Predicting the soil moisture retention curve, from soil

Hydrol. Earth Syst. Sci., 18, 4053–4063, 2014www.hydrol-earth-syst-sci.net/18/4053/2014/doi:10.5194/hess-18-4053-2014© Author(s) 2014. CC Attribution 3.0 License.

Predicting the soil moisture retention curve,from soil particle size distribution and bulk density datausing a packing density scaling factor

F. Meskini-Vishkaee1, M. H. Mohammadi1,2, and M. Vanclooster2

1Department of Soil Science, Faculty of Agriculture, University of Zanjan, Zanjan, Iran2Earth and Life Institute, Environmental Sciences, Univ. Catholique de Louvain, Croix du Sud 2, Bte 2,1348 Louvain-la-Neuve, Belgium

Correspondence to:F. Meskini-Vishkaee ([email protected])

Received: 25 September 2013 – Published in Hydrol. Earth Syst. Sci. Discuss.: 21 November 2013Revised: 28 May 2014 – Accepted: 20 August 2014 – Published: 15 October 2014

Abstract. A substantial number of models predicting the soilmoisture characteristic curve (SMC) from particle size distri-bution (PSD) data underestimate the dry range of the SMCespecially in soils with high clay and organic matter con-tents. In this study, we applied a continuous form of the PSDmodel to predict the SMC, and subsequently we developeda physically based scaling approach to reduce the model’sbias at the dry range of the SMC. The soil particle packingdensity was considered as a metric of soil structure and usedto define a soil particle packing scaling factor. This factorwas subsequently integrated in the conceptual SMC predic-tion model. The model was tested on 82 soils, selected fromthe UNSODA database. The results show that the scaling ap-proach properly estimates the SMC for all soil samples. Incomparison to the original conceptual SMC model withoutscaling, the scaling approach improves the model estimationson average by 30 %. Improvements were particularly signifi-cant for the fine- and medium-textured soils. Since the scal-ing approach is parsimonious and does not rely on additionalempirical parameters, we conclude that this approach may beused for estimating SMC at the larger field scale from basicsoil data.

1 Introduction

Increasing contamination of the groundwater resources hasprofoundly accentuated the need for accurate predictions ofsubsurface flow and chemical transport. Water flow and sub-

sequent chemical transport are largely determined by the soilhydraulic properties, such as the soil moisture characteristiccurve (SMC) (Wang et al., 2002; Mohammadi et al., 2009).Measuring the soil hydraulic properties is still difficult, labor-intensive, and expensive. Therefore, many researchers havemade an attempt to develop an indirect method as an al-ternative to the direct measurement of hydraulic properties.For the SMC, indirect methods are classified into conceptualmethods (Nimmo et al., 2007; Mohammadi and Vanclooster,2011), semi-physical methods (e.g., Arya and Paris, 1981;Haverkamp and Parlange, 1982; Wu et al., 1990; Smettenand Gregory, 1996) and empirical methods (e.g., Saxton etal., 1986; Schaap et al., 1998).

The semi-physical methods are mainly based on shapesimilarity between the SMC and the particle size distribution(PSD) curve (Zhung et al., 2001; Schaap, 2005; Haverkampet al., 2005; Hwang and Choi, 2006), implying that the pore-size distribution (PoSD) is closely related to the PSD (Aryaet al., 2008). Arya and Paris (1981) did a pioneering work(AP model) for the development of semi-physical models.They showed that the pore size, which is associated with apore volume, is determined by scaling the pore length, usinga scaling factor,α. They demonstrated that an average valueof 1.38 forα scales the pore lengths based on spherical par-ticles to natural pore lengths properly. However, later inves-tigations by Arya et al. (1982), Tyler and Wheatcraft (1989),Basile and D’Urso (1997) and Vaz et al. (2005) revealed thatα value varies between 1.02 and 2.97 for fine- and coarse-textured soils, respectively. A slight error in the estimation

Published by Copernicus Publications on behalf of the European Geosciences Union.

4054 F. Meskini-Vishkaee et al.: Predicting the soil moisture retention curve, from soil particle size distribution

of α may result in considerable error in predicting the SMC(Schuh et al., 1988). Schuh et al. (1988) found that the valueof α varies with soil texture and suction head, especially inthe wet range of sandy soils. Using three formulations ofα,Arya et al. (1999) modeled the parameterα as a functionof particle sizes and showed thatα was not constant. It de-creased with increasing particle size, especially for the coarsefractions. Tyler and Wheatcraft (1989) showed that the pa-rameterα is equivalent to the fractal dimension of a tortuousfractal pore.

Although the empirical methods have been developed ex-tensively (e.g., Puhlmann and von Wilpert, 2012), the perfor-mance of an empirical method will depend on the databasesbeing used for the model calibration and testing (Tietje andTapkenhinrichs, 1993; Kern, 1995; Schaap and Leij, 1998;Schaap et al., 2004; Haverkamp et al., 2005; Hwang andChoi, 2006; Weynants et al., 2009). Moreover, direct mea-surements of SMC are often integrated as predictor vari-ables of the continuous SMC function. Many attempts havebeen made to reduce the sensitivity of the indirect meth-ods to empirical and database-dependent parameters. For in-stance, Mohammadi and Vanclooster (2011) proposed a con-ceptual robust model (MV) that does not include an empiricalparameter and is independent of the databases that are be-ing used. The disadvantages of semi-physical or conceptualmodels such as the AP and MV models are the use of “bun-dle of cylindrical capillaries” (BCC) concept to represent thepore space geometry and the lack of consideration of surfaceforces (Or and Tuller, 1999; Tuller et al., 1999; Mohammadiand Meskini-Vishkaee, 2012). These conceptual problemsoften lead to the underestimation of the dry range of the SMC(Arya et al., 1999; Hwang and Choi, 2006; Mohammadi andVanclooster, 2011). Such underestimations would result inlarge modeling errors of hydraulic dependent soil functionssuch as mechanical resistance functions (Gras et al., 2010),plant water uptake functions (Ryel et al., 2002), and micro-bial activity functions (Jamieson et al., 2002; Santamarí a andToranzos, 2003), in particular in arid environments.

To predict continuous SMC, Naveed et al. (2012) pa-rameterized the van Genuchten model based on the SMCdata points predicted from organic matter, clay, silt, finesand and coarse sand content. Mohammadi and Meskini-Vishkaee (2013) integrated the MV model with the vanGenuchten (VG) model (van Genuchten, 1980) to predictthe continuous SMC curve (MV–VG model) from PSD data.They found that ignoring the residual moisture content (θr) isthe main source of systematic error in the MV model. Theyfurther tested and compared four approaches to predict theθr, and showed that the incorporation of predictedθr willimprove the MV–VG prediction results considerably. How-ever, the estimation ofθr has some limitations, due to thelack of a conceptual underpinning and the poor predictabil-ity of θr (Leij et al., 2002). Tuller and Or (2005) suggestedthat the introduction ofθr as a fitting parameter in most SMCmodels often makes the physical representation of key pro-

cesses in the dry soils vague. Moreover, they pointed out thatthe dry range of the SMC shows remarkable scaling behav-ior. Arya et al. (2008) developed a procedure to scale nat-ural pore lengths, directly from straight pore lengths. Theyshowed that the scaling approach is less sensitive to uncer-tainties in model parameters and provides better predictionsof the SMC, compared with the original AP model.

Kosugi (1996) showed that the SMC can be expressedby a lognormal pore-size distribution function, while Ko-sugi and Hopmans (1998) found that the set of scaling fac-tors is lognormally distributed when PoSD curve is lognor-mal. Havayashi et al. (2007) used the Kosugi model (Kosugi,1996) to evaluate the effectiveness of three kinds of scal-ing factors obtained by the microscopic characteristic length,standard deviation of pore-size distribution and the porosity.They indicated that, in the natural forested hillslope soils, thevariability in the SMC is characterized by variability in theeffective soil pore volume. Nasta et al. (2009) concluded thatthe scaling of the PSD curves provides for adequate charac-terization of the mean and variance of SMCs, which allowsfor characterization of the soil spatial variability.

Many researchers developed empirical models for express-ing the SMC since the parameters of these models do notaddress the physical significance of the medium. Hence thespatial variability in the pore structure of soils is not fullyunderstood (Havayashi et al., 2007). Likewise, the conven-tional scaling approaches are based on empirical curve fit-ting, without considering the physical meaning of the scalingfactor (Perfect, 2005; Millán and González-Posada, 2005).To apply these models, one needs to determine the scalingfactor, where the complexity of measurements of the pore-size and pore-volume distributions easily nullifies the esti-mation of the scaling factors. Nevertheless, some efforts havebeen made to relate the scaling factor to the soil texture (Tuliet al., 2001; Millán et al., 2003).

From this brief review we conclude that the scaling ap-proaches improve the modeling and prediction of the SMC.Yet, most scaling approaches imply empirical parameters,and a robust fully conceptual approach for the estimation ofthe SMC from easily measurable properties is still lacking.

The MV model underestimates the moisture content in thedry range of the SMC because of the simplified conceptual-ization of the pore geometry. In particular the packing param-eter does not effectively reflect the pore geometry. The gen-eral aim of this study is to improve the accuracy of the modelproposed by Mohammadi and Meskini-Vishkaee (2013) us-ing a scaling approach.

Therefore the objectives of this study are (i) to formulatea robust and physically based model to scale the SMC fromthe PSD and porosity, and (ii) to compare the model perfor-mance with the results from the existing MV–VG model, us-ing soils documented in the UNSODA database (Nemes etal., 2000). We also evaluate the overall model performancewith the results from the full empirically SMC predictionsoftware ROSETTA (Schaap et al., 2001).

Hydrol. Earth Syst. Sci., 18, 4053–4063, 2014 www.hydrol-earth-syst-sci.net/18/4053/2014/

F. Meskini-Vishkaee et al.: Predicting the soil moisture retention curve, from soil particle size distribution 4055

2 Theory

Because of the close similarity between the shapes of thePSD and SMC curves, many researchers expressed an SMCmodel in terms of a PSD model (Haverkamp and Parlange,1982; Fredlund et al., 2000; Zhuang et al., 2001). The SMCmodel developed by van Genuchten (1980) is very flexible,widely used and given by

Se =

[1

1+ (αh)n

]m

(1)

Se =θ − θr

θs− θr, (2)

whereθ (L3 L−3) is the soil moisture content,Se (−) is effec-tive saturation degree andθs (L3 L−3) andθr (L3 L−3) are sat-urated and residual soil moisture contents, respectively. Theparametersn, m, θr andα (L−1) are fitting coefficients, andh (L) is the suction head.

The suction head,hi (L), corresponding to the particle ra-dius of theith fractionRi (L) is given by (Mohammadi andVanclooster, 2011)

hi =0.543× 10−4

Ri

ζ, (3)

whereζ (−) is a coefficient depending on the state of soilparticles packing and is defined as

ζ =1.9099

1+ e, (4)

wheree (−) is the void ratio given by

e =ρs− ρb

ρs, (5)

where theρs (ML−3) andρb (ML−3) are soil particle andbulk densities respectively.

Arya and Paris (1981) suggested that the moisture content,θi (L3 L−3), can be obtained from PSD andθs (L3 L−3), as

θi = θs

j=i∑j=

wj ; i = 1, 2, 3, . . . , k, (6)

wherewj is the mass fraction of particles (−) in thej th par-ticle size fraction. Consider that

Pi =

j=i∑j=1

wj (7)

would result in

θi/θs = S, (8)

whereS (−) is the saturation degree andPi (−) is the cu-mulative mass fraction of soil particles. It is obvious that

if θr = 0, thenSe= S and subsequentlyS = Pi . Arya andParis (1981), however, ignored the residual moisture content,while it may be a considerable value for many types of soiland clayey soils in particular. Combining Eqs. (1) and (3)with Eq. (7) yields

Pi =

1

1+

(α 0.543×10−4

Riζ)n

m

. (9)

In Eq. (9), the cumulative mass fraction,Pi , is substitutedwith theSe in Eq. (1). Hence, fitting Eq. (9) to the PSD dataenables one to directly predict the SMC parameters (n, m andα). Moreover, these coefficients allow expression of the con-tinuous form of predicted SMC. Since assuming thatθr = 0would result in model underestimation in dry range of theSMC (Mohammadi and Meskini-Vishkaee, 2013), we devel-oped a conceptual scaling approach to reduce the model bias.

Scaling approach

Following Havayashi et al. (2007), we suggest that the poros-ity is an appropriate property for inferring a characteristicscaling factor. Since the soil porosity is linked to the packingparameter,ζ , in the MV model (Eq. 4), we hypothesize thatζ is the characteristic scale of the soil.

We assume that the reference soil is the one that consists ofuniform-size spherical particles that are arranged in randomclose packing state, leading to minimum porosity (known asthe Kepler conjecture in literature of crystallography). Liter-ature suggests that the porosity of this packing state is 0.259(Hopkins and Stillinger, 2009). Subsequently, the maximumvalue of packing parameter,ζmax, would equal 1.41432 forreference soil. Hence the scaling factor,λ, for each soil sam-ple can be suggested by

λ =ζ

ζmax. (10)

In general, the values of the pore-size distribution index,n

(Eqs. 1 and 9), andζ are large for the coarse-textured soilsand small for the fine-textured soils. We suggest that theλ

can scale the parametern, obtained from fitting Eq. (9) tothe PSD data, to then parameter in the SMC model (Eq. 1)(hereaftern∗) as follows:

n∗= λ · n, (11)

wheren∗ is scaled to the PoSD index in VG model. Hence,the modified model is

θ

θs=

[1

1+ (αh)n∗

]m

. (12)

In summary, given a knownθs we can calculateζ and sub-sequentlyλ using Eq. (10). The soil parametersα andm areobtained from fitting Eq. (9) to the PSD data, andn∗ is es-timated by Eq. (11), and consequently the SMC is predicteddirectly by Eq. (12).

www.hydrol-earth-syst-sci.net/18/4053/2014/ Hydrol. Earth Syst. Sci., 18, 4053–4063, 2014


Table 1.Textural classes and UNSODA codes for soils used for testing and evaluating the approach.

Textural Clay Clay Loam Silt Silty Silty Loamy Sand Sandy Sandyclass loam loam clay clay sand clay loam

loam loam

UNSODA 1400 3033 3221 2000, 3090 3030 3100 1160 1050, 1240, 1460 3202 1130codes 2340 1211 3213, 3261 1360 3101 2102 1464, 1466, 2100 3180

2361 1260 4042, 4070 1371 2103 3133, 3134, 3140 32002362 1261 4180, 4181 3130 3141, 3144, 3155 32904120 2530 2464, 1341 3150 3162, 3163, 31644680 3190 1342, 1350 3152 3165, 3172, 33404681 3191 1351, 1352 3160 4051, 4152, 42632360 3222 2001, 2002 3161 4272, 4282, 4441

2010, 2011 3170 4520, 4650, 40002012 3171

4251

3 Material and methods

A total of 82 soil samples, with a wide range of physicalproperties that contained at least five PSD data, were selectedfrom the UNSODA hydraulic properties database (Nemeset al., 2000). UNSODA is a database of basic soil and hy-draulic properties from 790 samples, gathered from all overthe world, and compiled by the US Department of Agricul-ture. All soils are summarized in Table 1.

In this procedure, volumetric moisture contents corre-sponding to theith fraction were computed using Eq. (6),and suction heads were predicted using Eq. (3), in which theparameterζ was calculated with Eq. (4). In this study, weassumed that the porosity is equivalent toθs. For soils thatneither provide a porosity nor aθs, the first point of the SMCdata that corresponds to the lowest suction head was used asθs (Chan and Govindaraju, 2004).

We fitted Eq. (9) to the PSD data. We used nonlinearregression analysis to fit Eq. (9) to the PSD, using Mat-lab 7.1 software (Matlab 7.1, The Mathworks Inc., Natick,MA, USA) and the Marquardt–Levenberg algorithm (Mar-quardt, 1963). We calculated for each soil the scaling factor,using either the bulk density or the available saturated soilmoisture content, and predicted the SMC.

For each prediction, the agreement between the predictedmoisture contentθi(p) and measured moisture contentθi(m)

was expressed in terms of the root mean square errors (RM-SEs), given by

RMSE=

√√√√ 1

N

n∑i=1

(θi(p) − θi(m)

), (13)

in which N is the number of observed data points. The rel-ative improvement (RI) resulting from the scaling approachrather than MV–VG model was calculated as follows (Mi-nasny and McBartney, 2002):

RI(%) =RMSEM − RMSES

RMSEM· 100, (14)

where RI is the relative improvement, RMSEM and RMSEsare RMSE values of the MV–VG model and the currentscaled model, respectively. Obviously, a negative RI valueindicates that the scaling approach would diminish the accu-racy level of the prediction of the SMC in comparison withthe MV–VG model.

We also fitted a cubic polynomial function to the over-all predicted data and calculated the area between the fittedpolynomial and the 1 : 1 line from the difference of the nu-merical integrals of these two functions (do Carmo, 1976).

Moreover, to consider and compare the reliability of thescaled MV–VG model with a fully empirical SMC predic-tion model, we compared the estimations of scaled MV–VGmodel with the estimations of the ROSETTA software. Inthis software, we used the SSCBD model option; that is, weused textural (sand, silt, clay) percentages and bulk densityas model predictors (Schaap et al., 2001).

4 Results and discussion

Table 2 gives the comparison between the MV–VG model,the ROSETTA model and the scaled MV–VG model in termsof RMSE,R2 and RI. Table 2 demonstrates the significantlyimproved accuracy of the scaled MV–VG approach as com-pared to the original MV–VG model and ROSETTA. TheRMSEs of the predicted and measured moisture contentsranged from 0.0223 to 0.1502 for the original MV–VG model(average 0.086), from 0.0169 to 0.1122 (average 0.0601)for the scaled approach and from 0.0188 to 0.2453 (aver-age 0.745) for the ROSETTA software. In terms of RM-SEs, the scaled approach performed better than Schaap etal. (1998) and the Schaap and Leij (1998) models with simi-lar predictor variables. The results showed that there is a sig-nificant difference between performance of scaled MV–VG



Tabl

e2.

Ave

rage

root

mea

nsq

uare

erro

r(R

MS

E),

coef

ficie

nts

ofde

term

inat

ion

(R2),

and

rela

tive

impr

ovem

ent(

RI)

com

pare

dto

the

MV

–VG

mod

elan

dhy

drau

licpa

ram

eter

sfo

rea

chso

ilte

xtur

algr

oup,

with

stan

dard

devi

atio

nsin

pare

nthe

ses.

The

low

erca

sele

tters

aan

db

indi

cate

sign

ifica

ntdi

ffere

nces

atP

<0.

05.

Soi

lN

umbe

rR

MS

ER

2R

IH

ydra

ulic

para

met

ers

text

ure

ofso

ilsva

lue

(%)

MV

–VG

Sca

ling

RO

SE

TTA

MV

–VG

Sca

ling

RO

SE

TTA

θ sα

mn

n∗

λ

mod

elap

proa

chm

odel

appr

oach

(L3

L−

3)

(L−

1)

(−)

(−)

(−)

(−)

Cla

y8

0.08

80.

041

0.11

500.

973

0.97

70.

9153

53.8

70.

510.

043

0.12

82.

457

1.46

70.

6228

(0.0

14)

(0.0

20)

(0.0

403)

(0.0

17)

(0.0

20)

(0.0

710)

(16.

74)

(0.0

4)(0

.085

)(0

.104

)(2

.195

)(1

.127

)(0

.070

6)

Cla

y1

0.02

70.

017

0.14

680.

725

0.87

20.

9790

38.1

50.

580.

002

0.24

81.

634

1.04

00.

6365

loam

Loam

80.

078

0.04

50.

0546

0.89

60.

913

0.89

8641

.04

0.45

0.04

20.

157

3.11

52.

187

0.69

70(0

.019

)(0

.015

)(0

.033

3)(0

.100

)(0

.088

)(0

.066

7)(1

7.74

)(0

.06)

(0.0

30)

(0.0

79)

(1.4

52)

(1.1

00)

(0.0

389)

Silt

190.

082

0.05

90.

0512

0.92

20.

950

0.95

9825

.63

0.44

0.01

90.

233

4.93

73.

637

0.71

89lo

am(0

.026

)(0

.020

)(0

.022

2)(0

.043

)(0

.033

)(0

.023

0)(2

0.28

)(0

.04)

(0.0

11)

(0.3

65)

(2.5

93)

(1.9

82)

(0.0

414)

Silt

y2

0.07

60.

061

0.08

680.

932

0.94

10.

9166

3.01

0.51

0.04

00.

195

1.57

31.

091

0.66

28cl

ay(0

.056

)(0

.022

)(0

.042

7)(0

.040

)(0

.010

)(0

.034

8)(4

3.07

)(0

.09)

(0.0

28)

(0.1

57)

(1.4

02)

(1.0

39)

(0.0

699)

Silt

y1

0.12

90.

093

0.10

800.

887

0.92

40.

9083

28.1

50.

430.

020

0.11

62.

548

1.87

00.

7339

clay

loam

Loam

y11

0.09

30.

060

0.08

620.

893

0.92

60.

9067

32.6

30.

400.

067

0.17

95.

488

4.04

80.

7339

sand

(0.0

37)

(0.0

22)

(0.0

359)

0.06

2(0

.038

)(0

.053

7)(1

1.64

)(0

.07)

(0.0

38)

(0.0

58)

(1.3

57)

(1.0

88)

(0.0

445)

San

d27

0.09

30.

073

0.02

540.

854

0.89

30.

8853

20.7

40.

370.

052

0.45

85.

592

4.38

00.

8228

(0.0

30)

(0.0

24)

(0.0

626)

(0.1

02)

0.08

10.

0760

(7.5

1)(0

.04)

(0.0

33)

(0.4

44)

(2.0

18)

(1.5

31)

San

dy1

0.08

40.

065

0.06

530.

957

0.96

70.

9702

23.0

70.

360.

043

0.05

48.

000

6.58

20.

7031

clay

(0.0

630)

loam

San

dy4

0.07

30.

035

0.07

760.

950

0.97

10.

9364

51.6

30.

460.

066

0.09

35.

676

3.98

00.

7641

loam

(0.0

14)

(0.0

15)

(0.0

547)

(0.0

28)

(0.0

08)

(0.0

278)

(18.

12)

(0.0

5)(0

.020

)(0

.032

)(1

.526

)(1

.137

)(0

.044

4)

Ave

rage

82

0.0

86a

0.0

60b

0.0

74

5a0

.89

80

.92

70

.92

76

30

.14

0.4

20

.04

40

.27

24

.72

63

.51

90

.71

77

(0.0

28

)(0

.02

4)

(0.0

41

7)

(0.0

84

)(0

.06

5)

(0.0

64

)(1

8.8

8)

(0.0

7)

(0.0

40

)(0

.33

8)

(2.3

29

)(1

.81

6)

(0.0

53

2)



approach and ROSETTA (p = 0.05). Despite the pure statis-tical and empirical nature of the ROSETTA approach, it pro-vides worse results than the approach based on the currentscaling technique. The improvement of the scaled approachis also reflected by RI in Table 2. Except for soils no. 3033(clay loam) and 3090 (silt loam), the scaling approach re-sulted in more accurate predictions for all soils. Table 2 alsoindicates that the scaling approach can improve the modelestimations of the original MV–VG model by 30 %.

For the fine- and medium-textured soils, the values of RIare larger than for the coarse-textured soil. This result wasexpected, because the MV and MV–VG models underes-timate the dry range moisture content for the fine-texturesoils (Mohammadi and Vanclooster, 2011; Mohammadi andMeskini-Vishkaee, 2013), and subsequently the scaling ap-proach was more effective for these soils.

We examined the possible relations between the RI andsoil physical properties. Among all parameters, the satu-rated moisture content and scaling factor show strong rela-tions with the RI. Figure 1a shows that the RI values in-crease significantly with the saturated moisture content of thesoils; that is, the scaling approach would more effectivelyimprove the model accuracy for the fine-texture soils withhigherθs. This result can be confirmed with Fig. 1b, whichshows that the scaling factor is inversely correlated with theIR factor (Fig. 1b). Indeed, the soils with high porosity com-monly have an abundant amount of clay materials and or-ganic matter, characterized with high surface energy. Theseattributes are the main sources of errors of the MV and MV–VG models.

Typical examples of measured vs. predicted SMCs withthe MV–VG model, the scaling approach and ROSETTA forclay, sandy loam, loam, and silt loam textures are presentedin Fig. 2a–f. For the clay (codes: 2340 and 4681), sandyloam (code: 3180 and 3200), loam (code: 3191) and silt loam(code: 3090) soils, the scaling approach fits the data well andperforms better than the MV–VG model in the entire rangeof the SMC. For the silt loam soil (code: 3090), the scalingapproach slightly overestimates the moisture content throughthe entire range of suction heads and the MV–VG model un-derestimates the moisture content at low suction heads. Fig-ure 2a–f show that the ROSETTA software performs worsein the wet part of SMC. Overall, the scaling approach per-forms better than MV–VG model and ROSETTA softwarefor all soil samples (Table 2), but the performance of scal-ing approach did not suitably respond for two soil samples(codes: 3033, 3090). The residual model error may be relatedto the simplified representation of the total porosity which isconsidered equal to the saturated volumetric moisture con-tent. The swelling properties and high organic carbon contentof these soils (> 4 %, 3.85 %, respectively) may partially be asource of these errors. We further suspect that the complexityof the relationship between PSD, PoSD and pore connectiv-ity can be effective in the model performance (Zhuang et al.,

Figure 1. The efficiency of scaling approach, % RI, defined withRMSE (Eq. 14) as function of(a) the saturated moisture content and(b) scaling factor for all soil samples.∗∗: significant atP = 0.01.

2001). The assumption of the similarity between PSD andPoSD does not perform equally well to all soils.

We tentatively conclude that the scaling of the PSD curvesusing the parameterζ generally performs better in predictingthe SMC as compared to the original MV–VG model. Theunscaled MV–VG model underestimates the moisture con-tent at high suction heads.

The most semi-physical based methods for predictingSMC rely on the use of empirical parameters to improve theSMC estimates from PSD (Lilly and Lin, 2004). Hydraulicproperties are indeed affected by both the soil texture and thesoil structure (Haverkamp et al., 2002). The MV–VG modeluses the packing parameter,ζ , derived from soil bulk densityas a metric of soil structure. Moreover, the scaling param-eter that is inferred from the packing state is integrated inthe scaled MV–VG model. Hence the soil structural featuresare integrated in the MV–VG model at two levels: first at theMV–VG model to convert moisture into pressure head and,second, to correct the SMC model prediction. The good per-formance of scaling approach in the wet range and dry rangeof SMC suggests a convenient of soil structural features inthe SMC prediction.

Figure 3 compares all estimated moisture contents, usingMV–VG model and scaling approach, respectively, with themeasured soil moisture content for all the 82 soil samples.The overall predictability of the two methods is evaluated by



Figure 2. Examples of measured vs. predicted soil moisture characteristic curve (SMC) for each texture using the integrated MV–VG model(Eq. 9), scaling approach (Eq. 12) and ROSETTA: for(a) clay soil,(b) sandy loam soil,(c) loam soil,(d) sandy loam soil,(e) clay soil and(f) silt loam soil.

comparing the experimental data and the predicted soil mois-ture content on a 1 : 1 plot. Linear regression of the measuredand estimated moisture contents, using the two methods forall the soil samples, showed that the slope values were 0.7675and 0.8484, and the coefficients of determination (R2) be-tween the estimated results and measured data for all soilswere 0.765 and 0.8565 for MV–VG model and scaling ap-

proach, respectively. Hence, the MV–VG model and the scal-ing approach underestimate the moisture content by about23 and 15 %, respectively, while the bias of the scaling ap-proach is smaller than the MV–VG model. Since it has beenreported that usual measurement method of SMC (pressureplate apparatus) is susceptible to some errors at high soilsuction heads (Campbell, 1988; Gee et al., 2002; Cresswell



Figure 3. Comparisons of the measured and estimated moisturecontents for 82 selected soils using the MV–VG model (Eq. 9) andscaling approach (Eq. 12). Dashed lines: the 1 : 1 line. Solid lines:linear-regression line. Solid green line: nonlinear-regression line.

et al., 2008). We suggest that a part of the underestimationin the dry range of SMC of our method is partially relatedto limitation of this method for measuring the SMC (Soloneet al., 2012). Regarding theR2, the scaling method still re-mains the most preferable method. Comparing the overallpredictability of the two methods, the correlation coefficientof linear regression should not replace the visual examinationof the data. We, therefore, use the cubic polynomial functionto adequately express the data variations. The fitted polyno-mial functions are drawn as shadowed green curves in Fig. 3.We further suggest that the area between the fitted curve andthe 1 : 1 line (AE) is an expression of the systematic error.The values of AE were 0.0369 and 0.0250 for the MV–VGmodel and the scaling approach respectively. It confirms thatthe level of systematic errors of the scaling approach is about33 % less than that of MV–VG model. This result can be con-firmed by the comparison of theR2 values obtained whenpredicting the SMC for each soil with two methods (Table 2columns 5 and 6). We conclude that the scaled PSD curvewill result in a more accurate prediction of the SMC as com-pared to the unscaled PSD data.

To scale the SMC, Tuller and Or (2005) used the soil spe-cific surface area (SA) and the thickness of water film to ex-press the moisture content in dry range of the SMC. Despitetheir reasonable model performance, the application of theirprocedure was limited due to difficulty in the measurementor the estimation of SA.

Havayashi et al. (2007) found that, in natural forested hill-slope soils, the variability in the SMC is scaled and charac-terized by the variability in effective porosity. Nevertheless,the determination of the effective porosity is also difficult.

The scaling factor proposed in the current study is definedby using the index of packing state which can be determinedeasily from the bulk density of soil and particle density. Ascompared to prediction models that rely on the measured at-tributes as suggested above, or prediction models that relyon measured SMC data, our approach is based on a robustmetric of the soil structure: the packing density. It does notrely on any other additional empirical parameter. The scaledMV–VG model is therefore very parsimonious and robust.We therefore conclude that the scaled MV–VG model maybe appropriate for predicting SMC from basic soil data.

5 Conclusions

Using a new scaling approach, the current study showed thatthe continuous form of SMC curve can be predicted fromknowledge of PSD, as modeled by the van Genuchten (1980)model and particle packing state. In this approach it was as-sumed that the scaling factor can be defined as the ratio ofpacking state of a soil sample and the packing state of a ref-erence soil. Results showed that the proposed approach canadequately predict the SMC of 82 soil samples selected fromthe UNSODA database. It was further found that the scal-ing approach provides better predictions of the SMC thanMV–VG model and ROSETTA software, especially in thedry range of the SMC. For soils for which the error was im-portant, we attributed the proposed scaling approach error tohigh organic carbon content and swelling properties of thesoil. Indeed, in these soils the soil pore structure and poros-ity is changing in time, leading to uncertainty in the scalingfactor based on the soil porosity.

In summary, we concluded that the main advantages ofthe proposed scaling approach as compared to many SMCprediction models are the following: (i) the applied scalingfactor is determined easily from soil bulk and particle den-sities; (ii) the scaling factor has physical meaning, whichdoes not depend on soil database and empirical parameters;(iii) the proposed approach predicts a continuous form of theSMC; and (iv) this approach estimates the SMC more appro-priately in comparison with many other models. Consider-ing that there is no further need for empirical parameters, weconclude that this approach may be useful in estimating theSMC for regional-scale soil hydrological studies.



Appendix A

Table A1. Symbols and abbreviations.

SMC soil moisture characteristic curvePSD particle size distributionPoSD pore-size distributionMV Mohammadi and Vanclooster (2011) modelBCC bundle of cylindrical capillariesVG model van Genuchten modelMV–VG model integrated the MV model with the van Genuchten modelAP Arya and Paris (1981)PTF pedotransfer functionRMSE root mean square errorθ the soil moisture contentSe effective saturation degreeθs saturated moisture contentsθr residual moisture contentsn fitting coefficientsm fitting coefficientsα fitting coefficientsh suction headξ a coefficient depending on the state of soil particle packinge the void ratioρs soil particle densityρb soil bulk densitywj the mass fraction of particles in thej th particle size fractionS saturation degreePi cumulative mass fraction of soil particlesRi particle radius of theith fractionβ scaling factorγ microscopic characteristic lengths of the referenceγ microscopic characteristic lengths of the subjected soilξmax maximum value of packing parametern pore-size distribution indexλ scaling factorn∗ scaled the PoSD index in VG modelθi(p) predicted moisture contentθi(m) measured moisture contentRI relative improvementRMSEM RMSE values of MV–VG modelRMSEs RMSE values of scaling approachAE area between the fitted curve and 1 : 1 line



Edited by: G. Fogg

References

Arya, L. M. and Paris, J. F.: A physicoempirical model to predictthe soil moisture characteristic from particle-size distribution andbulk density data, Soil Sci. Soc. Am. J., 45, 1023–1030, 1981.

Arya, L. M., Ritchter, J. C., and Davidson, S. A.: A comparison ofsoil moisture characteristic predicted by the Arya–Paris modelwith laboratory-measured data, AgRISTARS Tech. Rep. SM-L1-04247, JSC-17820, NASA-Johnson Space Center, Houston, TX,1982.

Arya, L. M., Leij, F. J., van Genuchten, M. T., and Shouse, P. J.:Scaling parameter to predict the soil water characteristic fromparticle-size distribution, Soil Sci. Soc. Am. J., 63, 510–519,1999.

Arya, L. M., Bowman, D. C., Thapa, B. B., and Cassel, D. K.:Scaling soil water characteristics of golf course and athletic fieldsands from particle-size distribution, Soil Sci. Soc. Am. J., 72,25–32, 2008.

Basile, A. and D’Urso, G.: Experimental corrections of simplifiedmethods for predicting water retention curves in clay-loamy soilsfrom particle-size determination, Soil Technol., 10, 261–272,1997.

Campbell, G. S.: Soil water potential measurement: an overview,Irrig. Sci., 9, 265–273, 1988.

Chan, T. P. and Govindaraju, R. S.: Soil water retention curves fromparticle-size distribution data based on polydisperse sphere sys-tems, Vadose Zone J., 3, 1443–1454, 2004.

Cresswell, H. P., Green, T. W., and McKenzie, N. J.: The adequacyof pressure plateapparatus for determining soil water retention,Soil Sci. Soc. Am. J., 72, 41–49, 2008.

do Carmo, M.: Differential geometry of curves and surfaces,Prentice-Hall, Englewood Cliffs, 98 pp., 1976.

Fredlund, M. D., Fredlund, D. G., and Wilson, G. W.: An equationto represent grain-size distribution, Can. Geotech. J., 37, 817–827, 2000.

Gee, G. W., Ward, A. L., Zhang, Z. F., Campbell, G. S., and Math-ison, J.: The influence of hydraulic nonequilibrium on pressureplate data, Vadose Zone J., 1, 172–178, 2002.

Gras, J.-P., Delenne, J.-Y., Soulie, F., and ElYoussoufi, M. S.: DEMand experimental analysis of the water retention curve in poly-disperse granular media, Power Tech., 12, 231–238, 2010.

Havayashi, Y., Kosugi, K., and Mizuyama, T.: Soil water retentioncurves characterization of a natural forested hillslope using ascaling technique based on a lognormal pore-size distribution,Soil Sci. Soc. Am. J., 73, 55–64, 2007.

Haverkamp, R., and Parlange, J. Y.: Comments on “A physicoem-perical model to predict the soil moisture characteristic fromparticle-size distribution and bulk density data”, Soil Sci. Soc.Am. J., 46, 1348–1349, 1982.

Haverkamp, R., Reggiani, P., and Nimmo, J. R.: Property-TransferModels. in: Methods of soil analysis, Part 4, edited by: Dane, J.H. and Topp, G. C., SSSA Book Series No. 5, SSSA, Madison,WI, 759–782, 2002.

Haverkamp, R., Leij, F. J., Fuentes, C., Sciortino, A., and Ross, P.J.: Soil water retention: I. Introduction of a shape index, Soil Sci.Soc. Am. J., 69, 1881–1890, 2005.

Hopkins, A. B. and Stillinger, F. H.: Dense sphere packings fromoptimized correlation functions, Phys. Rev. E, 79, 031123,doi:10.1103/PhysRevE.79.031123, 2009.

Hwang, S. I. and Choi, S. I.: Use of a lognormal distribution modelfor estimating soil water retention curves from particle-size dis-tribution data, J. Hydrol., 323, 325–334, 2006.

Jamieson, R. C., Gordon, R. J., Sharples, K. E., Stratton, G. W.,and Madani, A.: Movement and persistence of fecal bacteria inagricultural soils and subsurface drainage water: A review, Can.Biosyst. Eng., 44, 1–9, 2002.

Kern, J. S.: Estimation of soil water retention models based on soilphysical properties, Soil Sci. Soc. Am. J., 59, 1134–1141, 1995.

Kosugi, K.: Lognormal distribution model for unsaturated soil hy-draulic properties, Water Resour. Res., 32, 2697–2703, 1996.

Kosugi, K. and Hopmans, J. W.: Scaling water retention curves forsoils with lognormal pore-size distribution, Soil Sci. Soc. Am. J.,62, 1496–1505, 1998.

Leij, F. J., Schaap, M. G., and Arya, L. M.: Indirect methods, in:Methods of Soil Analysis, Part 4, edited by: Dane, J. H. and Topp,C. G., SSSA Book Series 5, SSSA, Madison, 1009–1045, 2002.

Lilly, A. and Lin, H.: Using soil morphological attributes and soilstructure in pedotransfer functions, in: Development of pedo-transfer functions in soil hydrology, edited by: Pachepsky, Ya.and Rawls, W., Dev. Soil Sci. 30, Elsevier, Amsterdam, 115–141,2004.

Marquardt, D. W.: An algorithm for least-squares estimation of non-linear parameters, SIAM J. Appl. Math., 11, 431–441, 1963.

Millán, H. and González-Posada, M.: Modelling soil water retentionscaling, Comparison of a classical fractal model with a piecewiseapproach, Geoderma, 125, 25–38, 2005.

Millán, H., González-Posada, M., Aguilar, M., Domínguez, J., andCespedes, L.: On the fractal scaling of soil data. Particle-size dis-tributions, Geoderma, 117, 117–128, 2003.

Minasny, B. and McBartney, A. B.: The neuro-m method for fittingneural network parametric pedotransfer functions, Soil Sci. Soc.Am. J., 66, 352–361, 2002.

Mohammadi, M. H. and Meskini-Vishkaee, F.: Predicting the filmand lens water volume between soil particles using particle sizedistribution data, J. Hydrol., 475, 403–414, 2012.

Mohammadi, M. H. and Meskini-Vishkaee, F.: Evaluation of soilmoisture characteristics curve from continuous particle size dis-tribution data, Pedosphere, 23, 70–80, 2013.

Mohammadi, M. H. and Vanclooster, M.: Predicting the soil mois-ture characteristic curve from particle size distribution with asimple conceptual model, Vadose Zone J., 10, 594–602, 2011.

Mohammadi, M. H., Neishabouri, M. R., and Rafahi, H.: Predictingthe solute breakthrough curve from soil hydraulic properties, SoilSci., 174, 165–173, 2009.

Nasta, P., Kamai, T., Chirico, G. B., Hopmans, J. W., and Romano,N.: Scaling soil water retention functions using particle-size dis-tribution, J. Hydrol., 374, 223–234, 2009.

Naveed, M., Moldrup, P., Tuller, M., Ferre, T. P. A., Kawamato,K., Komatso, T., and de Jong, L. W.: Prediction of the soil watercharacteristic from soil particle volume fractions, Soil Sci. Soc.Am. J., 76, 1946–1959, doi:10.2136/sssaj2012.0124, 2012.

Nemes, A., Schaap, M. G., and Leij, F. J.: The UNSODA unsat-urated soil hydraulic property database, version 2.0, availableat: http://www.ars.usda.gov/Services/docs.htm?docid=8967(lastaccess: November 2013), 2000.


http://dx.doi.org/10.1103/PhysRevE.79.031123

http://dx.doi.org/10.2136/sssaj2012.0124

http://www.ars.usda.gov/Services/docs.htm?docid=8967


Nimmo, W., Herkelrath, N., and Luna, A. M. L.: Physically basedestimation of soil water retention from textural data: generalframework, new models, and streamlined existing models, Va-dose Zone J., 6, 766–773, 2007.

Or, D. and Tuller, M.: Liquid retention and interfacial area invariably saturated porous media: Upscaling from single-pore tosample-scale model, Water Resour. Res., 35, 3591–3605, 1999.

Perfect, E.: Modeling the primary drainage curve of prefractalporous media, Vadose Zone J., 4, 959–966, 2005.

Puhlmann, H. and von Wilpert, K.: Pedotransfer functions for waterretention and unsaturated hydraulic conductivity of forest soils,J. Plant Nutr. Soil Sci., 175, 221–235, 2012.

Ryel, R. J., Caldwell, M. M., Yoder, C. K., Or, D., and Leffler,A. J.: Hydraulic redistribution in a stand of Artemisia tridentata:Evaluation of benefits to transpiration assessed with a simulationmodel, Oecologia, 130, 173–184, 2002.

Santamaría, J. and Toranzos, G. A.: Enteric pathogens and soil: Ashort review, Int. Microbiol., 6, 5–9, doi:10.1007/s10123-003-0096-1, 2003.

Saxton, K. E., Rawls, W. J., Romberger, J. S., and Papendick, R.I.: Estimating generalized soil-water characteristics from texture,Soil Sci. Soc. Am. J., 50, 1031–1036, 1986.

Schaap, M. G.: Models for indirect estimation of soil hydraulicproperties, in: Encyclopedia of Hydrological Sciences, edited by:Anderson, M. G. and Mc Donnell, J. J., Wiley, 1145–1150, 2005.

Schaap, M. G. and Leij, F. J.: Database related accuracy and uncer-tainty of pedotransferfunctions, Soil Sci., 163, 765–779, 1998.

Schaap, M. G., Leij, F. J., and van Genuchten, M. Th.: Neural net-work analysis for hierarchical prediction of soil water retentionand saturated hydraulic conductivity, Soil Sci. Soc. Am. J., 62,847–855, 1998.

Schaap, M. G., Leij, F. J., and van Genuchten, M. Th.: ROSETTA:A computer program for estimating soil hydraulic parameterswith hierarchical pedotransfer functions, J. Hydrol., 251, 163–176, 2001.

Schaap, M. G., Nemes, A., and van Genuchten, M. Th.: Comparisonof models for indirect estimation of water retention and availablewater in surface soils, Vadose Zone J., 3, 1455–1463, 2004.

Schuh, W. M., Cline, R. L., and Sweeney, M. D.: Comparison of alaboratory procedure and a textural model for predicting in situsoil water retention, Soil Sci. Soc. Am. J., 52, 1218–1227, 1988.

Smettem, K. R. J. and Gregory, P. J.: The relation between soil waterretention and particle-size distribution parameters for some pre-dominantly sandy Western Australian soils, Aust. J. Soil Res.,34, 695–708, 1996.

Solone, R., Bittelli M., Tomei, F., and Morari, F.: Errors inwater retention curves determined with pressure plates: Ef-fects on the soil water balance, J. Hydrol., 470–471, 65–74,doi:10.1016/j.jhydrol.2012.08.017, 2012.

Tietje, O. and Tapkenhinrichs, M.: Evaluation of Pedo-TransferFunctions, Soil Sci. Soc. Am. J., 57, 1088–1095, 1993.

Tuli, A., Kosugi, K., and Hopmans, J. W.: Simultaneous scaling ofsoil water retention and unsaturated hydraulic conductivity func-tions assuming lognormal pore-size distribution, Adv. Water Re-sour., 24, 677–688, 2001.

Tuller, M. and Or, D.: Water films and scaling of soil characteristiccurves at low water contents, Water Resour. Res., 41, W09403,doi:10.1029/2005WR004142, 2005.

Tuller, M., Or, D., and Dudley, L. M.: Adsorption and capillary con-densation in porous media: liquid retention and interfacial con-figurations in angular pores, Water Resour. Res., 35, 1949–1964,1999.

Tyler, S. and Wheatcraft, S.: Application of fractal mathematics tosoil water retention estimation, Soil Sci. Soc. Am. J., 53, 987–996, 1989.

van Genuchten, M. Th.: A closed-form equation for predicting thehydraulic conductivity of unsaturated flow, Soil Sci. Soc. Am. J.,44, 892–898, 1980.

Vaz, C., lossi, M. D. F., Naimo, J. D. M., Macero, A., Reichert,J. M., Reinert, D. J., and Cooper, M.: Validation of the Arya andParis water retention model for Brazilian soils, Soil Sci. Soc. Am.J., 69, 577–583, 2005.

Wang, Q. J., Horton, R., and Lee, J. A.: Simple model relating soilwater characteristic curve and soil solute breakthrough curve,Soil Sci., 167, 436–443, 2002.

Weynants, M., Vereecken, H., and Javaux, M.: Revisiting VereeckenPedotransfer Functions: Introducing a Closed-Form HydraulicModel, Vadose Zone J., 8, 86–95, 2009.

Wu, L., Vomocil, J. A., and Childs, S. W.: Pore size, particle size,aggregate size, and water retention, Soil Sci. Soc. Am. J., 54,952–956, 1990.

Zhuang, J., Jin, Y., and Miyazaki, T.: Estimating water retentioncharacteristic from soil particle-size distribution using a non-similar media concept, Soil Sci., 166, 308–321, 2001.


http://dx.doi.org/10.1007/s10123-003-0096-1

http://dx.doi.org/10.1007/s10123-003-0096-1

http://dx.doi.org/10.1016/j.jhydrol.2012.08.017

http://dx.doi.org/10.1029/2005WR004142

Documents

Predicting the soil moisture retention curve, from soil