Pedosphere 20(4): 456–465, 2010
ISSN 1002-0160/CN 32-1315/P
c 2010 Soil Science Society of China

Published by Elsevier Limited and Science Press

Estimation of the van Genuchten Soil Water Retention
Properties from Soil Textural Data∗1
B. GHANBARIAN-ALAVIJEH1,2 , A. LIAGHAT1 , HUANG Guan-Hua3,∗2 and
M. Th. VAN GENUCHTEN4
1 Department of Irrigation and Reclamation Engineering, University of Tehran, Karaj 31587-77871 (Iran)2 Department of Earth and Environmental Sciences, Wright State University, Dayton, OH 45435 (USA)
3 Department of Irrigation and Drainage Engineering, China Agricultural University, Beijing 100083 (China)
4 Department of Mechanical Engineering, Federal University of Rio de Janeiro, RJ 21945-970 (Brazil)(Received December 29, 2009; revised May 21, 2010)

ABSTRACTThe van Genuchten (vG) function is often used to describe the soil water retention curve (SWRC) of unsaturated
soils and fractured rock. The objective of this study was to develop a method to determine the vG model parameter mfrom the fractal dimension. We compared two approaches previously proposed by van Genuchten and Lenhard et al. forestimating m from the pore size distribution index of the Brooks and Corey (BC) model. In both approaches we used a
relationship between the pore size distribution index of the BC model and the fractal dimension of the SWRC. A dataset
containing 75 samples from the UNSODA unsaturated soil hydraulic database was used to evaluate the two approaches.
The statistical parameters showed that the approach by Lenhard et al. provided better estimates of the parameter m.
Another dataset containing 72 samples from the literature was used to validate Lenhard’s approach in which the SWRC
fractal dimension was estimated from the clay content. The estimated SWRC of the second dataset was compared withthose obtained with the Rosetta model using sand, silt, and clay contents. Root mean square error values of the proposed
fractal approach and Rosetta were 0.081 and 0.136, respectively, indicating that the proposed fractal approach performed
better than the Rosetta model.
Key Words:

fractal dimension, soil water retention curve, van Genuchten parameterization

Citation: Ghanbarian-Alavijeh, B., Liaghat, A., Huang, G. H. and van Genuchten, M. Th. 2010. Estimation of the van
Genuchten soil water retention properties from soil textural data. Pedosphere. 20(4): 456–465.

Many soil and water management and environmental protection practices require knowledge of theevolution of water and solutes in the subsurface. During the past several decades, a large number ofcomputer models have been developed to simulate water ﬂow and contaminant transport in saturated
and unsaturated soils and fractured rock. Their application is often restricted by a lack of hydraulicproperty information involving the soil water retention curve (SWRC) and the unsaturated hydraulicconductivity. Furthermore, due to inherent temporal and spatial variability of the hydraulic properties inthe ﬁeld, large numbers of samples are generally required to properly characterize the spatial distributionof the hydraulic properties.
Accurate characterization and estimation of the SWRC has been a major focus of research for morethan 60 years. Many empirical models (Gardner, 1958; Brooks and Corey, 1964; Campbell, 1974;
Clapp and Hornberger, 1978; van Genuchten, 1980; Hutson and Cass, 1987; Russo, 1988) have beendeveloped for modeling the SWRC. Unfortunately, estimating the coeﬃcients in these various models
remains diﬃcult and time-consuming. Several studies consequently have been carried out to predictthe coeﬃcients using pedotransfer functions (PTFs) (Cosby et al., 1984; Saxton et al., 1986; Vereecken
et al., 1989; Wösten et al., 1995; Wösten, 1997; Rawls et al., 2001), including neural network analyses(Schaap and Bouten, 1996; Schaap et al., 1998; Minasny and McBratney, 2007), and the use of tables∗1 Supported by the National Natural Science Foundation of China (Nos. 50979106 and 50779067).
∗2 Corresponding author. E-mail: ghuang@cau.edu.cn.

ESTIMATION OF WATER RETENTION PROPERTIES

457

based on soil texture (Rawls et al., 1982). Schaap et al. (2001) recently developed a computer program(Rosetta) that implements ﬁve PTFs for hierarchical estimation of the soil water retention and thesaturated and unsaturated hydraulic conductivity. The hierarchy in the PTFs allows predictions of thehydraulic parameters using limited (textural classes only) to more extended (texture, bulk density, andone or two water retention points) input data. Rosetta is based on neural network analyses combinedwith the bootstrap method, thus allowing the program to provide uncertainty estimates of the predicted
hydraulic parameters (Schaap et al., 2001). As noted by Giménez et al. (1997) most PTFs are essentiallyempirical, which causes their application to be restricted by the data upon which they were based.
Another approach of estimating the SWRC from the particle-size distribution is based on the premiseof shape similarity between the water retention curve and the cumulative particle-size distribution.
Several quasi-physical models (Arya and Paris, 1981; Haverkamp and Parlange, 1986; Fredlund et al.,
2002; Haverkamp et al., 2005; Leij et al., 2005) have been developed using this premise.
Considerable attention has been focused recently on the development of functional relationships
between the water content and the matric potential using fractal geometry models. Fractal models havesolid mathematical and physical bases in their description of the geometry of porous media. Threetypes of SWRC models have been proposed based on the fractal organization of soil structure. One
type is the mass fractal (Sierpinski carpet or Menger sponge) in which the fractal dimensions of mass,pore surface and pore volume have the same value (Rieu and Sposito, 1991; Perfect, 1999). A second
approach is based on the fractal surface (de Gennes, 1985; Toledo et al., 1990) without considering thescaling of mass. A third approach considers the fractal pore-size distribution without any assumption
about the geometry of mass and the solid-pore interface (Tyler and Wheatcraft, 1990; Pachepsky et al.,
1995; Perrier et al., 1996).
Perfect (1999) presented a SWRC function based on the capillary equation and a randomized Menger
sponge algorithm with upper and lower scaling limits. The theoretical equation was ﬁtted to water
retention data of six soils collected by Campbell and Shiozawa (1992). While all of the ﬁts were excellent,
reﬁtting the same data over restricted subsets of the matric potential caused the fractal dimension to
become greater than the theoretical limit of 3.0. Perfect (2005) later presented a modiﬁcation of Rieu
and Sposito’s (1991) prefractal water retention equation to accommodate hysteresis during monotonic
drying. Model testing consisted of ﬁtting the modiﬁed equation to previously published water retention
data for diﬀerent randomized Sierpinski carpets and soils. A more generalized model for the SWRC
was recently developed by Bird et al. (2000) based on the pore-solid fractal (PSF) distribution. The
PSF model displays symmetry between the solid and pore phases (Bird et al., 2000) and includes as
special cases several existing models, such as the Brooks and Corey (BC) model (Brooks and Corey,1964). Theoretically, the PSF model provides a direct method to estimate the SWRC fractal dimension
from the particle-size distribution. However, Bird et al. (2000) used only two datasets to test theirmodel. Using data from the unsaturated soil hydraulic database (UNSODA), Huang and Zhang (2005)
used the PSF model to obtain a method of estimating the fractal dimension by using either clay content
or soil textural class.
As an alternative to the BC model, van Genuchten (1980) proposed an empirical SWRC model whichassumes a more or less sigmoidal shape and includes three shape parameters (two if the parameters are
restricted to certain values). As a special case of the PSF model, the BC model can be related to thevan Genuchten (vG) model when certain relationships are assumed between the vG and BC model
parameters (van Genuchten, 1980; Lenhard et al., 1989; Morel-Seytoux et al., 1996). Since the power ofthe BC model is a function of the fractal dimension of the PSF model, this implies that the vG model
parameters can also be expressed as functions of the fractal dimension. However, to our knowledge noliterature exists where fractal methods have been used to estimate vG model parameters. Thus, theobjective of this study was to develop a fractal method for estimating the vG model parameters andthe soil water retention curve, and to compare results with those obtained with the Rosetta software,
which has been used widely in the literature.

458

B. GHANBARIAN-ALAVIJEH et al.

THEORYThe vG model has the following form (van Genuchten, 1980):
Se =1[1 + (αh)n ]m(1)

where Se is the eﬀective saturation and calculated by Se = (θ − θr )/(θs − θr ), where θ is the watercontent (cm3 cm−3 ), θr is the residual water content, and θs is the saturated water content, and where
h is the matric potential (kPa), and α (kPa−1 ), n and m (m = 1 − 1/n) are empirical parameters.
Assuming no residual water, the BC model (Brooks and Corey, 1964) has the following form: h −λ
θ
=
θs
hmin

(2)

where λ is an empirical parameter often referred to as the pore size distribution index, and hmin is
the air entry value (kPa). Assuming the fractal pore-size distribution to be that of a Sierpinski carpet,
Tyler and Wheatcraft (1990) proposed a SWRC model without a lower cut-oﬀ of scale, which has aform similar to the BC model: h D−E
θ
=
(3)
θs
hmin
where E is the Euclidian dimension, being equal to 2 or 3 for two- and three-dimensional systems,
respectively, and D is the fractal dimension of the SWRC. Based on the assumptions of a Sierpinski
carpet, the fractal dimension D can be deﬁned as (Mandelbrot, 1983):
D=

log[(bi )E − z]log(bi )(4)

where b is a scale factor and z is the number of removed squares in each iteration i of the Sierpinski
carpet.
Eq. 3 has a similar form as the BC model. The fractal dimension D could explain the empirical
exponent λ (Huang et al., 2006). We note that the corresponding fractal model is only a distribution
model without considering the solid phase. The power-law form of the SWRC is a special case of thepore-solid fractal model (Bird et al., 2000; Huang and Zhang, 2005), and has been presented also by deGennes (1985), Hunt and Gee (2002), and Xu (2004).
Lenhard et al. (1989) related the BC and vG models by equating the diﬀerential ﬂuid saturation
capacities, dSe /dh, of the two models to obtain the following relationship between the BC model poresize distribution index λ and the vG model parameter m:λ=m(1 − Se1/m )
1−m

(5)

To deﬁne a single “eﬀective” λ, the authors proposed using Se equal to 0.5 in accordance with thesuggestion by van Genuchten (1980) that the best location for evaluating dSe /dh is midway between
the saturated and residual water contents. Employing this approach, Eq. 5 becomes:λ=m(1 − 0.51/m )
1−m

(6)

Comparing Eqs. 2 and 3 and considering a three-dimensional Euclidian domain, one has:
λ=3−D
Substituting Eq. 7 in Eq. 6 leads to:(7)

ESTIMATION OF WATER RETENTION PROPERTIES

D =3−

459

m
(1 − 0.51/m )
1−m

(8)

Based on Eq. 8, the value of m can be estimated from the fractal dimension D of the SWRC. Anotherway to estimate m from D is using one of the van Genuchten (1980) restrictions on the value of m.
When assuming m = 1 − 1/n, we have:
λ=n−1

(9)

which results from equating the BC and vG models at large values of h such that (αh)n  1 in Eq. 1,
and hence λ = mn. The equation alternatively follows from Eq. 5 by setting Se = 0. Eq. 9 was used by
Ma et al. (1999). Substituting Eq. 9 into 7 and using m = 1 − 1/n, one has:
m=

3−D
4−D

(10)

A relationship between hmin (kPa) and the vG model parameter α (kPa−1 ) was suggested by Lenhard
et al. (1989) as follows:
1/λ

α=

Sx
(S −1/m − 1)1−mhmin x(11)

where Sx is a match-point eﬀective wetting phase saturation and is given by the empirical equation:
Sx = 0.72 − 0.35 exp(−n4 )(12)Based on an analysis of 24 compacted clay samples, Tinjum et al. (1997) presented another empirical
relationship between hmin (kPa) and α (kPa−1 ):α = 0.078(hmin )−1.26(13)

MATERIALS AND METHODS
The measured SWRC data of 75 soil samples (i.e., dataset 1) with a wide range of textures (as shownin Table I) were taken from the UNSODA database of Leij et al. (1996) to analyze the relationships
between the fractal dimension and the soil hydraulic parameters, especially m in Eqs. 8 and 10.
TABLE I
Soil properties of the 75 samples from the unsaturated soil hydraulic database (Leij et al., 1996) and the ﬁtted van
Genuchten model (1980) parameters m and α
TextureNo. of samples

D a)
Max

hmin b)
Min

Max

m
Min

Max

α
Min

Max

0.47
0.16
0.11
0.07
0.11
0.08
0.09
0.09
0.05
0.05
0.02

kPa−1
1.28
0.20
1.30
0.30
5.61
0.09
1.60
0.20
0.24
0.01
1.97
0.09
1.47
0.13
1.51
0.01
2.46
0.21
0.10
16.30
0.04
50.30
0.01

kPa
Sand
Loamy sand
Sandy loam
Sandy clay loam
Sandy clay
Loam
Silt loam
Silty clay loam
Clay loam
Silt
Silty clay
Clay

6
8
6
5
3
8
7
10
5
1
8
8

2.668
2.817
2.915
2.946
2.965
2.920
2.907
2.947
2.967
2.802
2.960
2.969

2.245
2.493
2.667
2.877
2.922
2.861
2.744
2.819
2.851
2.832
2.941

a) D is the fractal dimension of soil water retention curve; b) h

1.73
1.21
1.61
2.03
19.05
2.68
2.37
4.42
1.24
0.92
4.07
3.74

0.36
0.29
0.12
0.28
1.49
0.24
0.39
0.49
0.32
0.02
0.02

0.85
0.68
0.73
0.39
0.26
0.34
0.29
0.48
0.26
0.25
0.50
0.13

min is the air entry value.

Min

460

B. GHANBARIAN-ALAVIJEH et al.

The vG model parameters m, α and θr were calculated directly by ﬁtting the vG model to themeasured data using the RETC (RETention Curve) program (van Genuchten et al., 1991), given thecondition that m = 1 − 1/n. The fractal dimension and air entry value were determined directly byﬁtting the Tyler and Wheatcraft’s (1990) model (i.e., Eq. 3) to the measured data by nonlinear leastsquares optimization method using the MATLAB software (MathWorks, 2005). The calculated fractal
dimension was subsequently used to estimate the values of m of the 75 soil samples using Eqs. 8 and
10. Values for α were estimated by Eqs. 11 and 13, in which m and hmin were the directly determinedvalues. The estimated m and α values were subsequently compared with the directly calculated m andα values.
Another 72 soil samples (referred to as dataset 2) representing 11 diﬀerent soil textures, collectedfrom the literature (Puckett et al., 1985; Leij et al., 1996; Huang et al., 2006), were used to further
test the various methods for estimating m and α. Selected properties of these samples are presented inTable II.
TABLE II
Soil properties of 72 samples collected from the literature and the ﬁtted van Genuchten model (1980) parameters m andα
TextureNo. of samplesmClay content
Max

Min

Max

α
Min

Max

g kg−1
Sand
Loamy sand
Sandy loam
Sandy clay loam
Loam
Silt loam
Silty clay loam
Clay loam
Sandy clay
Silty clay
Clay

2
10
11
15
7
5
8
6
5
2
1

18
108
178
349
260
270
390
348
421
460
452

Min
kPa−1

14
23
70
208
122
120
280
304
352
420
-

0.61
0.61
0.58
0.46
0.49
0.20
0.30
0.39
0.33
0.09
0.34

0.55
0.25
0.11
0.06
0.19
0.12
0.12
0.05
0.09
0.08
-

0.28
0.45
0.50
0.40
0.50
0.98
0.90
0.50
0.50
0.65
0.09

0.27
0.24
0.13
0.15
0.18
0.15
0.10
0.08
0.18
0.55
-

We evaluated two methods for estimating m when m = 1 − 1/n. One approach (method 1) was
based on Eq. 8 with known D values estimated by a relationship proposed by Huang and Zhang (2005):
D = a0 +1 − ea1 Cp
a2 (1 + ea1 Cp ) + a3 (1 − ea1 Cp )(14)

where Cp is the soil clay percentage, and a0 , a1 , a2 and a3 are constant coeﬃcients equal to 2.35,
0.0822, −0.497 and 1.238, respectively. Eq. 14 was based on data from 148 disturbed and undisturbed
soil samples. Method 2 used the Rosetta software assuming known clay, silt and sand percentages.
The methods used for estimating α were: (1) Eq. 11 with hmin determined from the literature (Rawls
et al., 1982) for diﬀerent soil textures, and λ and m from Eqs. 7 and 8, respectively, while the fractal
dimension was estimated from the clay content using Eq. 14; and (2) the Rosetta software using sand,silt and clay contents. Consequently, the estimated vG model parameters using methods 1 and 2 as wellas the measured saturated water content were used to estimate the soil water retention curve.
For the estimation and validation procedures, the following statistical measures were used to evaluate
the results: mean of residuals (MR), root mean square error (RMSE), and Akaike’s information criterion(AIC), which are calculated as follows:
j


MR =

i=1

(Pi − Oi )j

(15)

ESTIMATION OF WATER RETENTION PROPERTIES

461


 j


(Pi − Oi )2
 i=1
RMSE =jj
AIC = jln i=1(Pi − Oi )2
j

(16)


+ 2q

(17)

where Oi and Pi are the observed and predicted values, respectively; j is the number of observations;and q is the number of model parameters.
RESULTS AND DISCUSSION
For dataset 1 (Leij et al., 1996), the estimated m values were obtained with Eqs. 8 and 10. A
comparison of the estimated m values with the directly calculated values as obtained with RETC isshown in Fig. 1. The two equations both underestimated m as evidenced by the negative MR values
listed in Table III. Predictions of m using Eq. 8 were slightly better than those obtained with Eq. 10.
This result is further conﬁrmed by the relatively low RMSE and AIC values obtained with Eq. 8 asshown in Table III. Similar results were also found by Ma et al. (1999).

Fig. 1 Comparison of estimated van Genuchten model parameter m values obtained with Eqs. 8 and 10 and estimatedα values based on Eqs. 11 and 13 with directly calculated values using the RETC program for 75 soil samples from the
UNSODA unsaturated soil hydraulic database.
TABLE III
Statistical parameters for diﬀerent methods used to estimate the van Genuchten (1980) model parameters m and α (kPa−1 )for the 75 soil samples from the unsaturated soil hydraulic database
Parameter

Model

Mean of residuals

Root mean square error

Akaike’s information criterion

m

Eq. 8
Eq. 10Eq. 11
Eq. 13−0.120−0.125
0.100
3.666

0.176
0.183
3.266
14.760−258.7−252.9
174.4
391.6

α

462

B. GHANBARIAN-ALAVIJEH et al.

In Eq. 8, the only parameter for estimating m is the fractal dimension D (0 < D < 3). We foundthat when D ranged from 0 to 3, the value of m ranged from 0.84 to 0.0001. The value of m hence did
not cover the entire range of m values (0 < m < 1). The values of m were found to be underestimatedusing Eq. 8. Also, the van Genuchten approximation, Eq. 9, assumed that the value of (αh)n was muchhigher than 1 so that it was possible to ignore the term 1 in the vG model. By ignoring this factor
in Eq. 1, to achieve a constant value for eﬀective saturation, the value of m or n was underestimated.
Therefore, Eq. 9 inherently underestimated the m values. The α values were also estimated with Eqs. 11
and 13 for dataset 1. A comparison of estimated α values with directly calculated results using RETC
is shown in Fig. 1. Eq. 11 with λ = 3 − D was found to give a more reasonable estimation for the αvalues. As shown in Table III, the RMSE values for Eq. 11 were signiﬁcantly lower than those of Eq. 13.
This conﬁrmed that Eq. 11 with the given hmin and m values estimated α values better than Eq. 13.
Dataset 2 was used for further validation of the proposed methods for estimation of m and α. A
comparison of m values estimated with the two methods against the directly calculated m values isshown in Fig. 2. We found that Rosetta (method 2) provided better estimates of the m values thanEq. 8 (method 1). As shown in Table IV, method 2 had lower AIC values than method 1. This implies that the neural network approach of Schaap et al. (2001) is more appropriate for estimating thevG model parameter m than the developed fractal approach. As shown in Fig. 2 and by the MR values inFig. 2 Comparison of estimated van Genuchten model parameter m values obtained with method 1 based on Eq. 8 andmethod 2 using the Rosetta software, and estimated α values obtained by method 1 based on Eq. 11 and method 2 using
the Rosetta software with directly calculated values using the RETC program for 72 soil samples collected from theliterature.
TABLE IV
Statistical parameters for diﬀerent methods used to estimate the van Genuchten model parameters m and α (kPa−1 ) andthe soil water retention curve (SWRC) for 72 soil samples collected from the literature
Parameter

MethodMean of residual

Root mean square error

Akaike’s information criterionmMethod 1 (Eq. 8)
Method 2 (Rosetta)
Method 1 (Eq. 11)
Method 2 (Rosetta)
Method 1 (Eqs. 8 and 11)
Method 2 (Rosetta)−0.1300.028
0.202−0.078−0.041−0.111

0.179
0.141
0.337
0.236
0.081
0.136

−245.7−275.7−150.4−201.7−3 099.3−2 456.1

αSWRC

ESTIMATION OF WATER RETENTION PROPERTIES

463

Table IV, the Rosetta software (method 2) overestimated the m values. An overestimation of m wasalso noted by Schaap et al. (2001). They found that Rosetta often underestimates the water content. A
sensitivity analysis of the vG model showed that the water content was underestimated when the shapeparameter n was overestimated. Overestimation or underestimation of the m values with Eq. 8 may be
attributed to underestimation or overestimation of the fractal dimension when using Eq. 14.
Dataset 2 was also used to validate diﬀerent methods for estimating α. Fig. 2 compares α values
estimated by methods 1 and 2, with the directly calculated RETC values. We found that the α values
estimated with Rosetta (method 2) were in a more restricted range compared with values estimated withmethod 1. However, both methods were unable to estimate α accurately. As shown by the MR values inTable IV, methods 1 and 2 overestimated and underestimated the α value, respectively. Judging fromthe AIC values, method 2 estimated the α values better than method 1. To estimate α using method
1, the values of m, λ and hmin should be estimated. The poor results obtained with method 1 may be
due to errors in estimating these parameters. The values of hmin were determined using soil texturalclass PTFs developed by Rawls et al. (1982). These PTFs may introduce additional uncertainty sincehmin is more related to soil structure and the pore size distribution. It may be better to estimate themaximum pore size from the maximum particle size. Using the relationship between the pore-size andparticle-size distributions as proposed by Arya and Dierolf (1992), hmin could be estimated using theYoung-Laplace equation and the maximum pore size.
We also found that the ﬁtted α (kPa−1 ) and hmin (kPa) were closely related. For dataset 1, weestablished the relationship:
α = 0.3126h−1.2366
min(R2 = 0.88, m = 1 − 1/n)

(18)

Eq. 18 is similar to Eq. 13, with only a small diﬀerence in the coeﬃcients. However, Eq. 18 was basedon 75 soil samples, whereas Eq. 13 was established using only 24 compacted clay specimens.
The estimated vG model parameters m and α obtained with the diﬀerent methods, in conjunction
with measured saturated water contents, were next used to estimate water contents at diﬀerent matricpotentials and compared with measured values. Fig. 3 shows the estimated soil water contents using
diﬀerent methods versus the measured values. The MR values in Table IV show that the Rosetta
software underestimated the water contents. Similar results were obtained by Schaap et al. (2001). The
other calculated statistical parameters presented in Table IV (i.e., the RMSE and AIC values) showedthat method 1 based on Eqs. 8 and 11 performed better than method 2, in which the Rosetta software
was used to estimate the soil water contents. The RMSE value of method 2 was 68% larger than thatof method 1, thus indicating that the proposed fractal approach provided better predictions of the soil
water retention curve than the Rosetta software using only soil textural properties such as sand, silt,
and clay contents.

Fig. 3 Comparison of estimated soil water content obtained with method 1 based on Eqs. 8 and 11 and method 2 using
the Rosetta software, with measured values of 72 soil samples taken from the literature.

464

B. GHANBARIAN-ALAVIJEH et al.

It should be pointed out that the Rosetta software can lead to better estimates of the vG model
parameters than the developed fractal approach (Table IV), even though the RMSE and AIC values
showed that predictions of the water content using Rosetta were not as good as those obtained withmethod 1 (Table IV). This is due to the fact that estimation of the water content is a result of aninteraction between the estimated vG model parameters, such as between α and m.
CONCLUSIONS
We used a method proposed by Lenhard et al. based on relationships between the vG and BC
models, and relationships between the BC hydraulic parameters and the pore-solid fractal dimension, toestimate the vG model parameters and subsequently the soil water retention curve. Two datasets, one
from the UNSODA database and one from the literature, were used to evaluate and validate the proposed methods. The approach allowed us to establish physical relationships that predict the vG model
parameters from the fractal dimension and air entry value. The proposed method was found to performbetter than the Rosetta software for the predictions of the soil water retention curve. However, none of
the methods we investigated was able to obtain reasonable predictions for the vG model parameter α.
REFERENCES
Arya, L. M. and Dierolf, T. S. 1992. Predicting soil moisture characteristics from particle-size distribution: An improved
method to calculate pore radii from particle radii. In van Genuchten, M. Th., Leij, F. J. and Lund, L. J. (eds.) Proceedings of the International Workshop on Indirect Methods for Estimating the Hydraulic Properties of Unsaturated
Soils. University of California Press, Riverside, CA. pp. 115–125.
Arya, L. M. and Paris, J. F. 1981. A physicoempirical model to predict soil moisture characteristics from particle-size
distribution and bulk density data. Soil Sci. Soc. Am. J. 45: 1023–1030.
Bird, N. R. A., Perrier, E. and Rieu, M. 2000. The water retention function for a model of soil structure with pore andsolid fractal distributions. Eur. J. Soil Sci. 51: 55–63.
Brooks, R. H. and Corey, A. T. 1964. Hydraulic Properties of Porous Media. Hydrology Paper No. 3. Colorado State
University, Fort Collins, CO.
Campbell, G. S. 1974. A simple method for determining unsaturated hydraulic conductivity from moisture retention data.
Soil Sci. 117: 311–314.
Campbell, G. S. and Shiozawa, S. 1992. Prediction of hydraulic properties of soils using particle size distribution and bulk
density data. In van Genuchten, M. Th., Leij, F. J. and Lund, L. J. (eds.) Proceedings of the International Workshop
on Indirect Methods for Estimating the Hydraulic Properties of Unsaturated Soils. University of California Press,
Berkeley. pp. 317–328.
Clapp, R. B. and Hornberger, G. M. 1978. Empirical equation for some soil hydraulic properties. Water Resour. Res.
14: 601–604.
Cosby, B. J., Hornberger, G. M., Clapp, R. B. and Ginn, T. R. 1984. A statistical exploration of the relationships of soil
moisture characteristics to the physical properties of soil. Water Resour. Res. 20: 682–690.
de Gennes, P. G. 1985. Partial ﬁlling of a fractal structure by a wetting ﬂuid. In Adler, D., Fritzsche, H. and Ovshinsky,
S. R. (eds.) Physics of Disordered Materials. Plenum Press, New York. pp. 227–241.
Fredlund, M. D., Wilson, G. W. and Fredlund, D. G. 2002. Use of the grain-size distribution for estimation of the soil-water
characteristic curve. Can. Geotech. J. 39: 1103–1117.
Gardner, W. R. 1958. Some steady-state solutions of the unsaturated moisture ﬂow equation with application to evaporation from a water table. Soil Sci. 85: 228–232.
Giménez, D., Perfect, E., Rawls, W. J. and Pachepsky, Y. 1997. Fractal models for predicting soil hydraulic properties: areview. Eng. Geol. 48: 161–183.
Haverkamp, R. and Parlange, J.-Y. 1986. Predicting the water retention curve from particle-size distribution: I. Sandy
soils without organic matter. Soil Sci. 142: 325–339.
Haverkamp, R., Leij, F. J., Fuentes, C., Sciortino, A. and Ross, P. J. 2005. Soil water retention: I. Introduction of a shapeindex. Soil Sci. Soc. Am. J. 69: 1881–1890.
Huang, G. and Zhang, R. 2005. Evaluation of soil water retention curve with the pore-solid fractal model. Geoderma.
127: 52–61.
Huang, G., Zhang, R. and Huang, Q. 2006. Modeling soil water retention curve with a fractal method. Pedosphere. 16:
137–146.
Hunt, A. G. and Gee, G. W. 2002. Water retention of fractal soil models using continuum percolation theory: Tests ofHanford site soils. Vadose Zone J. 1: 252–260.
Hutson, J. L. and Cass, A. 1987. A retentivity function for use in soil-water simulation models. J. Soil Sci. 38: 105–113.

ESTIMATION OF WATER RETENTION PROPERTIES

465

Leij, F. J., Alves, W. J., van Genuchten, M. Th. and Williams, J. R. 1996. Unsaturated Soil Hydraulic Database,
UNSODA 1.0 User’s Manual. EPA Report 600/R96/095. USEPA, Ada, OK.
Leij, F. J., Haverkamp, R., Fuentes, C., Zatarain, F. and Ross, P. J. 2005. Soil water retention: II. Derivation andapplication of shape index. Soil Sci. Soc. Am. J. 69: 1891–1901.
Lenhard, R. J., Parker, J. C. and Mishra, S. 1989. On the correspondence between Brooks-Corey and van Genuchten
models. J. Irrig. Drain. E. 115: 744–751.
Ma, Q. L., Hook, J. E. and Ahuja, L. R. 1999. Inﬂuence of three-parameter conversion methods between van Genuchten
and Brooks-Corey functions on soil hydraulic properties and water-balance predictions. Water Resour. Res. 35:
2571–2578.
Mandelbrot, B. B. 1983. The Fractal Geometry of Nature. W. H. Freeman, San Francisco.
MathWorks. 2005. Installation Guide for Windows. Version 7.1.
Minasny, B. and McBratney, A. B. 2007. Estimating the water retention shape parameter from sand and clay content.
Soil Sci. Soc. Am. J. 71: 1105–1110.
Morel-Seytoux, H. J., Meyer, P. D., Nachabe, M., Tourna, J., van Genuchten, M. Th. and Lenhard, R. J. 1996. Parameter
equivalence for the Brooks-Corey and van Genuchten soil characteristics: Preserving the eﬀective capillary drive.
Water Resour. Res. 32: 1251–1258.
Pachepsky, Y. A., Shcherbakov, R. A. and Korsunskaya, L. P. 1995. Scaling of soil water retention using a fractal model.
Soil Sci. 159: 99–104.
Perfect, E. 1999. Estimating soil mass fractal dimensions from water retention curves. Geoderma. 88: 221–231.
Perfect, E. 2005. Modeling the primary drainage curve of prefractal porous media. Vadose Zone J. 4: 959–966.
Perrier, E., Rieu, M., Sposito, G. and de Marsily, G. 1996. Models of the water retention curve for soils with a fractal
pore size distribution. Water Resour. Res. 32: 3025–3031.
Puckett, W. E., Dane, J. H. and Hajek, B. F. 1985. Physical and mineralogical data to determine soil hydraulic properties.
Soil Sci. Soc. Am. J. 49: 831–836.
Rawls, W. J., Brakensiek, D. L. and Saxton, K. E. 1982. Estimation of soil water properties. T. ASAE. 25: 1316–1320.
Rawls, W. J., Pachepsky, Y. and Shen, M. H. 2001. Testing soil water retention estimation with the MUUF pedotransfer
model using data from the southern United States. J. Hydrol. 251: 177–185.
Rieu, M. and Sposito, G. 1991. Fractal fragmentation, soil porosity, and soil water properties: I. Theory. Soil Sci. Soc.
Am. J. 55: 1231–1238.
Russo, D. 1988. Determining soil hydraulic properties by parameter estimation: On the selection of a model for thehydraulic properties. Water Resour. Res. 24: 453–459.
Saxton, K. E., Rawls, W. J., Romberger, J. S. and Papendick, R. I. 1986. Estimating generalized soil-water characteristics
from texture. Soil Sci. Soc. Am. J. 50: 1031–1036.
Schaap, M. G. and Bouten, W. 1996. Modeling water retention curves of sandy soils using neural networks. Water Resour.
Res. 32: 3033–3040.
Schaap, M. G., Leij, F. J. and van Genuchten, M. Th. 1998. Neural network analysis for hierarchical prediction of soil
water retention and saturated hydraulic conductivity. Soil Sci. Soc. Am. J. 62: 847–855.
Schaap, M. G., Leij, F. J. and van Genuchten, M. Th. 2001. Rosetta: A computer program for estimating soil hydraulicparameters with hierarchical pedotransfer functions. J. Hydrol. 251: 163–176.
Tinjum, J. M., Benson, C. H. and Blotz, L. R. 1997. Soil-water characteristic curves for compacted clays. J. Geotech.
Geoenviron. 123: 1060–1069.
Toledo, P. G., Novy, R. A., Davis, H. T. and Scriven, L. E. 1990. Hydraulic conductivity of porous media at low watercontent. Soil Sci. Soc. Am. J. 54: 673–679.
Tyler, S. W. and Wheatcraft, S. W. 1990. Fractal processes in soil water retention. Water Resour. Res. 26: 1047–1054.
van Genuchten, M. Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil
Sci. Soc. Am. J. 44: 892–898.
van Genuchten, M. Th., Leij, F. J. and Yates, S. R. 1991. The RETC Code for Quantifying the Hydraulic Functions of
Unsaturated Soils. EPA Report 600/2-91/065. US Salinity Laboratory, USDA, ARS, Riverside, CA.
Vereecken, H., Maes, J., Feyen, J. and Darius, P. 1989. Estimating the soil moisture retention characteristics from texture,bulk density and carbon content. Soil Sci. 148: 389–403.
Wösten, J. H. M. 1997. Pedotransfer functions to evaluate soil quality. In Gregorich, E. G. and Carter, M. R. (eds.) Soil
Quality for Crop Production and Ecosystem Health. Elsevier, Amsterdam. pp. 221–246.
Wösten, J. H. M., Finke, P. A. and Jansen, M. J. W. 1995. Comparison of class and continuous pedotransfer functions togenerate soil hydraulic characteristics. Geoderma. 66: 227–237.
Xu, Y. 2004. Calculation of unsaturated hydraulic conductivity using a fractal model for the pore-size distribution.
Comput. Geotech. 31: 549–557.