Journal&dogy
ELSEVIER

Journal of Hydrology 199 (1997) 295-3 18

Comparison of three hydraulic property measurement methods
Dirk Mallama**, Diederik Jacques’, Peng-Hsiaug Tseng’ b,Martinus Th. van Genuchtenb, Jan Feyena‘institute for Land and WaterManagement, Faculty of Agriculttual and Applied Biological Sciences,

University of Luuven, Decosterstraat 102 B-3000 Leuve~ Belgium
bU.S. Salinity L.ubomtory,USDA-ARS, 450 West Big Springs Road, Riverside, CA 92507, USA
Received 5 February19%; revised 18 October 1996; accepted 21 November 1996

Hydraulic functions of soils may differ depending on the different measuring methods used. The
performance of three different methods for measuring soil-hydraulic properties of a heterogeneous
field were evaluated. The experiments were conducted using three different sizes of undisturbed soilcores collected systematically along a 31 m long transect of a well drained sandy loam soil having
three soil horizons (Ap, O-O.25 m; Cl. 0.25-0.55 m; C2,0.55-1.00 m). The laboratory studies
involved: (1) detailed unsteady drainage-flux experiments perfotmed on fifteen columns of 1 mlength and 0.3 m diameter; (2) combined crust test and hot-air methods applied to thirty columns of
0.2 m length and 0.2 m diameter and to a subset of sixty cylinders of 0.1 m length and 0.045 m
diameter, respectively, taken from the Ap horizon; and (3) desorption experiments carried out on a
total of one hundred eighty cores of 0.05 1 m length and 0.05 m diameter collected evenly from thethme horizons. Mean soil hydraulic properties were inferred from experimental data characterizing
either selected depths or the soil profile as a whole. The results revealed considerable differencesamong estimated mean soil properties as obtained with different measuring techniques. Although theapplication of scaling theory substantially reduced variation in the measured pressure heads (h) andconductivities (K), the results revealed that scaling parameters determined from soil pressure head
were not identical to scaling factors determined from hydraulic conductivity. The results also showthat K scaling factors in general were much more variable than h scaling factors, and that theobserved variability in scaling factors also depend upon the measurement technique used. 0 1997
Elsevier Science B.V.
Keywords: Soil hydraulic properties; Measurement techniques; Similar-media scaling; Spatial
variability; Variably-saturated flow

+ comsponding

author.

’Resent addmxx Institute of Geophysics and Planetary Physics, Univmity of California. Riverside, CA
92521-0412, USA.
CQ22-1694/97/317.00Q 1997- Elsevier Science B.V. All rights ttxerved
PII SOO22-1694(96)03331-8

2%

D. Mallants et al./Jounvll of Hydrology 199 (1997) 295-318

1. Introduction
Soil variability is a major factor impeding the accuracy of measured soil hydraulicproperties, i.e. the soil water retention characteristic, B(h), and the unsaturated hydraulicconductivity curve, K(8) or K(h), where 8 is the volume fraction of water and h is thepressure head of the soil. The effects of spatial variability in hydraulic properties on
subsurface water flow (including infiltration and drainage), and the transport of surfaceapplied chemicals, have been widely studied (Warrick et al., 1977; Hopmans, 1992;
Mallants et al., 1996c). A major problem related to field-scale water flow and solutetransport modeling is how to deal with variability in the soil hydraulic properties.
One popular way for estimating the hydraulic properties of unsaturated soils has been to
statistically relate B(h) and K(B) or K(h) to soil texture, bulk density, organic matter
content, clay mineralogy, and soil structure (for an overview see van Genuchten et al.
(1992)). Although attractive from a practical and economical point of view, the accuracyof such methods is often incommensurate with the type of problem being investigated(Among many others, W&ten and Bouma, 1992; Espino et al., 1995). Another approach isto predict the hydraulic conductivity from. more easily measured soil water retention datausing closed-form analytical expressions (van Genuchten, 1980; Mualem, 1992). Rredictive models of this type tend to work reasonably well for coarse- and medium-textured
soils, but often fail to represent the unsaturated hydraulic conductivity of structured soils,
as was demonstrated in a recent study by Mallants et al. (1996a).
The use of direct measurements, whether in-situ or ex-situ, generally remains the only
viable means to characterize f?(h) and K(8) or K(h), and to fully explore the reliability of
indirect methods such as those described above. In situ measurement procedures often
used are the steady flux method (Hillel and Gardner, 1970) and the unsteady drainage
method (Green et al., 1986). A typical example of the former is the introduction of a low
permeability crust to achieve a steady water flux. The unsteady drainage flux method, on
the other hand, comprises the simultaneous monitoring of soil water content and pressure
head following the commencement of water redistribution from an initially saturated soil
profile. Although the drainage method has been mainly used for field conditions (FlUhler
et al., 1976a; Rose et al., 1965; Sisson and van Genuchten, 1991), this method can beapplied equally well to laboratory soil columns (Richards and Weeks, 1953; Mallants et al.,1996a). Ex-situ determination of the hydraulic properties using undisturbed soil columns
has progressed through a variety of methods, including the crust test (Bouma et al., 1983),
the sorptivity method (Dirksen, 1979). the hot-air method (Arya et al., 1975). the evaporation method (Boels et al., 1978), and desorption tables (Hillel, 1980). Comparison of
hydraulic properties determined by means of in-situ field and ex-situ laboratory techniques
has revealed significant differences for fine-textured soils (Fltihler et al., 1976b). although
better agreement often has been noted for relatively coarse-textured media (Roulier et al.,1972). Because the unsaturated soil hydraulic functions are necessary inputs to numericalsimulation models evaluating alternative soil and water management practices, there is aneed to continuously assess the accuracies and limitations of in-situ and ex-situ methods.
A potentially attractive method to describe variability of laboratory or field-measured
hydraulic properties involves the use of scaling (Miller and Miller, 1955a, b; Warrick et al.,
1977; Sharma and Luxmoore, 1979; Ahuja et al., 1984; Hopmans, 1987). Because

D. Mallants et alJJounm1 of Hydrology

199 (1997)

295-318291

frequency distributions of scaling factors can be conveniently incorporated into numericalfield-scale or even watershed-scale flow and transport models, their application hasreceived considerable attention in the literature (Warrick and Amoozegar-Fard, 1979;
van Gmmen et al., 1989; Hopmans, 1992). Still, the appropriateness of doing so has notbeen fully investigated.
The purpose of this study is to compare the performance of three measurement techniques for quantifying the soil-hydraulic properties of a heterogeneous soil profile. Data of8(h) and K(h) obtained from a detailed unsteady drainage experiment on large undisturbed
soil columns will be analyzed and compared with those obtained from two other laboratory
methods, i.e. desorption of 100 cm3 soil cores to determine B(h), and the crust test/hot-air
method to measure K(h). Scaling is subsequently used to simplify the description of spatial
variability in the measured soil hydraulic properties and to obtain average functions ofconductivity, K,, and pressure head, h,, in terms of soil water content, 13.The suitability ofapplying scaling to characterize spatial variability of soils is examined and discussed. The
set of scaling factors obtained here together with the average hydraulic functions will be
used in a future study to predict field-scale water flow processes.

2. Materials and methods
2.1. Transect lay-out
The experimental field site was located in Bekkevoort, near Leuven (Belgium). The site
involves a well-drained, macroporous sandy loam (Mallants et al., 1996b). Soil cores ofdifferent dimensions were collected along a 31 m long transect. A total of 30 columns of
1 m length and 0.3 m diameter, 30 columns of 0.2 m length and 0.2 m diameter, and 180
soil cores of 0.05 1 m length and 0.05 m diameter were used for the laboratory experiments.
Three soil horizons (Ap, O-O.25 m; Cl, 0.25-0.55 m; C2,0.55-1.0 m) were contained in
the columns of 1 m length, whereas the columns of 0.2 m length were collected only in the
Ap horizon. Sixty cores of 0.051 m length were collected in each soil horizon, giving a
total of 180 samples. The sampling interval between columns of the same size was
generally 1 m, with sampling locations of the two smaller columns located in between
those of the columns of 1 m length. Details of the sampling scheme are given in Mallants
et al. (1996a).
2.2. Method 1: unsteady drainage jux method
The soil water retention curve, 8(h), and the hydraulic conductivity curve, K(8) or K(h),were characterized using the unsteady drainage-flux method (Green et al., 1986) as applied
to a subset of 15 columns of 1 m length. Saturated soil columns were allowed to drain for aperiod of 30 days while the outflow was recorded continuously. Soil water contents andpressure heads during redistribution were measured simultaneously using an automated
9O-channel time-domain reflectometry (TDR) system and tensiometers, respectively,
located at depths of 0.05, 0.15, 0.30, 0.45, 0.60, and 0.80 m (Mallants et al., 1996a).
Observations of 8 and h were taken at 0.5 h intervals for the first seven hours of drainage,D. Mallants et aUJouma1 of Hydrology 199 (1997) 295-318

298

and at 1.8, and 12 h intervals as the experiments progressed. When measurements of Bandh are made at different depths and times during the drainage experiment, K(8) at various
depths in the soil profile may be found by time integration of the Richards’ equation(Green et al., 1986) with a zero flux condition assumed at the soil surface (z = 0), i.e.
iJedz=K(e)~[h(z,t)-21;

ZI

where zi is an arbitrary depth in the soil, z is vertical coordinate (positive downward), and t
is time.
2.3. Method 2: combined crust rest and hot-air method
A second method used to obtain the unsaturated hydraulic conductivity was by combining the suction crust test (Bouma et al., 1983) and the hot-air method (Arya et al.,1975). The crust test was performed on thirty soil columns of 0.2 m length and 0.2 m
diameter collected from the Ap horizon located in between the large columns of 1 mlength. We first measured the saturated hydraulic conductivity, K,, by putting the saturated
soil cylinder on a perforated disk to which a filter cloth was attached to prevent soil fromfalling out. One tensiometer was installed at 0.1 m below the soil surface and was connected to a pressure. transducer. Water was ponded on the soil surface to a depth of 0.01 m.
Values of K, were obtained by measuring outflow with time when pressure was zero. The
unsaturated hydraulic conductivity was subsequently measured on the same cores after acrust was constructed (Mallants et al., 1996a). Unlike the classical crust test, crusts were
not removed but installed on top of an old crust. In this way, the initial contact between
crust and soil was not disturbed.
From each column of 0.2 m length, two cylinders of 0.1 m length and 0.045 m diameter
were taken to perform the hot-air method. In total 60 cores were exposed to a hot-air
blower for 11 min. The soil was then sliced into 0.001-0.002 m increments near the
exposed end and 0.007-0.01 m increments at the non-exposed end. Weighing of all
sections was done before and after oven drying. Volumetric water content of each section
was determined assuming a uniform bulk density along the core. Soil water diffusivity,
D(0), was computed for each core using the water content distribution following enhanced
evaporation (Arya et al., 1975; Anderson and Cassel, 1986). From the soil-water diffusivity, D(e), we calculated the unsaturated hydraulic conductivity, K(B), by multiplying
D(e) by the soil-water capacity, C(0) = de/d$, in the pressure range of h = 0 to h =-30000 cm (pF = 4.5).
2.4. Method 3: &sorption

on core samples

As described by Mallants et al. (1996c), 8(h) curves were obtained by desorption of soilcores of 0.051 m length and 0.05 m diameter from saturation to h = -16000 cm, using
standard techniques (Klute, 1986). Three groups of 60 retention curves (180 in total)
representing three horizons were analyzed. The pooled data of the sixty locations wereD. Mallants et alJJouma1 of Hydrology

199 (1997) 295-318

analyzed in terms of the empirical retention model of van Genuchten (1980), i.e.

es-4
e=er+

[1 +(alhl)qm

where the subscripts r and s refer to residual and saturated water contents, respectively,
and a a-‘), n, and m are curve-shape parameters. Assuming m = 1 - l/n, the hydraulicconductivity function, K(h) can be described by combining Mualem’s pore-size distribution model with Eq. (2):K(h)=K (l-larhl~[l+larhl”]-m)2I[ 1 + ICYhI”]“’(3)

or in terms of the effective saturation, S, = (0 - ery(e, - erj, (0 5 S, 5 1):
K(S,)=K,S,‘[l -(l-S;‘“)“]2(4)

where T is a pore-interaction parameter often considered to be 0.5 (Mualem, 1976).
2.5. Scaling of soil hydraulic properties
The description of spatially variable soil hydraulic functions can be simplified by
invoking scaling. In this approach, spatial variability is described by a set of scale factors,
a,, which relate the soil hydraulic functions at each spatial location r to a single curve orsome representative mean (Warrick et al., 1977). Scaling theory is based on the similar
media concept which assumes that porous media differ only in the scale of their internal
microscopic geometries, while their porosities are assumed identical (Miller and Miller,
1955a, b).
By means of scaling, the soil pressure head, h, at any spatial location r can be related tosome mean pressure head, h,, as
hr=hm/or(5)

where r = l,...,R is spatial location. In a similar way, the hydraulic conductivity, K,., can be
related to a mean K,:
K, = K,,,cY;

0%

In the case of similar media, Eqs. (5) and (6) hold for the complete range of water contents
at which h, and K, have been measured. Since field soils generally have spatially variable
porosities, h, and K, are not expressed here in terms of the volumetric water content, 8, but

in terms of the effective saturation, SC,assuming that 8, = 0, such that S, = 818,.
The method used here for scaling soil water retention data is the same as initiallyintroduced by Warrick et al. (1977). but with some minor modifications. The scaling
factors, a,, were obtained by minimizing the sum of squares:
SS= ,7[MSe,9 - arhr(Se,
i)]’

(7)

where &,(S,,J is the mean pressure head at a given degree of saturation, and i = l,..., I(r)are pressure head steps at location r. Differentiation of Eq. (7) with respect to aI and300

D. Makmts et aNJouma1 of Hydrology 199 (1997) 295-318

setting each of the equations equal to zero, lead to a linear system of equations which issolved subject to the constraint(ar,+fX~+...+cr/J/R=l(8)

Estimates of h, as a function of S, were obtained by employing a third degree polynomial
similar to that used by Hopmans (1987), i.e.
ln(h,)=ao+alS,+a2SZ+a3S,3(9)

where uO,..., a3 are constants. Eq. (9) will be applied to all h(S,) data of all R locations.
An iterative procedure was developed to minimize Eq. (7) subject to Eq. (8). An
estimate of h, was first made by applying Eq. (9) to the unscaled data. Next, best-fit
values of CX,were obtained using the procedures described above to scale the h(S,) data.
Eq. (9) was subsequently applied to the scaled data and the procedure repeated until SS
reached its minimum.
The method used for scaling hydraulic conductivity is identical to the procedures
developed by Warrick et al. (1977) whereby the mean K, was calculated at all R locations
according to the regression equation
In(K,)=bo+b,S,+bzS,2+b3S,3(10)

where bs, bt,...,bs are constants. Estimates of ln(K,) were used to obtain approximate
values of Q,’ according to:(11)which were normalized to give the Q, values:
cq=Ra,‘/

Za,’
r

(12)

Eqs. (11) and (12) were solved iteratively to obtain optimal values for the constants of Eq.
(10). This procedure resulted in a minimum sum of squares between K, and K,i. The sumof squares is now defined as:
SS = I: [In&, i) + 2ln(cf,) - ln(&, i)]*
r.i

(13)

the following discussion we compare results in terms of the mean squared error, MSE =SSldf, with &being the degrees of freedom, rather than SS, since the number of pressureInsteps is not identical for different depths to which the method is applied.
3. Results and discussion
3.1. Unsteady drainage jlux method
Data of 8 and h at five selected depths and ten observation times (t = 0.5,1,2,4,16,32,
64, 128, 256, and 512 h) were analyzed to evaluate their probability density functions(PDF). The results revealed that a normal PDF better described the data than a log-normal

D. Mallants et alJJouma1 of Hydrology

199 (1997) 295-318

301

Table 1
Statistics of observed soil water content as a function of elapsed time at five soil depths along a 3 1m long transect
during a drainage experiment.
Time

Soil depth
(m)

Mean
(cm3 cm”)

StalKhideviation(cm’ cmJ)

cv

01)

(S)

Maximum(cm’ cm”)

Minimum(cm3 cm”)

1

0.05
0.15
0.30
0.45
0.60
0.05
0.15
0.30
0.45
0.60
0.05
0.15
0.30
0.45
0.60
0.05
0.15
0.30
0.45
0.60
0.05
0.15
0.30
0.45
0.60

0.319
0.334
0.336
0.350
0.361
0.354
0.319
0.325
0.339
0.353
0.321
0.306
0.313
0.325
0.342
0.314
0.302
0.306
0.317
0.335
0.310
0.299
0.300
0.308
0.329

0.035
0.027
0.017
0.018
0.016
0.025
0.023
0.012
0.016
0.016
0.021
0.015
0.009
0.014
0.015
0.021
0.016

9.3
8.2
5.1
5.1
4.5
6.9
7.1
3.7
4.8
4.6
6.6
5.1
3.1
4.3
4.5
6.6
5.2
2.9
3.8
4.5
7.0
5.7
2.9
3.6
4.5

0.424
0.388
0.370
0.393
0.387
0.393
0.376
0.345
0.363
0.378
0.365
0.331
0.328
0.337
0.366
0.359
0.326
0.320
0.331
0.361
0.354
0.323
0.313
0.327
0.355

0.302
0.279
0.300
0.314
0.334
0.298
0.279
0.300
0.299
0.316
0.292
0.276
0.297
0.290
0.307
0.279
0.268
0.292
0.288
0.307
0.268
0.262
0.284
0.287
0.301

4

32

128

512

0.012
0.015
0.022
0.017
0.008
0.011
0.015

PDF. Statistics of the soil water content at five selected depths and observation times arelisted in Table 1. The coefficient of variation (CV) was generally the highest at theshallowest observation depth, and lowest at 0.3 m depth, thus suggesting a more homogeneous structure of the Cl horizon. The CV generally decreased slightly when the soil
dried, a result of the decrease, though small, of the standard deviation with time.
Table 2 gives the statistics of the pressure head at the same selected depths and times asin Table 1. The surface horizon (0.05 m and 0.15 m deep) shows the most variation in thepressure head, whereas the second horizon (0.30 m and 0.45 m deep) generally shows theleast. All CVs decrease considerably with time.
Soil water retention curves were obtained from simultaneous measurements of 8 and hat the five soil depths. The van Genuchten soil water retention model (Eq. (2)) was used toanalyze the unscaled data for each depth. Estimated model parameters were obtained byusing the non-linear parameter optimization program RETC (van Genuchten et al., 1991).
The retention parameters thus obtained will be compared with retention parameters
obtained with scaled data (to be discussed later). Preliminary estimates with Eq. (2)indicated that values of 0, were generally zero, and very close to zero, probably because

302

D. Mallantset alJJouma1 of Hydrology 199 (1997) 295-318

Table 2
Statistics of the soil water pressure head, h, as a timction of elapsed time measured at five soil depths along a 31 mlong transect during a drainage experiment
Time

Soil depth

Mean

01)

(m)

(cm)

standard
deviation

cv

Maximum

Minimum

@I

(cm)

(cm)

70.9
62.8
51.4
31.4
48.5
54.4
37.0
29.1
35.1
26.2
25.3
26.5
15.2
19.3
26.5
21.6
23.4
15.7
14.1
22.9
17.4
22.9
12.4
11.7
18.8

31.0
31.0
33.0
27.0
24.0
52.0
44.0
44.0
38.0
33.0
79.0
69.0
58.0
43.0
34.0
94.0
85.0
76.0
56.0
43.0
103.0
95.0
85.0
71.0
60.0

1.0
0.0
2.0
11.0
3.0
1.0
1.0
16.0
7.0
8.0
14.0
10.0
35.0
19.0
15.0
23.0
18.0
45.0
32.0
18.0
42.0
21.0
54.0
36.0
27.0

(cm)
1

4

32

128

512

0.05
0.15
0.30
0.45
0.60
0.05
0.15
0.30
0.45
0.60
0.05
0.15
0.30
0.45
0.60
0.05
0.15
0.30
0.45
0.60
0.05
0.15
0.30
0.45
0.60

14.4
14.6
15.3
16.3
13.4
29.7
28.4
26.3
23.2
23.8
60.2
51.7
43.8
35.4
27.6
74.4
63.9
58.4
45.3
33.4
88.9
18.5
70.2
55.9
44.0

10.2
9.2
8.8
5.1
6.5
16.2
10.5
7.7
8.1
6.2
15.2
13.7
6.7
6.8
1.3
16.1
14.9
9.2
6.7
1.7
15.5
18.0
8.7
9.8
8.3

of the relatively narrow pressure head range used in our experiments, A fixed value of
8, = 0 was therefore assumed for cases similar to those used by Greminger et al. (1985) andW&ten and van Genuchten (1988), unless stated otherwise. Two cases were considered:(i) the relationship m = 1 - l/n was assumed, leading to the parameter vector { Qs,txrn); and(ii) the parameters IZand m were both assumed to be variable, giving the parameter vector(O,, a, n, m). Keeping m and II independent adds additional flexibility to the parameter
optimization process, but its use is recommended only for data showing little scatter andcovering a wide range of pressure head data (van Genuchten et al., 1991). In our study,
both cases described the data equally well. Fig. l(a) shows the soil water retention curve
Eq. (2) fitted to all data points for the 0.3 m depth with parameter values for the condition
where m = 1 - l/n. Measured values of B exhibited a range of approximately 20.02 cm3
cmS3 about the fitted curve. We note the minor differences between the two fitted curves
representing the cases where m and II were both variable and m = 1 - l/n. For the casewhere no restrictions were put on m and n we found correlations between the estimatedvalues of m and n parameters to be between -0.98 and -1.00. This strong correlation
justifies the restriction m = 1 - l/n. Table 3 gives the estimated parameters of Eq. (2) forL,0

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305

D. Mallamet al./Joutml of Hydrology199 (1997) 295-318

obtained from the retention data, irrespective of the depth. A simultaneous fit of both Eqs.
(2) and (3) to the observed (8,h) and (K,h) data, with weighting coefficient Wl in theobjective function (van Genuchten et al., 1991) set to unity, resulted in estimated K(h)curves which deviated considerably from those obtained with separate fits, and gave
visibly poor descriptions of the data. Using values between l’and 0.1 for W 1 did improve
the fitting of the retention data somewhat, but not for the conductivity data. For thesereasons, parameter values for the simultaneous fit were not given in the corresponding
tables. Including the connectivity parameter T in the parameter optimization process did
also not improve the results in any of the above cases, presumably due to the scatter in thedata and the limited range of conductivities.
3.2. Combined crust test and hot-air method
Fig. 2 shows a combination of all data points obtained from the crust test and hot-air
method. At saturation, the range in K, values was more than two orders of magnitude.
Between saturation and h = - 32 cm, the range of variation in crust-based K values
decreased from almost three orders of magnitude to one order of magnitude. However,

1410DATA PaNrs
I
0

1

2

3

Pressure head, kg(h)Fig. 2. Observed (mcrhod 2) and fitted hydraulic conductivity..

4

I

5

306

D. Mallantset al./Joumal of Hydrology199 (1997) 295-318

K values obtained from the hot-air method showed a much larger variation than K values

from the crust method. This is likely due in part to the fact that the hot-air method is adiffusivity-based approach which requires retention (soil-water capacity) data to convert
diffusivities to conductivities. The retention parameters used for this purpose wereobtained independently from the soil cores discussed above, thereby possibly introducing
additional spatial variability (Mallants et al., 1996c). Note that Mallants et al. (19961~)
illustrated that a different functional form for C(O), e.g. by considering a multimodal
retention function, may lead to different conductivities, especially in the dry range. The
solid line in Fig. 2 represents the best fit of Mualem’s conductivity model with estimatedvalues for the parameter vector ((Y,n ) given in Table 4. In the estimation process the valueof the weighting coefficient for the data from the crust test, excluding K,, was increased
from 1 to 4, thus accounting for the fact that the crust test is generally judged to be morereliable than the hot-air method. Using different weightings for the crust-based K values
also accounts for the smaller number of observations compared to the hot-air data.
1EM

lE+O

1E-l

1E-l

1E-2

1E-2

lE-3

1E-S

lE-4

lE-4
lE-5
lE-8

0.45-m DEPTH

1E-7

I

0

I

1

P~8urka~

I

1

I

1

80

h (cm)

I

100

E

z

g
-u
#
L

lE+O
1E-l
lE-2
lE-3
lE-4
lE-5
lE-8

0

20

40

80

80

loo

Pre88w-e head, -h (cm)Fig. 3. Comparison betweendifferentmethodsto derive the unsaturated hydraulic conductivity: (i) gravitydrainage eXperiment (method 1). (ii) combined crust test/hot-air method (method 2), end (iii) indirectly estimatedfrom nmion parametersdetermined on small soil cams (method 3).

D. Mallanu et al./Joumai 6f Hydrology 199 (1997) 295-318

301

Fig. 3 presents the conductivities for all three methods. The K values obtained withmethod 2 (crust/hot-air data) compared favorably with those based on (analytic prediction)
method 1 in the pressure range of 0 to -50 cm, but were higher in the range of -50 to-100 cm (Fig. 3, upper left panel). The maximum difference in conductivity between
methods 1 and 2 in the pressure range considered here was about 1.5 orders of magnitude.
Possible errors introduced in the hot-air measurements are related to violations of its
assumptions. These assumptions include a homogeneous soil sample and representative
retention curves obtained from independent measurements. Furthermore, water redistribution after the heating is stopped and before the sample is sliced into pieces is assumed to be
negligible. van Grinsven et al. (1985) analysed the effect of the temperature gradient on
thermal vapor and liquid flow as well as on the viscosity of water. Their results indicatedthat the errors may be large but seem to compensate each other. An evaluation of thetemperature effects on the calculated conductivities for the hot-air method is beyond thescope of this paper.
3.3. Desorption on core samples
For each horizon, Eq. (2) was fitted to the pooled data of sixty locations. Estimated soil
water retention parameters are summarized in Table 3. In only one instance was theestimated parameter B, different from zero, i.e. 8 r = 0.021 for 0.90 m depth. The estimatedcurves, together with their 90% confidence intervals, were given in Pig. 4 and comparedwith the estimated retention curves obtained with the drainage method. The 90% confidence intervals for the fitted curves correspond to + 1.65 times the standard deviations of
the differences between fitted and measured values of 8 at nine values of h. Retention
curves obtained from the drainage method (fitted to the retention data separately) at the
0.3 m and 0.45 m soil depths fell entirely in the 90% confidence interval of the predictedcurve derived from the desorption method. Such an agreement between methods 1 and 3
did not hold for the Ap and C2 horizons, for which the drainage method gave consistently
lower water contents than method 3 for the entire range of measured h. We believe that theobserved disparities between the retention curves of methods 1 and 3 may have beencaused by air entrapment during initial saturation of the large columns.
Predicted unsaturated conductivity curves (Eq, (3)) using method 3 considerably overpredicted the conductivities obtained with the drainage method (Fig. 3). Fig. 3 further
reveals that measured conductivity curves (method 1) pertaining to the same horizon, i.e.
0.05 m and 0.15 m for Ap and 0.3 m and 0.45 m for Cl, agreed fairly well. Moreover, acomparison of conductivities for the first horizon indicates that K values obtained with the
combined crust test and hot-air method (method 2) were always lower than those obtainedwith method 3. Indirect estimates of K (method 3) are influenced by inaccuracies connected with the determination of the retention curve and assume the adequacy of thecapillary model theory (Mualem, 1976). The latter assumption, which gives a very simplified picture of actual soils, generally does not work well when water flow throughmacropores and interaggregate regions occurs, as was most likely the case in our soil. It
is clear from these comparisons that different measurement methods yield different
conductivities. Stolte et al. (1994) found that diffisivity based methods, e.g. the hot-air
method, do not compare well with methods that directly yield a conductivity curve,

308

D.Mallants
etalfJoumalofHydrology
199(1997)295-318- -.- -0.45-mDEPrn- -<:*-0.3-m
DEPlH

0.48
0.44
0.42
0.40
0.38
0.38
0.34

&q(a),

,

,

,
100

E
8

O

0.32
0.30
0.28
0.28 )

I

0

1

100

Pressure tEki, h (cm)Pressure h&J, -h (cm)

$
0.48
0.48
0.44
0.42
0.40
0.38
0.38
0.34
0.32
0.30
0.28

- - g - _ _ _

-w-__

w

0.28 1,

0

100

Pressure II%,-h (cm)Fig.
4.Comparisonbetweendifferent methodsto estimate the van Genuchtenretention model usingretention datafrom drainage experiment. Estimation based on retemtiondata measud on small soil cores is also shown,
together with the 90% confidence interval (shaded area).

especially in coarse-textured materials. Furthermore, the large variability in K values
obtained from hot-air measurements (Fig. 2) may indicate its inappropriateness for spatial
variability studies. Introducing several improvements to the hot-air method such as those
discussed by van Grinsven et al. (1985) may reduce some of the scatter in the data.
Note that methods 2 and 3 have a much wider application range than method 1. Also, thesample volume and experimental efforts for methods 2 and 3 are much smaller than that of
method 1. Therefore, from a practical point of view, methods 2 and 3 might be preferredover method 1 for routine investigations.
3.4. Scaling retention and conductivity data
All three measurement methods discussed above show a considerable variability in the

D.MollantsetaUJourmlofHydrology199(1997)295-318

309

retention and/or conductivity data. To reduce this variability, in an attempt to obtain an
average curve characteristic for the entire dataset, scaling was applied to the water
retention and hydraulic conductivity data obtained with the three measuring methods.
Fig. 5 gives the unscaled and scaled retention data from the drainage experiments when
all five soil depths are combined (i.e. pooled data). The solid line in Fig. 5(a) is the b&t fitof Eq. (9) for the unscaled data. At h = -50 cm, the width of the scattering range in S, isabout 0.33. After scaling, the data are confined to a fairly narrow band around the mean

E

3

3
1al

0.7

0.6

0.5

SCALED

1.0

b?

0.0

k = a@p5.10 - 247.13G+ 333.m qhw!3

0.5

I
0

I

I
20

40

I
50

I
60

s@q

I
loo

I
120

140

Press6e head, h (cm)Fig. 5. Unscaled (a) and acaled (b) soil water retention data for all five soil depths.

310

D. Malhnts

et alJJouma1of Hydrology

199(1997)

295-318

Table 5
Efficiency of scaling
MCtbOdDepthW

scaled II(&)
Reduction in MSE

scaled K(S,)
r*

WI

1

2
3

Reduction in MSE
66
70
82

48

0.722
0.765
0.793
0.665
0.587
0.642

7

0.899

-

0.05
0.15

50
49

0.30
0.45

60
44

0.60

49

pool4
o-o.2

pooled ’

r*

W)
0.807
0.859
0.918

81

0.919

86
76
77

0.917
0.863
0.885

’Datafor threedepths(0.10.0.50.0.90 m) was combined.

h,(S,) curve, although a considerable amount of the original variability is still present (e.g.
a band width of 0.22 in 5, remains at h = -50 cm). For this case, the MSE reduced from
0.481 for the unscaled data to 0.252 after scaling.
The pooled unscaled K values for all five depths together with the best-fitted thirddegree polynomial are given in Fig. 6(a). Scaling was again very effective in reducing thevariability (Fig. 6(b)) and now narrowly confined the scaled K values around the mean
curve K,(S,). A reduction of the h4SE of 76% was obtained. The reduction in variability
was somewhat less significant at saturation, presumably due to the large variability in K, of
our macroporous soils (Mallants et al., 1997).
Table 5 summarizes the efficiency of scaling in terms of the percent reduction in themean sum of squared error of the retention and hydraulic conductivity data for all threemeasuring methods. Notice the consistently higher reduction in h4SE values for scaling the
conductivity data in comparison with the scaled retention data. This somewhat artificial
behaviour is partly due to the use of logarithmic transformed conductivity data in thecalculation of the SS. Therefore, a comparison of scaling of retention and conductivity dataon the basis of the MSE values can only be of limited use.
Differences in scaling efficiency between different depths are fairly small for both theretention and conductivity data (Table 5). Notice, however, that reduction in lUSE for
K(S,) increases with depth indicating that the conductivity data are better scaled at greater
depths, presumably because of reduced spatial variability and less macropores (Mallants etal., 1996c). Table 5 also shows the r2 values obtained from a regression analysis between
unscaled and estimated values of h, or K, using the optimized mean curve h, or K, (Eqs.
(9) and (10)). Higher r2 values were obtained for scaling the conductivity data as comparedto the retention data. Hence our scaling procedure was more effective in reducing thevariability in hydraulic conductivity data than in retention data.
Probability plots or fiactile diagrams of the scaling factors (Y, and their logarithmictransformed values, lncx, calculated from water retention data and conductivity data
revealed a better description with a log-normal distribution than with a normal distribution (Mallants, 1996). These
results are consistent with earlier findings of Wanick et al.

D. MaUants et alJJouma1 of Hyhlogy
lE+l

UNSCALEDW-

lE+O- _/$,,I exp t-30.8+ 183.3Se of=
lE-1 3- 241.8 S&

z
.&
*s
s

311

I99 (1997) 295-31800

i125.2.$q olE-2 1_
lE-3

E
y

lE-4

2

lE-5
0

lE-8

ALL5SOlLDEMliS

.

’

1E-7
0.S

0.8

0.7

0.8

0.8

1.0

0.0

1.0

K, = exp(-128.7+ 480.4s,- 878.9 Se2 + 328.1Se3 ]

0.5

0.6

0.7

0.8

Effective saturation, S,
Fig. 6. Unscakd (a) and scaled @I)hydraulic cunductivity data for all five soil depths.

and Hopmaus (1987). Statistics of the log-normal distributed scale factors
obtained from the retention and conductivity data for the three measming methods arelisted in Table 6, with p and a2 referring to the mean and variance of lnar, When estimatedfrom the retention data (method l), the highest variability in scaling factors was found at
the 0.15 m depth (variance of a, = 1.02) and the lowest variability at 0.45 m (varianceof a, = 0.18). The skewness was also found to be the highest and lowest at these two
depths. Scaling the retention data from all five soil depths together resulted in a coefficient(1977)

312

D. Mallantset aNJouma1 of Hydrology199 (1997) 295-318

Table 6
Statistics of scale factors Q, estimated from retention and conductivity data
Method1

2
3

De@(m)

0.05
0.15
0.30
0.45
0.60
pooled co-o.2pooled fscaled K(S,)

Scaled II(&)
Mean’

Varb

CVc

Skew. *

Mean’

Var. bCV’

Skew. *

0.99
1.03
1.04
1.00
0.99
1.01
0.98

0.35
1.02
0.36
0.18
0.21
0.31
0.16

59
98
57
42
46
55
41

1.97
3.89
1.89
1.33
1.48
1.82
1.29

1.30
2.84
1.79
1.12
1.00
1.43
0.99
-

22.7
478
41.9
2.24
0.64
24.2
2.08
-

365
1510
360
134
80
340
145
-

59.6
3488
57.5
6.43
0.605
49.5
7.38
-

’Mean = exp@ + 0*/2).
bvariance = exp(& + uZ)[exp@) - 11.
’CV = 100[exp(0*)- I]“.
* Skewness coefficient = 3 CV + (CV)‘.
’Data for five depths was combined.
‘Date for threedepths(0.10,0.50,0.90m) was combined.

of variation of 55%. Scale factors calculated from conductivity data for method 1 exhibited amuch larger variability in comparison with the 01, values estimated from retention data
(Table 6). The extremely large variation may be due to the small number of observations
from which we estimated the statistics. Scaling of hydraulic conductivity data for all five
depths combined resulted in a CV of 340% for ar,..This value is considerably larger thanthe values previously found by Warrick et al. (1977) and Hopmans (1987). Although h(S,)
and K(S,) were both obtained from simultaneous measured (0, h) data during the drainage
experiment, the estimated conductivity showed a much higher variability in scale factors.
This may be due in part to errors introduced in the calculation of the conductivities basedon a too coarse grid of vertically spaced (6, h) data. Also, K values are more sensitive tochanges in S, than h values. In a previous study by Ahuja et al. (1984), scale factors werealso found to be more variable for saturated hydraulic conductivity than for water
retention.
Table 6 shows that scale factors calculated from h(S,) and K(S,) may not be. identical. If
soils behaved as Miller-Miller type similar media, scale factors obtained from retention
and conductivity data (Eqs. (5) and (6)) should follow closely the 1:l line (Luxmoore andSharma, 1980; Hopmans, 1987). This was not the case for our soil (Mallants, 1996). As aresult, two different sets of scale factors must be estimated.
Unsaturated hydraulic conductivity data determined by means of the combined crust/
hot-air technique (method 2) for the Ap horizon were also scaled using the techniques
described above. Scaling was again found to be very effective in reducing spatial variability, with the mean sum of squares reduced from 7.84 to 1.77 (Table 5). Values of themean, variance, CV, and skewness based on the log-normal distribution for (r, are given
in Table 6. The variance of 2.08 was considerably smaller than the variance obtained withmethod 1 for the same observation depth (0.05 and 0.15 m). The larger variability ofscaling factors for method 1 may be the result of the experimental methodology and/or the313

D. Mallants et aUJoumal of Hydrology 199 (1997) 295-318

computational procedure used in the calculation of the hydraulic conductivity. Especiallythe limited vertical resolution of the measured (B,h) data may have been insufficient toaccurately evaluate Eq. (1) in the calculation of K.
Scaling of the retention data obtained from desorption on small soil cores was previously performed by Jacques et al. (1996). Table 6 shows a reduced variability in the hscaling factor for method 3 in comparison with method 1. The larger variability of scaling
factors for method 1 in comparison with those for methods 2 and 3 may be the result of thelarger experimental errors associated with the drainage data.
Scaled retention and conductivity dam obtained from method 1 were subsequently used
to estimate parameters (cr. n) in the van Genuchten retention model (Eq. (2)) and parameters {n, m, K,} for the Mualem conductivity model with r = 0.5. Two cases wereconsidered: the first data set contained the data scaled separately by depth whereas thesecond set was obtained after splitting the scaled pooled data into five groups according totheir measurement depth.
Another choice to obtain the mean van Genuchten curve together with a set of scaling
parameters would have been to fit the van Genuchten model directly to the unscaled data.
This method was presented by Hopmans (1987) (his Method III) and compared to threeother methods. However, Hopmans found that use of the method of Wanick et al. (1977)(Hopmans’ Method I and IV) was superior to Hopmans’ Method III, because the polynomials better described the scatter in the data. This is not surprising, as the polynomial,
generally of the third or fourth degree, has a higher flexibility in fitting highly variable
data.
Results for the estimated soil water retention parameters {ar, n} are summarized inTable 7. As could be expected, the retention data scaled by depth could be better describedthan the pooled data. The estimated retention curves shown in Fig. 7(a) for the 0.05 and0.15 m depth are based on the parameter values from Table 7 (scaling by depth) andestimated values for 0, taken from Table 3. We assumed 8, = 0 at all depths. Differences
between the fitted water retention curves for the unscaled (Table 3) and scaled retention
data appear mainly as a result of having a steeper slope (higher n values), especially for the
0.05 m and 0.60 m depth.
Similar results were obtained for the scaled conductivity data (Table 8), although thedifferences in performance and estimated parameter values between the two data sets were
smaller in comparison with the retention data. The results given here are for the case wheren and m are treated as independent variables. Parameter estimations with the constraint m =Table I
EAnated pammeters of Eq. (2) using scaled retention data (tn = 1 - l/n)

DepthCm)

0.05
0.15
0.30
0.45
0.60

Scaling by depth

Scaling pooled data

cc (cm-‘)n

I.1

u (cm-‘)

n

r*

0.019
0.044
0.037
0.027
0.026

1.433
1.113
1.115
1.183
1.198

0.736
0.836
0.891
0.888
0.846

0.019
0.027
0.032
0.018
0.014

1.471
1.038
1.153
1.246
1.325

0.686
0.825
0.869
0.848
0.779

D. Mallants et aLLJournal of Hydrology 199 (1997) 295-318314

0.30 0.25 -

u‘

lE-1

Q
‘a

lE-2

s
8
Y
‘3
e
u
P
%

‘z
2

-

_

O.l&mDEPTH-

0.05-m DEPTH-o.oknmPlH
lE-3

---c-

0.15-m DEPnl

/

‘E-4

1E-5

-unseabdconducthMydata

lE6
1E-7

Fig. 7. (a) Estimatedretentionfunctionsbesedon unscaled
andscateddrainagedata.(b) Estimatedconductivity
funclion~ bawd on unscaledand scaleddrainagedata.

1 -‘l/n were also carried out but the goodness of fit in terms of r2 was reduced somewhat ascompared to the variable case (results not further shown here). Differences in estimatedhydraulic conductivity between the unscaled and scaled data (scaling by depth) are considerable as exemplified for the 0.05 and 0.15 m depths (Fig. 7(b)). In all cases, using thescaled K(S,) data resulted into much lower conductivities in comparison with the unscaled

D. Malkmts et aLLJournal of Hydrolosy 199 (1997) 295-318

315

Table 8Estimatedparametersof Mualem’sconductivity model using scaled conductivity data

Depth(m)Scaling pooled data

Scaling by depth
?I

ma

K*

2

n

m’

0.05

0.15
0.30
0.45
0.60

8.09
1.26
7.34
6.69
3.34

0.010
0.030
0.004
0.005
0.009

0.15
107.5
8.92
2.69
2.34

KS

t*

(cm h-‘)(cm h-l)
0.746
0.846
0.916
0.883
0.899

7.08
1.23
7.57
7.02
2.48

0.011
0.034
0.004
0.005
0.014

0.27
52.8
10.4
4.46
9.14

0.714
0.828
0.910
0.874
0.892

’Variable tn case.

data. This is because the estimated II parameter generally was much larger for the scaled
data than for the unscaled data (Table 8).

4. conclusions
Both water retention and hydraulic conductivity functions are influenced by themeasurement method. Comparison of soil water retention functions calculated from agravity-drainage experiment involving 15 undisturbed soil cores of 1 m length takenfrom a layered sandy loam with those obtained from desorption on small soil cores showedthat the former gave consistently lower water contents in the Ap and C2 horizon. These
disparities could be the result of air entrapment during initial saturation of the large
columns. The agreement between retention functions was relatively good for the Cl
horizon.
predicted unsaturated conductivities using estimated van Genuchten parameters fromsmall soil cores were higher than conductivities calculated from the gravity-drainage
experiment. This is partly due to the above-mentioned difference between the retention
curves for both methods, i.e. for given pressure head values, lower water contents formethod 1 will result in lower unsaturated conductivities compared to method 3. Anotherreason may be the inadequacy of the capillary model theory to describe the hydraulicconductivity of our structured soil. A good agreement in the pressure head range of 0 to
-50 cm was found when the unsaturated hydraulic conductivity relationship computedfrom the drainage data, was compared with K values obtained from combined crust/hot-air
data. However, the combined crust/hot-air method gave higher values in the range of -50
to -100 cm. These disparities together with the high variability in conductivities derived
from hot-air measurements may be attributed to the fact that diffusivities had to betranslated to conductivities. There also is a difference in application range: (i) a rather
narrow range for the gravity-drainage method; (ii) a wide range for the crust/hot-air
method and the desorption method. Finally, experimental efforts for (i) are much larger
than for (ii), especially in view of spatial variability studies.
Scaling soil water retention and hydraulic conductivities was found to be very effective
in coalescing large data sets, irrespective of the measuring technique. Variability of316

D. Mallants et al./Journal of Hydrology 199 (1997) 295-318

scaling factors was much higher for scaled K thanfor scaled h data. No relationship was
found between h and K scaling factors obtained from the drainage experiment, indicatingthat for our soil different sets of scaling parameters have to be determined for hydraulicconductivity and pressure head. K scaling factors obtained from the combined crust/hot-air
data were less variable in comparison with those obtained from the scaled drainage data.
We also found that h scaling factors determined from retention data measured on small soilcores showed less variability than those obtained from the drainage data. These results
indicate that statistics of scaling factors are also influenced by the measurement methodused.
For the shallower two depths, fitting of the van Genuchten function to scaled retention
data obtained from the drainage experiment resulted in CTvalues that were considerably
different from those estimated from unscaled retention data. For the remaining depths,
differences between the estimated van Genuchten parameters were smaller when using
either scaled or unscakl retention data. A comparison between estimated conductivity
functions using either unscaled or scaled K data(from drainage experiment) revealed large
differences for all depths considered. Mean retention and conductivity functions obtainedby direct fitting of unscaled data or alternatively by fitting scaled data will be used forsolutions of water flow problems in heterogeneous soils. An evaluation of the effect ofusing different sets of hydraulic functions in calculations of water flow problems is the
subject of a future study.
Acknowledgements
The authors give special thanks to the anonymous reviewers for their comments which
improved the readability of the paper. We also thank Nobuo Toride for discussions on an
earlier version of this manuscript.
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