments such as ONESTEP[6], SFIT [9], and HYDRUS-1D [23]. Some programs
are designed for speci®c experiments only (e.g., ONESTEP [6]), while others are more versatile (e.g., SFIT [9],HYDRUS-1D [23]). Of the above codes, only HYDRUS-1D allows simultaneous inversion of soil hydraulic and solute transport parameters, including
situations involving linear and nonlinear solute transport during either steady-state or transient water ¯ow.
The objective of this study is to determine soil hydraulic and solute transport parameters of a Tottoridune sand using various steady-state and transient water¯ow and solute transport laboratory column experiments. The transient and steady-state tests involve in®ltration at dierent rates. Parameters determined usingdierent analytical and parameter estimation approaches will be compared. We also discuss the application and calibration of a four-electrode sensor tomeasure the bulk soil electrical conductivity. The
measured bulk soil electrical conductivity is a variable
that integrates information on both water ¯ow andsolute transport and can thus be bene®cially used toestimate simultaneously soil hydraulic and solutetransport parameters. We show that the measured bulk
soil electrical conductivity is especially advantageouswhen used for the simultaneous estimation of soil hydraulic and solute transport parameters.
2. Theory
2.1. Water ¯ow
Variably-saturated water ¯ow in porous media is
usually described using the Richards equation


oh h
o
oh

K h  K h ;
1
ot
oz
oz
where t is time and z is depth (positive upward), and h
and h denote the volumetric water content and the soil
water matric pressure head, respectively. The Richards
equation can be solved numerically when the initial andboundary conditions are prescribed and two constitutive
relations, i.e., the soil water retention, h(h), and hydraulic conductivity, K(h), functions, are speci®ed. The
soil water retention curve in this study is described using
the van Genuchten analytical expression [32]Se h h h ÿ hr1
n m:
hs ÿ hr
1  jahj 2

The hydraulic conductivity function is described using
the capillary model of Mualem [14] as applied to the van
Genuchten function [32]

ÿ
m 2
:
3
K h  Ks Se` 1 ÿ 1 ÿ Se1=m

M. Inoue et al. / Advances in Water Resources 23 (2000) 677±688

679

In Eqs. (2) and (3), hr and hs denote the residual andsaturated volumetric water contents, respectively; Se iseective saturation, Ks the saturated hydraulic conductivity, ` a pore connectivity coecient, and a, n and m(  1 ) 1/n) are empirical coecients.

The water ¯ow and solute transport equations subject toinitial and boundary conditions were solved numericallyusing the HYDRUS-1D code [23].

2.2. Solute transport

The general approach of parameter estimation involves the minimization of a merit, goal or objectivefunction that considers all deviations between the measured and simulated data, with the simulated results
controlled by the adjustable parameters to be optimized[26]. The objective function OF(b) for the transient ¯owexperiments is given by

Solute transport in variably-saturated porous media
is described using the convection±dispersion equation


oRh C
o
oC
ovh C

hD
;
4
ÿ
ot
oz
oz
oz
where C is the solute concentration, R the retardation
factor, D the eective dispersion coecient, and v is thepore water velocity. The retardation factor R and thedispersion coecient D are de®ned as
R1

qb Kd
;
h

D  kjvj;

5
6

where Kd is the linear adsorption distribution coecient,
qb the bulk density, and k is the longitudinal dispersivity. Eq. (6) assumes that molecular diusion is insignificant relative to dispersion.
2.3. Initial and boundary conditions
The initial condition for each in®ltration experiment
was obtained by establishing steady-state downwardin®ltration with a constant water ¯ux and a constantsolute concentration. Then, at some time t  ti , bothmatric pressure head and solution concentration wereconstant with depth
h z; ti   hi ;C z; ti   Ci :7The upper boundary conditions (z  0) for the in®ltration experiments are given byoh 1  qtop t;
ÿK
oz
8
oC
 qC  qtop t Ctop t;ÿ hDoz
where qtop and Ctop are, respectively, water ¯ux andsolute concentration applied at the soil surface.
A zero matric pressure head gradient (free drainage,q  )K) and a zero concentration gradient are used as
the lower boundary conditions (at z  )L) for water¯ow and solute transport, respectively,
 
oh
 0;
oz zÿL


9
oC
 0:
oz zÿL

2.4. Parameter optimization

OF b  Wh

N1
X

hm ti  ÿ ho ti ; b

2

i1

 Wq

N2
X
qm ti  ÿ qo ti ; b2 ;

10

i1

where Wh , and Wq are normalization factors for matricpressure head and ¯ow rate, respectively, with eachfactor being inversely proportional to their measurement variance; N1 , and N2 the number of observations
for matric pressure head and ¯ux, respectively; and b isthe vector of optimized parameters. The subscripts mand o refer to the measured and optimized values. The
weighted least-squares estimator of Eq. (10) is a maximum-likelihood estimator as long as the weights containthe measurement error information of particularmeasurements.
The objective function for the transport part of thetransient experiments is given by
OF b  WEC

N3X
ECa;m ti  ÿ ECa;o ti ; b2 ;

11

i1

where ECa is the bulk electrical conductivity, WEC its
normalization factor, and N3 is the number of electricalconductivity measurements. The objective function forsimultaneous optimization of soil hydraulic and solutetransport parameters combines objective functions
Eqs. (10) and (11).
The Levenberg±Marquardt method [11,27,34] (as incorporated in the HYDRUS-1D code [23]) was used tominimize the objective function OF(b). The parameter
vector b includes the parameters a, n, hr , hs , Ks , l, and k.
Each inverse problem was restarted several times withdierent initial estimates of optimized parameters andthe run with the lowest value of the objective function
was assumed to represent the global minimum. The soil
bulk electrical conductivity, ECa , was calculated in the
HYDRUS-1D code from calculated values of the solution electrical conductivity, ECw , and the water content,h (see Section 3.1).

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M. Inoue et al. / Advances in Water Resources 23 (2000) 677±688

3. Materials and methods
3.1. Electrical conductivity measurements
Nondestructive methods for direct measurement ofsoil salinity include buried porous electrical conductivity
sensors, four-electrode probe systems, electromagnetic
induction sensors, and time domain re¯ectometry systems [16,18]. These methods all measure the bulk soil
solute concentration rather than the solution concentration of individual ions [19]. The four-electrode probe
is used for measurement of solute concentrations when
rapid measurements are needed; this method is well
suited for measuring both water ¯ow and solute transport variables simultaneously during transient in®ltration and/or evaporation. The main disadvantage of the
four-electrode sensor is that soil-speci®c calibration isrequired.
The four-electrode sensor developed for this study is
described in detail by Shiozawa et al. [21]. The sensor
consists of four stainless steel rods of 1 mm outside diameter, which are inserted parallel in the center of anacrylic cylinder ring of 20 mm length and 50 mm inside
diameter (Fig. 1). The two inner and two outer stainlesssteel rods are spaced 8 and 16 mm, respectively.
The ratio of the electric current (I) ¯owing throughthe outer electrodes to the voltage dierence (V2 ) between the two inner electrodes is measured. The ratio I/
V2 is inversely proportional to the electrical resistance ofthe measured medium, or proportional to its electricalconductivity (EC). The magnitude of the electric current(I) through the two outer electrodes is obtained fromI  V1 /Rf , where Rf is a known resistance inserted in
the circuit (Fig. 1). The ratio V1 /V2 is automaticallymeasured using a 21X micro-datalogger (Campbell
Scienti®c) which also supplies the required AC of 750
Hz. The proportionality constant between the output
value V1 /V2 and the bulk EC depends on the shape andconstruction of the sensor, and is determined by measuring known EC-values of various water solutions at aknown reference temperature. This was done with all
three sensors using sodium chloride solutions in therange between 0.005 and 0.2 mol/l.
The four-electrode sensors were calibrated in mixtures of Tottori dune sand and sodium chloride solu21X datalogger

Rf

∼

V1

Cylinder ringtions [4,21]. The measured bulk soil electricalconductivity (ECa ) depends on the electrical conductivity of the soil solution (ECw ), the volumetric watercontent (h), the dry bulk density (qb ), and temperatureof the water solution and the soil (T). After washing thesand with distilled water, clay and silt fractions were
removed, and the pure sand samples were oven-dried.
Clay and silt fractions were removed in order to prevent
potential clogging and permeability changes of the porous plate by transport of these ®ner fractions during therelatively fast in®ltration experiments. Known volumesof NaCl solution with known concentrations were added
to the sand to obtain the desired water content and salt
concentration values. In all, four-electrode probe calibration was carried out for NaCl concentrations of
0.005, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.08, 0.1, 0.2, 0.5,
and 1.0 mol/l, and for volumetric water content values
ranging from 0.0287 to 0.454 cm3 /cm3 , including fullsaturation. The mixtures were kept in a vinyl bag at aconstant temperature of 25°C for two days. A total of 96
prepared soil samples were packed uniformly in columnsof 50 mm inside diameter and 60 mm height, and thesurfaces leveled and covered to prevent evaporation.
After measurement of ECa with the four-electrode sensor, the volumetric water content and the dry bulk
density of each soil sample were determined from ovendrying. The average value of the dry bulk density (qb )was 1.45  0.02 g/cm3 at an average soil temperature ofT  25.0  0.5°C.
We assumed that the relation between ECa and ECw
can be described by the following relationship, which
neglects the surface conductance of the solid phase [17]ECa
 a ECw  h  b:h

The surface conductance, associated with the exchangeable ions at the solid±liquid interface, can beneglected only for sandy soil. Rhoades and Oster [18]discussed relations between ECa and ECw for situations
when the surface conductance cannot be neglected, suchas for silt and clay fractions. The ®tted relations of ECa
versus h are shown in Fig. 2 for various solution concentrations; symbols in the ®gure are experimental values and the lines represent the ®tted Eq. (12). Fitted
values for a and b were 1.45 and 0.102, respectively, witha correlation coecient value of 0.998.
To convert ECw to concentration, the following experimentally derived power relationship was used [4]C  0:008465  ECw1:073 :Electric mini-tensiometer

V2

Fig. 1. The four-electrode sensor and the electric circuit.

12

13

Using Eq. (12), the water content can be determined if
the pore water salinity, ECw , is known, or alternatively
ECw can be computed from the measured ECa and aknown h. Average P
relative errors were calculated using
the expression 100
jYm ÿ Ye j=Ye =n, where n, Ye andYm represent the number of experimental data, andM. Inoue et al. / Advances in Water Resources 23 (2000) 677±688

681

Stop cock

Microtube pump
13cm

23cm

Four-electrode sensor

Electric mini-tensiometerTwo-electrode sensor

Fig. 2. Calibration of the four-electrode sensor for dune sand. Lines
represent ®tted Eq. (2) for dierent concentrations (dierent electricalconductivities of the soil solution as given in the legend), with thehighest concentration at the top and lowest at the bottom.
Electrical balance

estimated and experimental values, respectively. We
obtained average errors of 2.41% and 5.60% for thevolumetric water content and the soil water electricalconductivity, respectively. Hence, the four-electrodeprobe can be conveniently used to measure either watercontent h or concentration C.
3.2. Steady-state in®ltration experiments
Four-electrode probes and mini-tensiometers wereused to determine the soil water retention curve fromsteady-state downward in®ltration into the washed
Tottori sand. The experimental setup consisted of a soilcolumn of 50 mm diameter and 30 cm height, and abalance to measure the drainage rate (Fig. 3) [34]. The
soil column was packed under wet condition at the bulk
density of 1:50  0:03 g/cm3 . Two mini-tensiometerswith pressure transducers and four-electrode sensors
were installed at the 13 and 23 cm depths, and an additional tensiometer at the 27 cm depth. A 5 mm thickcoarse-sintered glass plate with a saturated conductivity
value of 0.000405 cm/s was placed at the bottom.
Using the syringe pump a constant in®ltration rate ofa NaCl solution having a constant concentration of 0.02
mol/l was applied to the soil surface. After establishing a
steady drainage rate at saturation, the in®ltration rate
was ®rst decreased in steps (to obtain drying retentiondata) and then increased (providing wetting retentiondata). Variations in the soil water matric pressure head(h) and the bulk electrical conductivity (ECa ) with time

Fig. 3. Schematic illustration of the column experimental setup.

were measured using the mini-tensiometers and the fourelectrode sensors, respectively. The matric pressure headat the bottom of the column was continuously adjusted
to the value monitored by mini-tensiometer at the 23 cm
depth. Since the solution concentration (C) was known
and constant, ECw was determined directly from Eq.
(13). Hence, the water content, h, could be estimateddirectly using Eq. (12) and the four-electrode sensor's
measurement of ECa .
We then conducted a series of eight steady-state ¯owexperiments with steady-state in®ltration ¯uxes (i)varying between 0.022 and 0.00036 cm/s (®rst column ofTable 1). Using the steady-state data of the four-electrode sensors and mini-tensiometers at column depths of
13 and 23 cm, unsaturated hydraulic conductivities, K,
were calculated with Darcy's law from the known
steady-state water ¯uxes and measured total hydraulichead gradients, dH/dz. Since ¯ow is steady state, thewater ¯ux is equal to the in®ltration rate at any column
depth. The results in Table 1 show that the hydraulicgradient dH/dz is near unity during each of the experiments, and that unsaturated hydraulic conductivity K isa rapidly decreasing function of h. The values of v, h, h,K, and dH/dz, presented in Table 1 are average values
between depths of 13 and 23 cm.
Once steady-state water ¯ow was established over theentire soil column, the NaCl concentration of the in¯ow

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M. Inoue et al. / Advances in Water Resources 23 (2000) 677±688

Table 1
Estimation of unsaturated hydraulic conductivities, dispersivities, and dispersion coecients from steady-state water ¯ow experimentsai (cm/s)v (cm/s)h (cm3 /cm3 )h (cm)

dH/dz (cm/cm)

K (cm/s)k (cm)

D (cm2 /s)

0.02190.01850.01510.00869
0.00506
0.002270.00105
0.000360.0727
0.06680.05830.03820.02490.01340.00750.0030

0.301
0.277
0.259
0.227
0.203
0.169
0.140
0.121

)11.9
)11.2
)12.6
)13.7
)12.4
)14.1
)15.5
)18.0

1.036
1.063
0.928
1.008
0.968
1.004
0.945
1.020

0.02110.01740.01630.008620.005220.002260.001110.00035

0.157
0.162
0.232
0.304
0.280
0.278
0.272
0.190

0.01140.01080.01350.01160.00700.00370.00200.00056ai ± In®ltration rate, v ± pore water velocity, h ± volumetric water content, h ± matric pressure head, dH/dz ± total head gradient, K ± hydraulicconductivity, k ± dispersivity, D ± dispersion coecient.

tained when we analyzed breakthrough curves at depthsof 13 and 30 cm. Although pore water velocities varied
more than one order of magnitude (0.0030±0.0727 cm/s),
dispersivity values were reasonably close (0.157±0.304
cm). The dispersivities were computed using Eq. (6)from the estimated pore water velocities and ®tted dispersion coecients.
3.3. Transient in®ltration experiments
The scanning wetting and drying water retention
curves were obtained by matching matric pressure head
and water content values measured during the transientin®ltration experiments. From the simultaneous
measurements at the 23 cm depth during three wetting(applied ¯ux varying between 0.4 and 3.2 ml/min) andtwo drying (¯ux varying between 3.2 and 0.4 ml/min)processes, the hysteretic relationships between h and hcould be determined as shown in Fig. 5.
We subsequently conducted three in®ltration experiments (Experiment I, II, and III) to simultaneously
measure the hydraulic and solute transport variables. In
each of the three experiments, the initial steady-state

0 .4

3

θ [cm /cm ]1

3

z = 15 cm
z = 23 cm
z = 30 cm

0 .5
i = 0 .0 0506 c m /s3

θ = 0 .2 03 cm /c m3v = 0 .0 24 9 cm /s
2

D = 0.00 70 cm /s

0
0

50 0

1 0 00Tim e

1 50 0t

2 00 02 50 0[s]

Fig. 4. Breakthrough curves of NaCl solution measured using fourelectrode (at depths of 13 and 23 cm) and two-electrode (at a depth of
30 cm) sensors in the unsaturated sand column and ®tted using the
CXTFIT code.

Volumetric water content

Relative concentration (C-C 1)/(C 2-C 1)

solution was changed from 0.02 to 0.1 mol/l. The EC in
the soil column was measured at two depths using thefour-electrode sensors, while the EC of the euent wasmeasured with a two-electrode sensor in the out¯owtube at the bottom of the soil column (Fig. 3). As an
example, we present here the experimental results for asteady-state water ¯ux of i  0.00506 cm/s, corresponding to a soil water matric pressure head h of )12.4 cm, amean pore water velocity v of 0.0249 cm/s, and an average volumetric water content h of 0.203 cm3 /cm3 . The
breakthrough curves using relative concentration for thethree depths of 13, 23, and 30 cm are shown in Fig. 4.
Dispersion coecient, D, dispersivity values, k, andmean pore water velocity, v, were determined by ®tting
the analytical solution of the convection±dispersion
equation to the measured concentration values using the
CXTFIT program [31]. The average volumetric watercontent, h, was then calculated from a steady-state water¯ux, i, and a mean pore water velocity, v (h  i/v). Dispersivity values estimated from each of the eight steadystate experiments for the breakthrough curves measured
at the 23 cm depth are presented in Table 1. Similardispersivity values, as those give in Table 1, were ob-

Ste ad y -st at e da ta
0 .3

We ttin g sc an nin g d a ta
D ry ing sca nn in g d ata

0 .2

0 .1

0
0

-1 0

- 20

S oil water m atric pressure he ad

-3 0

-40

h [cm]

Fig. 5. Wetting and drying soil water retention curves obtained using
measurements of mini-tensiometer and four-electrode sensor at a depthof 23 cm during steady-state and transient in®ltration experiments.

M. Inoue et al. / Advances in Water Resources 23 (2000) 677±688

683

Table 2
Summary of transient in®ltration experiments
Experiment no.

I (cm/s)C (mol/l)

h2 i (cm)Measured variablesIii  0.00033
if  0.00278

Ci  0.02
Cf  0.1)24.7

h1 , h2 , h3 , q, ECa1 , ECa2IIii  0.000321
if  0.002581

Ci  0.02
Cf  0.1)26.8

h2 , ECa2IIIii  0.000312
if  0.002632

Ci  0.1
Cf  0.02)27.2

h2 , ECa2

in®ltration rate (i) of about 0.00032 cm/s was increased
to about 0.0026 cm/s at time t  1000 s. For Experiments
I and II, the solute concentration, C, was increased from0.02 to 0.1 mol/l simultaneously with the in®ltration rateincrease. The solute concentration was decreased from
0.1 to 0.02 mol/l at t  1000 s in Experiment III. Table 2
summarizes the applied in®ltration rates and soluteconcentrations, combined with the type of measurements, and at which depth they were taken. Subscripts
1, 2, and 3 refer to depths of 13, 23, and 27 cm,
respectively, and the subscripts i and f denote theinitial and ®nal stages of the in®ltration experiment,
respectively.

4. Results and discussion
4.1. Experimental data

0

3

0.4

0.2

0.3

0.2

0.1

0.1

1000

2000

T ime

t

3000

0
4000

3
3

3
3

[cm /cm ]3[cm /cm ]C [mol/dm ]θθC [mol/dm ]ECa [dS/m]

0.5

Volumetric water content

-10

0.3
h [cm]

N aCl concentration-20

EC a [d S/m ]-30

B ulk so il ele ctrica l co n du ctiv ity

Soil water matric pressure headh [cm ]

Observed values of h, ECa , and calculated values of hand C versus time at the 23 cm depth for Experiment II
are shown in Fig. 6. The values of h and ECa were directly measured with the mini-tensiometer and the fourelectrode sensor, respectively. The values of in h withtime were computed from the measured h values, while[s]Fig. 6. Variation of h, ECa , h, C with time at a depth of 23 cm in thesoil column after changing both q (from 0.000321 to 0.00258 cm/s) andC (0.02 to 0.1 mol/dm3 ) at t  1000 s (Experiment II).

those of C were calculated from the measured ECa andthe estimated h using Eqs. (12) and (13). As shown in
Fig. 6, the water content h and the solution concentration C at the 23 cm depth started increasing at about
1250 and 2400 s, respectively. Hence, the concentration
front lags behind the moisture front. This observation iswell known and occurs because new in®ltrating water
displaces old water initially present in the porous media,
and because of diusion, mixing and equilibration ofsolute between the new and old water. The old waterwith the initial solute concentration is thus pushed in
front of the advancing new water with the new soluteconcentration. This process results into separate waterand solute fronts, each traveling at dierent speeds depending on the ratio of the initial and ®nal water contents [5,28]. The higher in®ltration rate during thesecond stage increased h2 from )26 to about )13, corresponding to an increase in h from 0.12 cm3 /cm3 to0.19 cm3 /cm3 . The ECa curve in Fig. 6 also clearly shows
the combined control of both water content and solution concentration on the bulk electrical conductivity
curve. This combined eect of h and C on ECa results in
two fronts, the ®rst one at about 1250 s caused bywetting front, and the second one starting at about 2500
s responding to the concentration front.
The results of Experiment III for the 23 cm depth arepresented in Fig. 7. As the water content increased because of the higher in®ltration rate, the bulk electricalconductivity, ECa , increased during the time interval
between 1250 and 2300 s. However, the decreasing solute concentration at about 2300 s caused the ECa todecrease after that time. As in Experiments I and II, theinitial part of the ECa curve re¯ects mainly changes in h,whereas the second part of the curve characterizes solutetransport properties.
4.2. Inverse analysis
4.2.1. Hydraulic parameters
We next used the parameter optimization features if
HYDRUS-1D to analyze the h and C distributions. To
better understand the possible in¯uence of the porous
plate at the bottom of the column, we compared optimization results for pro®les with a plate (using a two-

3
3

3

3

[cm / cm ]3

C [m ol/dm ]
0.4

0.2

0.3

0.2

0.1

0.1

0

1000

2000

Tim e

0
4000

3000

t

θ

θ

3

EC a [dS/m]

0.5

Volumetric water content-10

C [mol/dm ]Bu lk soil electr ical c on duc tivity

Soil water matric pressure head

-20

0.3
h [cm]

NaCl concentration

EC a [dS/m ]-30[cm /cm ]M. Inoue et al. / Advances in Water Resources 23 (2000) 677±688

h

[cm]

684

[s]

-0.004-30h-20Effluent flux

-0.002-10
M easured q(t)-0.001

M easured h1(t)M easured h2(t)M easured h3(t)
0

0
0

10 00

20 00

Tim e

3 000

t

400 0

Soil water pressure head

-0.003q[cm/s]

Exp. I[cm ]Fig. 7. Variation of h, ECa , h, C with time at a depth of 23 cm in thesoil column after changing both q (from 0.000312 to 0.00263 cm/s) andC (0.1 to 0.02 mol/dm3 ) at t  1000 s (Experiment III).

5 000

[s]

Fig. 8. Measured and optimized out¯ow rates and matric pressure
heads for Experiment I. Either a one-layer (solid lines) or two-layer(dashed lines) models were used in the inverse analysis.

layer model) and without a plate (using a one-layer
model). The columns for this purpose were ®rst schematized as a two-layered pro®le with the Tottori sandrepresenting the ®rst layer and the sintered glass plate
representing the second layer. The saturated hydraulicconductivity of the plate was independently measured
and found equal to 0.000405 cm/s. Fig. 8 shows themeasured and optimized ¯ow rates, q(t), and the matricpressure heads h1 (t), h2 (t), h3 (t) for depths of 13, 23, and27 cm of Experiment I. Results reveal a very good
agreement between the measured and optimized valuesof both the out¯ow rate (q) and matric pressure head(h). Only the arrival of the water content front to tensiometer at depth of 27 cm lags behind the model prediction. The inversely obtained parameters are listed inTable 3 (®rst row).
Optimized ¯ow rates and matric pressure head values
for the 23 cm depth of Experiment I using the one-layer
model and the free drainage bottom boundary condition
are also shown in Fig. 8. The correspondence between
measured and optimized values for both out¯ow rates
and matric pressure heads are similar to those when a
two-layer model was used. The optimized parameters
obtained with the one-layer model for Experiment I arealso listed in Table 3 (second row). The optimized
parameters are also very similar with those obtainedusing the two-layer model and 95% con®dence intervals
for all optimized parameters overlapped. Only the lparameter diers by about one-third between the twosolutions, but that has no signi®cant consequence on theprediction of the hydraulic conductivity function within
the measurement range (Fig. 9). Fig. 9 compares the
optimized soil water retention and hydraulic conductivity functions for both the one-layer and two-layer
models, and independent data obtained using the steady-state and transient in®ltration methods. Notice that
all optimized functions closely approximate the independently measured data within the water content rangeof the transient experiments (0.1 < h < 0.2) and thatlarger dierences exist only beyond the measurement
range. Since results for two- and one-layered systems were similar, we conclude that the sintered glass
plate did not signi®cantly in¯uence ¯ow at the measurement depths (13 and 23 cm) used in optimization
calculations.
4.2.2. Transport parameters
Next, we optimized solute transport parameters forExperiment I using the objective function of Eq. (11),and ®xing the already optimized soil hydraulic parameters for water ¯ow. The measured and optimized bulk

Table 3
Optimization results for transient ¯ow and transport experiments
Experiment
I (two-layer)
I (one-layer)
I (two-layer)
I (one-layer)
II (one-layer)
III (one-layer)

Optimized parameter valueshr (cm3 /cm3 )hs (cm3 /cm3 )a (cmÿ1 )n

Ks (cm/s)

l

0.0
0.0

0.385
0.345

0.163
0.164

2.05
1.89

0.090
0.070

)0.66
)1.05

0.027
0.057

0.310
0.372

0.138
0.117

2.01
2.42

0.045
0.0514)1.16)0.813k (cm)

0.243
0.246
0.221
0.190

-4

M e asu rem en t rang e
10
0. 5

0
0

0.1

0. 2

0 .3

Volumetric water content

0 .4

θ

3

-5

10

λ =0. 19 0 (Exp III on e laye r m od el)-3m easured D (v)

10

-4

10

-3

5

0 .3

M eas u red

EC a 1( t)M eas u red

EC a 2( t)O ptim ized E C a 1(t )0 .2

O ptim ized E C a 2(t )
0 .1

0
0

10 00

2 00 0Time

3 00 0t

4 00 050 00[s]Fig. 10. Measured and optimized bulk electrical conductivities forExperiment I. A one-layer model was used in the inverse analysis.

soil electrical conductivities, ECa1 (t) and ECa2 (t), at
depths of 13 and 23 cm, respectively, are presented inFig. 10. As discussed above, the initial small increase of
the bulk soil electrical conductivity corresponds with thearrival of the moisture front, whereas the second, steeper
and larger increase corresponds with the concentration
front. Excellent agreement was obtained between themeasured and optimized ECa -values. The optimized
dispersivities k were equal to 0.246 and 0.243 cm using aone- and two-layer models, respectively. This shows thatthe porous plate also did not signi®cantly aect thedispersivity. Fig. 11 compares the optimized dispersioncoecients with the independently measured values
obtained with the steady-state column experiments(Table 1). Again, notice the excellent agreement between

10-1[cm /s]0.8
Measured h(t)

h

0 .4

5

v

-40

Exp. I

0 .5-2

Fig. 11. The dispersion coecient D as function of the mean porewater velocity v obtained by inverse optimization and from analysis ofsteady-state data.

[c m]

0 .6

10

M ea n pore water velocity

Optimized h(t)Soil w ater m atric pre ssure head

ECa [dS/m]

10

3[cm /cm ]Fig. 9. Comparison between soil hydraulic properties estimated inversely for Experiment I using either a one- (solid lines) or two-layer(dashed lines) models and those obtained from analysis of steady-state
and transient in®ltration data.

Bulk soil electrical conductivity

λ =0. 22 1 (Exp.II o ne la yer m o de l)-2[dS /m]10

λ =0. 24 6 (Exp.I o ne layer m o del)Measured ECa(t)-30Exp. IIECah( θ ) Tran sie n t (w et sc an )-10

λ =0. 24 3 (Exp.I tw o lay er m odel)

0.6

Optimized ECa(t)

-20

0.4

-10

0.2

0

1000

2 000

T im e

30 00

t

0
4000

Bulk soil ele ctr ica l conductivity

-3

D

10

Dispersion coefficientK ( θ ) S teady -s tate dataKh ( θ ) S tead y- s tate d ata
-20

685

-1

-2

Hydraulic conductivity

Soil water matric pressure head

10

-30

10

2

-1

[cm /s]10[cm/s]

-40

h

[cm]

M. Inoue et al. / Advances in Water Resources 23 (2000) 677±688[s ]Fig. 12. Measured and optimized matric pressure heads and bulk
electrical conductivities for Experiment II. A one-layer model was usedin the inverse analysis.

dispersion coecients obtained from transient andsteady-state experiments.
4.2.3. Simultaneous optimization of hydraulic and transport parameters
The next case involves transient Experiment II, again
with an increasing in®ltration ¯ux and increasing soluteconcentration, for which hydraulic and solute transportparameters were optimized simultaneously using the
one-layer model. The optimized and measured matricpressure heads and bulk soil electrical conductivities forthe 23 cm depth are presented in Fig. 12. As indicatedearlier, this ®gure also clearly shows that the positions of
the wetting and solute fronts are distinct, with theirpositions being a function of the initial water content.

10

-30

10

-1

Measured ECa(t)0.6

Optimized ECa(t)

-20

0.4

-10

0.2

0

1000

2 000

T im e

30 00

t

0
4000

[dS/m]
ECa

Optimized h(t)-30Bulk soil electrical conductivity

Exp. III[s ]Fig. 14. Measured and optimized matric pressure heads and bulk
electrical conductivities for Experiment III. A one-layer model wasused in the inverse analysis.

shown in Fig. 14. Again, excellent agreement exists between the optimized and measured soil water matricpressure heads, h(t), and bulk soil electrical conductivities, ECa (t). Similarly as for Experiment II, the water
front moved about 2.7 times faster than the concentration front. The optimized soil hydraulic functions arecompared in Fig. 13 with optimization results based onExperiment II, and with the independently measured
retention and hydraulic conductivity data. While there isa very good correspondence between the hydraulicconductivities obtained from the dierent experiments,
the optimized retention curve for Experiment III isslightly shifted towards higher water contents as compared with the curve for Experiment II. The parameter
estimates are again presented in Table 3. The optimized
dispersivity value, however, compares again very wellwith the independently measured values and with the
optimized values from the other in®ltration experiments(Fig. 11).

K

-2

K ( θ ) S te ady -state dat ah( θ ) S tead y-s tate d ata

-20

10

-3

h( θ ) Tran s ien t (w et s can )

10

-10

-4

M e as ure m en t r an g e
10
0. 5

0
0

0.1

0. 2

0 .3

Volumetric water content

0 .4

θ

3

Hydraulic conductivity

Soil water matric pressure head

0.8
Measured h(t)[cm/s]

-40

h

[cm]

This means that the center of the concentration front
should arrive approximately 2.7 times later than thewetting front at some location in the column. The
measured centers of the wetting and concentration
fronts (de®ned here as the arithmetic averages of theinitial and ®nal values of the matric pressure head andthe concentration, respectively) reached the 23 cm depth
at 640 and 1780 s after the change in the in®ltration rate,
respectively. Thus, the concentration front arrived about
2.78 times later than the wetting front, which agrees wellwith the estimated value of 2.7, according to Eq. (14).
Therefore, information contained in the two fronts does
not overlap and can be used to advantage for thesimultaneous estimation of soil hydraulic and solutetransport parameters. Fig. 13 compares the inversely
estimated hydraulic functions against the independently
measured data. Notice again the excellent correspondence of the optimized retention curve and the transient
wetting data within the experimental range. Optimization results are also presented in Table 3.
Finally, we optimized simultaneously the hydraulicand solute transport parameters to data from Experiment III, i.e., the experiment in which the in®ltration
rate was increased (from 0.000312 to 0.00263 cm/s) andthe solute concentration decreased (from 0.1 to 0.02
mol/l) at t  1000 s. The comparison of measured andsimulated h and EC data at the 23 cm column depth is-40hSince the initial (hi ) and ®nal (hf ) water contents wereabout 0.12 and 0.19, and the initial (ii ) and ®nal (if )in®ltration rates about 0.00032 and 0.00258, respectively, one may expect the wetting front (fw ) to move
about 2.7 times faster than the concentration front (fs ):
if ÿ ii
hf
:
14
fs :
fw 
if
hf ÿ hi[c m]

M. Inoue et al. / Advances in Water Resources 23 (2000) 677±688

Soil w ater m atric pre ssure head

686

-5

3

[cm /cm ]Fig. 13. Comparison between soil hydraulic properties estimated inversely for Experiments II (solid lines) and III (dashed lines) using a
one-layer model and obtained from analysis of steady-state and transient in®ltration data (data points).

5. Summary and conclusions
Matric pressure head and solute concentration were
simultaneously measured during in®ltration of sodium
chloride solution using mini-tensiometers with pressuretransducers and four-electrode sensors. Both the tensiometers and four-electrode sensors proved to be usefulfor measuring the rapid changes in matric pressure headand electrical conductivity during the simultaneous
movement of water and solute in the sandy soil columns.
The bulk soil electrical conductivity (ECa ) depends on
the solution electrical conductivity (ECw ) and the volumetric water content (h). After calibrating the functional
relation of ECa  f(ECw , h) for each soil, ECw can beobtained from the measured ECa and the known h.

M. Inoue et al. / Advances in Water Resources 23 (2000) 677±688

Alternatively, h can be obtained from the measured ECa
and a known ECw . In either case, ECa can be measured
accurately using the calibrated four-electrode sensor.
The more complex functional relation would be neededfor ®ner textured soils [18].
The Levenberg±Marquardt algorithm [11] in combination with the HYDRUS-1D (version 2.0) [23] code
was used to inversely estimate the unsaturated soil hydraulic and solute transport parameters from transientmatric pressure head, soil bulk electrical conductivityand ¯ux measurements. Experiments were carried out
with both increasing and decreasing solute concentrations following a sudden increase in the in®ltration rate
in a sand-®lled column. Optimization results werecompared with independently measured soil water retention, unsaturated hydraulic conductivity and solute
dispersion data. The optimized values corresponded wellwith those measured independently within the experimental range. Larger dierences between ®tted
functions existed only outside the measurement range.
However, it is generally accepted that parameters obtained with parameter estimation are to be used only
within the measurement range for which they were
determined [24].
Since the apparent electrical conductivity, ECa , integrates information about both water ¯ow and solutetransport, information imbedded in ECa proved to bevery useful for the simultaneous estimation of the soil
hydraulic and solute transport parameters. In®ltration
experiments produced two distinctive fronts for water
movement and solute transport, and thus the eects ofwater content and concentration changes could be well
distinguished. Application of the parameter estimation
technique, which combines a numerical solution of thegoverning partial dierential equations with the Levenberg±Marquardt method, to transient water ¯ow andsolute transport data resulted in accurate estimation of
the soil hydraulic and solute transport parameters. The
combined use of transient ¯ow and transport data for
estimation of the soil hydraulic and solute transportparameters resulted in substantial timesaving as compared to steady-state methods. The presented methodwith simultaneous measurement of water ¯ow and solute transport variables using a four-electrode probe andcoupled estimation of soil hydraulic and solute transportparameters can be a very ecient method for multi-dimensional ®eld experiments as well.

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