A study of hysteresis models for soil-watercharacteristic curves
Hung Q. Pham, Delwyn G. Fredlund, and S. Lee Barbour
Abstract: A review of hysteresis models for soil-water characteristic curves is presented. The models can be catego-rized into two groups: (i) domain models (or physically based models) and (ii) empirical models. Some models are ca-pable of predicting scanning curves, while other models are capable of predicting the boundary wetting curve and theboundary drying curve. A comparison of the ability of five selected models to predict the boundary wetting curveshowed that the Feng and Fredlund model with enhancements by Pham, Fredlund, and Barbour appears to be the mostappropriate model for engineering practice. Another comparison among five physically based models for predictingscanning curves showed that the Mualem model-II gives the best overall prediction of scanning curves. The studyshowed that taking the effect of pore blockage into account does not always give a better prediction of hysteretic soil-water characteristic curves. A scaling method for estimating the initial drying curve, the boundary wetting curve, andthe boundary drying curve is also presented in the paper.
Résumé : On présente une revue de modèles d’hystérèse pour les courbes caractéristiques sol-eau. Les modèles peu-vent être catégorisés en deux groupes : (i) modèles de domaines (ou modèles basés sur la physique) et (ii) modèlesempiriques. Certains modèles sont capables de prédire des courbes de balayage alors que d’autres sont capables de pré-dire les courbes d’humidification et de séchage à la frontière. Une comparaison de la capacité de cinq modèles sélec-tionnés pour prédire la courbe d’humidification à la frontière montre que le modèle de Feng et Fredlund bonifié parPham, Fredlund et Barbour semble être le modèle le plus approprié pour la pratique de l’ingénieur. Une autre compa-raison entre cinq modèles basés sur la physique pour prédire les courbes de balayage ont démontré que le modèle deMualem donne globalement la meilleure prédiction des courbes de balayage. L’étude a montré que prenant en comptel’effet du blocage des pores ne donne pas toujours une meilleure prédiction des courbes caractéristiques d’hystérèsesol-eau. On présente aussi dans cet article une méthode d’échelle pour estimer la courbe initiale de séchage, la courbede mouillage à la frontière et la courbe de séchage à la frontière.
Mots clés : courbe caractéristique sol-eau, modèle d’hystérèse, comparaison, courbe à la frontière, courbe de balayage,sols non saturés.
[Traduit par la Rédaction] Pham et al. 1568
Introduction
The functional relationship between water content and soilsuction, referred to as the soil-water characteristic curve(SWCC), plays a central role in understanding the behaviorof an unsaturated soil. Soil-water characteristic curves havebeen used to estimate the hydraulic conductivity, shearstrength, volume change, and aqueous diffusion functions ofunsaturated soils (Fredlund and Rahardjo 1993; Vanapalli etal. 1996; Barbour 1998; Lim et al. 1998; Fredlund 2000).
The SWCC is hysteretic, (i.e., the water content at a givensuction for a wetting path is less than that for a drying path).The names given to the hysteretic branches of the SWCCsare shown in Fig. 1. Aside from the primary curves; the ini-
tial drying curve, the boundary wetting curve, and theboundary drying curve, there are an infinite number of scan-ning curves inside the hysteresis loop. The difference in wa-ter content between the wetting and drying processes isbelieved to be caused by the following (Klausner 1991):(i) Irregularities in the cross-sections of the void passages
or the “ink-bottle” effect (Haines 1930).(ii) The contact angle being greater in an advancing menis-
cus than in a receding meniscus.(iii) Entrapped air, which has a different volume when the
soil suction is increasing or decreasing.(iv) Thixotropic regain or aging due to the wetting and dry-
ing history of the soil.The hysteretic nature of the SWCC has been known for a
long time but in many routine engineering and agricultureapplications the SWCCs are often assumed to be nonhys-teretic since the measurement of a complete set of hystereticSWCCs is extremely time consuming and costly, and it hasbeen difficult to represent these curves in a simple mathe-matical form for use in analyses.
The objectives of the paper include: (i) a review of themost appropriate hysteresis models from the literature that
Received 7 September 2004. Accepted 8 July 2005. Publishedon the NRC Research Press Web site at http://cgj.nrc.ca on8 November 2005.
H.Q. Pham,1 D.G. Fredlund, and S.L. Barbour.Department of Civil Engineering, University of Saskatchewan,57 Campus Drive, Saskatoon, SK S7N 5A9, Canada.
predict the boundary wetting curve and scanning curves; and(ii) development of a simple method for estimating thehysteretic curves. These objectives are addressed through areview of the historical development of hysteresis models; acomparison of selected models using datasets collected fromthe literature; and a statistical analysis of the relationshipsamong the primary hysteretic curves.
Historical development of hysteresismodels
The various models used to predict hysteretic SWCCs canbe classified into two categories: physically based models(domain models) and empirical models. The review andclassification of 28 hysteresis models found in the researchliterature are presented in this section.
Numerous researchers have used domain models to pres-ent the hysteresis phenomena including the Enderby (1955)diagram, Preisach (1935) diagram, and Néel (1942, 1943)diagram. The Néel (1942, 1943) method has been used ex-tensively by Everett (1954, 1955), Poulovassilis (1962),Philip (1964), Topp (1971a, 1971b), Mualem (1973), andParlange (1976, 1980) to represent the hysteresis of theSWCC. The original Néel (1942, 1943) diagram has alsobeen used to describe the hysteresis associated with theshape of magnetic hysteresis curves at low field strength(Everett 1955).
The Néel diagram assumes that the soil pore exists in oneof two states; full of water or empty. The state of a pore canbe characterized by two values of soil suction; namely,(i) drying soil suction, ψd; and (ii) wetting soil suction, ψw.When soil suction increases to the drying soil suction of thepore, ψd, then the pore is fully drained. When soil suctiondecreases to the wetting soil suction of the pore, ψw, then thepore is filled. A domain is composed of a group of poresthat are controlled by two small suction ranges; namely,(i) wetting soil suction from ψw to (ψw + dψw) and (ii) dry-ing soil suction from ψd to (ψd + dψd). A porous body is asystem made up of domains.
When the behavior of the domain is not a function of theadjacent domains, the domain is said to be “independent.”
The behavior of the particular pore depends only on a rangeof soil suctions. Therefore, the model is called an “inde-pendent domain” model. A “dependent domain” model takesinto account the effect of the surrounding pore blockagesagainst the entry of water or air into the pore. Dependentmodels can be developed from an independent model. Inother words, an independent model can be obtained from asimplified dependent model. Dependent models that havebeen developed from independent models include: Everett(1967), Topp (1971a), Poulovassilis and Childs (1971),Mualem and Dagan’s model-III (1975), Poulovassilis andEl-Ghamry (1978), Mualem and Miller’s model-IIIexpl(1979), and Mualem (1984b).
According to the theory of domain models, the water con-tent in a soil can be described using a three-dimensional wa-ter distribution diagram, as plotted in Fig. 2b. When soilsuction increases to ψmax, the water content in the domain isa minimum, θmin, and when soil suction decreases to ψmin,the water content is a maximum, θu (Fig. 2a). After the ini-tial drying process, the maximum water content in the soil isno longer the water content at saturation due to entrappedair. The points (ψmax, θmin) and (ψmin, θu) define the meetingpoints of the two boundary curves at high and low soilsuctions, respectively. The water distribution function, f, at apoint (ψ d
1 , ψ w1 ) represents the volume of a group of pores in
the domain that has a drying soil suction of ψ d1 and a wetting
soil suction of ψ w1 . It explains how water content in a do-
main can be specified as a function of two independent vari-ables, ψd and ψw. All the domains in the system can beidentified according to the specified soil suction range of ψwand ψd as shown by a three-dimensional Néel diagram(1942, 1943).
The drying soil suction, ψd is always larger than the wet-ting soil suction, ψw; therefore, all domains in the system arelocated in the triangle, ABC. The values of the functionf (ψd, ψw) outside the triangle ABC are equal to zero. Themaximum water content in the soil (i.e., the water content atwhich the two boundary curves converge at low soil suc-tion), θu, can be expressed as follows (Fig. 3):
[1] θ θ ψ ψ ψ ψψ
ψ
ψ
ψ
u d w d w, d dmin
max
min
max
= + ∫∫min ( )f
The conventional notations used by researchers for de-scribing the water content along the boundary SWCCs andthe scanning SWCCs are adopted in this paper:
θd(ψ) is the water content on the boundary dryingcurve at a soil suction of ψ,
θw(ψ) is the water content on the boundary wettingcurve at a soil suction of ψ,
θd(ψ1, ψ) is the water content at a soil suction of ψ alongthe primary drying scanning curve starting at asoil suction of ψ1 on the boundary wettingcurve, and
θw(ψ2, ψ) is the water content at a soil suction of ψ alongthe primary wetting scanning curve starting ata soil suction of ψ2 on the boundary dryingcurve.
Everett (1954, 1955) used the Néel (1942, 1943) diagramto describe the wetting and the drying processes in a soil
Fig. 1. Commonly used definitions for hysteretic soil–water hys-teresis curves.
(Fig. 3a). When the soil is drying, the vertical line passingthrough a suction of ψ moves from left to right. Similarly,when the soil is wetting, a horizontal line passing through asoil suction of ψ is moving downward. The filled areas in
Fig. 3a and 3b describe groups of pores that are filled withwater. The water content along the boundary drying curve atany specified soil suction, ψ, can be calculated as follows(Fig. 3b):
Fig. 2. Schematic illustration of volumetric water distribution in the soil. (a) Boundary drying and wetting curves. (b) Perspective viewof water content distribution as a function of the ψd, ψw plane for the Néel (1942, 1943) diagram.
Fig. 3. Boundary drying and boundary wetting processes in a soil and the corresponding Néel (1942, 1943) diagram. (a) Boundarydrying and wetting curves. (b) Néel (1942, 1943) diagram for boundary drying process. (c) Néel (1942, 1943) diagram for boundarywetting process.
[2] θ ψ) θ ψ ψ ψ ψψ
ψ
ψ
ψ
d d w d w, d dmax
min
max
( ( )min= + ∫∫ f
The water content along the boundary wetting curve atany specified soil suction, ψ, can likewise be calculated asfollows (Fig. 3c):
[3] θ ψ) θ ψ ψ ψ ψψ
ψ
ψ
ψ
w d w d w, d dmaxmax
( ( )min= + ∫∫ f
The water content along the drying scanning curves start-ing at a soil suction of ψ1 on the boundary wetting curve canbe calculated as follows (not shown in the figure):
[4] θ ψ ψ) θ ψ ψ ψ ψ ψψ
ψ
ψ
ψ
d w d w d w, d d( , ( ) ( )1 1
11
= − ∫∫ f
The water content along the wetting scanning curve start-ing at a soil suction of ψ2 on the boundary drying curve canbe calculated as follows (not shown in the figure):
[5] θ ψ ψ) θ ψ ψ ψ ψ ψψ
ψ
ψ
ψ
w d d w d w, d d( , ( ) ( )2 2
22
= + ∫∫ f
It is assumed that at soil suctions higher than ψmax and atsoil suctions lower than ψmin, all hysteretic SWCCs are coin-cident. The system is calibrated once the values of the func-tion, f(ψd, ψw), within the areas of the triangle, ABC, areknown.
Mualem (1974) proposed a diagram that is similar to theNéel (1942, 1943) diagram. Two variables; namely, normal-ized neck pore diameter, r , and normalized body pore diam-eter, ρ, are used to define points corresponding to the dryingsoil suction variable, ψd, and the wetting soil suction vari-able, ψw, on the Néel (1942, 1943) diagram shown in Fig. 4.The normalized pore diameter R can be defined as follows:
[6] RR R
R R= −
−min
max min
where R is the pore diameter, Rmin is the minimum pore di-ameter in the domain, and Rmax is the maximum pore diame-ter in the domain.
Using the Mualem (1974) diagram, the water contentalong the boundary drying curve, θd, at any specified soilsuction equivalent to R , can be calculated as follows(Fig. 4b):
Fig. 4. Boundary drying and boundary wetting processes in a soil and the corresponding Mualem (1974) diagram. (a) Boundary dryingand wetting curves. (b) Mualem (1974) diagram for boundary drying process. (c) Mualem (1974) diagram for boundary wetting process.
[7] θ ρ ρ ρ ρ0
d d d d d( ) ( , ) ( , )R f r r f r rR R
R
= +∫∫ ∫∫0
1
0
1
The water content along the boundary wetting curve, θw,at any specified soil suction equivalent to R , can likewise becalculated as follows (Fig. 4c):
[8] θ ρ ρ0
w d d( ) ( , )R f r rR
= ∫∫0
1
Hysteresis in the SWCC is well described using either ofthe above two diagrams; however, to calibrate the model, thevalues of the water distribution, f(ψd, ψw), at any pair of soilsuctions (ψd, ψw) (or equivalent normalized pore diameters(r , ρ) must be known. Poulovassilis (1962) was the first re-searcher to solve the hysteresis problem for an actual soil us-ing the Néel (1942, 1943) diagram. Poulovassilis (1962)divided the entire range of soil suctions (i.e., between thepoints of convergence on the boundary hysteresis curves)into n intervals of soil suction (Fig. 5). To calibrate themodel completely, values for each small square in Fig. 5must be known. There are (n2 + n)/2 unknowns, while thetwo boundary curves could provide only 2n equations.Therefore, additional scanning curves need to be measuredto calibrate the model. Poulovassilis (1962) found that in ad-dition to the two boundary curves, a series of the scanningwetting curves must be measured to predict drying scanningcurves and vice versa. The same problem also occurs for theMualem (1974) diagram.
Philip (1964) was the first researcher to propose “a simi-larity hypothesis” to simplify the required data for calibration.The hypothesis states that “the distribution of geometricalrelationships between wetting and drying meniscus curva-tures is independent of pore size” and can be expressedmathematically as follows:
[9] fl h
( )( ) ( / )ψ ψ ψ ψ ψ
ψd ww d w
w
, =
where l(ψw) and h(ψd /ψw) are two functions of soil suctions.Philip (1964) used the Néel (1942, 1943) diagram to de-
scribe hysteresis of the SWCC. With the similarity hypothe-sis, the Philip (1964) model requires only two boundaryhysteresis curves for calibration. Mualem (1973) proposed adifferent similarity hypothesis, which states: “The water dis-tribution function can be presented as a product of two inde-pendent distribution functions.” Mathematically, the Mualem(1973) similarity hypothesis can be expressed as follows:
[10] f(ψd, ψw) = h(ψd)l(ψw)
where l(ψw) and h(ψd) are two functions of soil suctions.The Mualem (1973) similarity hypothesis has been ap-
plied in a series of hysteresis models proposed by Mualem,including Mualem (1973, 1974, 1977, 1984a, 1984b), andprovides significant improvement in the practical applicationof the domain hysteresis models. The domain hysteresismodels for the SWCC can now be divided into four maingroups: (1) independent models that apply the similarity hy-pothesis; (2) dependent models that apply the similarity hy-pothesis; (3) independent models without the similarity
hypothesis; and (4) dependent models without the similarityhypothesis. A flow chart describing the historical develop-ment of 17 domain physically based hysteresis models forSWCCs is shown in Fig. 6.
Besides a number of domain models, there are several em-pirical models for hysteretic SWCCs (i.e., Hanks et al. 1969;Dane and Wierenga 1975; Scott et al. 1983; Jaynes 1984;Nimmo 1992; Kawai et al. 2000; Feng and Fredlund 1999;Karube and Kawai 2001). These models are simply based onfitting the observed shape of the hysteretic SWCCs to a se-lected equation using empirical parameters. The empiricalhysteresis model for the SWCC can be divided into two sub-groups: (1) models that simply use the same curve fittingequation for both the wetting and drying curves but adjustthe value of the parameters in each equation independently;and (2) models that rely on relationships between the twoboundary curves based on specified points or slopes taken atspecified points. A flow chart describing the historical devel-opment of 12 empirical hysteresis models for SWCCs isshown in Fig. 7. A summary of 28 hysteresis models is pre-sented in Table 1.
Comparison of the models
Several studies have compared the predictions of the vari-ous hysteresis models (Jaynes 1984; Viaene et al. 1994).However, these comparisons have been based on relativelyfew models and datasets. In this paper, a dataset of 34 soilshave been compiled and are used to compare selected mod-els. Two comparisons are made in this paper: (1) a compari-son of five models to predict the boundary wetting curve;and (2) a comparison of five other physically based modelsin predicting scanning curves.
Comparison criteriaThe following comparison criteria are used: R squared and
absolute percentage deviation (APD). The two criteria areapplied to 100 points on the predicted hysteresis curves.These 100 points are determined by dividing each hysteresis
Nonindependent domain model Boundary hysteresis loop and one family ofscanning curves.
7 1971a Topp A dependent domain model – extensionof Everett (1954–1955) model
Boundary hysteresis loop, one family ofscanning and a scanning curve of theother family.
8 1973 Mualem (model-I) Further simplification to Philip’ssimilarity hypothesis
Boundary hysteresis loop
9 1974 Mualem (model-II) Using the Mualem (1974) diagram Boundary hysteresis loop10 1975 Dane and Wierenga
(point model)Empirical model Boundary hysteresis loop
11 1975 Mualem and Dagan(model-III)
Extension of Mualem’s model-II (1974)to a dependent domain model
Boundary hysteresis loop and one primaryscanning curve
12 1976 Parlange Further simplification of Mualem’ssimilarity hypothesis
One branch of boundary hysteresis loop andtwo meeting points
13 1977 Mualem (universalmodel)
Further simplication of Mualem model-II(1974)
One branch of boundary hysteresis loop atwater contents at two meeting points
14 1978 Poulovassilis andEl-Ghamry
Extension of Poulovassilis and Childs(1971) to a dependent domain model
Boundary hysteresis loop and one family ofscanning curves
15 1979 Mualem and Miller(model-IIIexpl)
Improvement of model-III Boundary hysteresis loop and one primaryscanning curve
16 1983 Scott et al. (Scaling-down)
Empirical model using a curve fittingequation.
Boundary hysteresis loop
18 1984a Mualem Independent domain model (improvementof the Mualem (1977) universal model)
Boundary drying curve and a wettingscanning curve
17 1984b Mualem Dependent domain model (improvementof the Mualem (1977) universal model)
Boundary hysteresis loop
19 1984 Jaynes (slope model) An improvement of the Dane andWierenga (1975) model
Boundary hysteresis loop
20 1988 Hogarth et al. Improvement of the Parlange (1976,1980) model
Boundary drying curve and the water entryvalue
21 1992 Nimmo A semi-empirical model Boundary drying curve and two points onthe boundary wetting curve
22 1995 Liu et al. An application of the Hogarth et al.(1988) model
A family of drying scanning curves passingthrough a point at low water content(residual water content)
23 1999 Feng and Fredlund Curve fitting method Two parallel boundary curves; appliedsuccessfully for ceramic stones
24 2000 Kawai et al. Curve-fitting model using the Brooks andCorey (1964) equation
Boundary hysteresis loop
25 2001 Karube and Kawai(hyperbola) model
An improvement of the Kawai et al.(2000) model
Boundary hysteresis loop
26 2003 Wheeler et al. Scanning curves are straight lines Boundary hysteresis loop27 2003 Hayashida et al. Empirical model Boundary hysteresis loop28 2003 Pham et al. Improvement of Feng and Fredlund
(1999) modelOne boundary curve and two specified
points on the other boundary curve29 2004 This paper Simplified Feng and Fredlund (1999)
modelOne boundary curve and one specified
point on the other boundary curve
Table 1. Summary of hystersis models.
curve into 99 equally spaced suction values distributed alongan arithmetic scale (Fig. 8). The two criteria were calculatedas follows:APD (%):
[11] APD (%) Abs pr ms
ms
=−⎡
⎣⎢
⎤
⎦⎥
=∑ w i w i
w ii
( ) ( )
( )1
100
R squared:
[12] R2 =
w i w i
w i w ii i
ims pr
ms pr
100( ) ( )
( ) ( )−
⎡
= =
=
∑ ∑∑ 1
100
1
100
1
100
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
−
⎡
⎣⎢⎢
⎤
⎦⎥⎥=
=
∑
2
2 1
1002
1
10
w i
w ii
ipr
pr
100( )
( )0
1
1002
∑∑
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
−
⎡
⎣⎢⎢
⎤
⎦⎥⎥=
w i
w ii
ms2
ms
( )
( )
100i=∑
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
1
100
where wpr is the predicted water content at a suction, andwms is the measured water content at a suction.
Collection of the hysteretic SWCCThirty-four experimental datasets of hysteretic SWCCs for
different soils were collected from the research literature.Only soils having both boundary hysteresis curves were se-lected. The hysteretic SWCCs were selected based on thecompleteness of the data and the range of soil suctions. Thecurves presented by the researchers as best representing theirmeasured data were scanned and digitized. Information onthe 34 soils is presented in Table 2 and includes six glassbeads, a ceramic stone, 14 sands, six sandy loams, five siltloams, and two clay loams. Most collected soils are sandy innature. It is assumed that there is negligible volume changefor all the soils upon wetting and drying processes; there-fore, the collected soil datasets can be converted and used interms of gravimetric water content and degree of saturation.
Comparison of models on predicting the boundarywetting curve
Previous comparison studies have focused primarily onthe scanning curves (Jaynes 1984; Viaene et al. 1994). Mostmodels require at least two boundary curves to predict scan-ning curves. Five hysteresis models for predicting theboundary wetting curve were selected for evaluation in thisstudy. These models were selected primarily because of theirrelative simplicity and the limited amount of data requiredfor calibration. The following models are compared: Mualem(1977); Mualem (1984a); Hogarth et al. (1988); Feng andFredlund (1999); and a simplified version of the Feng andFredlund (1999) model. The first three models: Mualem(1977), Mualem (1984a), and Hogarth et al. (1988) are do-main models that utilize a similarity hypothesis. A brief de-scription of each of these models is presented below.
Mualem (1977) modelThe Mualem (1977) universal model is a domain model
that requires the boundary drying curve and water content attwo meeting points. The meeting points are selected wherethe boundary drying and the boundary wetting curves co-alesce. The Mualem (1977) universal model is a simplifica-tion of the Mualem (1974) model-II, which uses the Mualem(1974) diagram to describe the hysteretic process. TheMualem (1977) universal model uses the term effective de-gree of saturation to describe the amount of water in the soil.The effective degree of saturation Se can be calculated usingeq. [13].
[13] S e min
u min
= −−
( )( )
θ θθ θ
where θu is the volumetric water content at the meeting pointof the two boundary curves at low soil suction (i.e., close tothe air entry value); and θmin is the volumetric water contentat the meeting point of the two boundary curves at high soilsuction (i.e., close to the residual soil suction).
The boundary wetting curve can be calculated from theboundary drying curve as follows:
[14] S Swe
de( ( )] /ψ) = 1 − [1 − ψ 1 2
where S de is the effective degree of saturation along the
boundary drying curve; and S we is the effective degree of sat-
uration along the boundary wetting curve.
Mualem (1984a) independent modelThe Mualem (1984a) independent model was meant to be
an improvement of the Mualem (1977) model. In addition tothe boundary drying curve, a wetting scanning curve is neededto predict the boundary wetting curve. Mualem (1984a) pre-sented mathematical transformations to obtain the relation-ships between a scanning curve and the two boundary curves.Considerable mathematical effort is required to obtain therelationships between the scanning curves and the twoboundary hysteresis curves. Details of the Mualem (1984a)model can be found in the original paper.
The predicted boundary wetting curve consists of two parts;from zero soil suction to the suction at the starting point ofthe scanning curve, and values of soil suction greater thanthe suction at the starting point of the scanning curve. The
Fig. 8. Schematic illustration of dividing a hysteresis curve ac-cording to the soil suction range.
first part of the boundary wetting curve is predicted usingboth the boundary drying curve and the scanning curve,while the second part is predicted using only the boundarydrying curve (Mualem 1977). The water content along theboundary wetting curve can be expressed as follows:
[15] S
S
Swe
wse
sm
de
smmin smfor
(
( ,[ ( )] /
ψ) =1 − 1 − ψ ψ)
1 − ψψ ψ ψ
1 2≤ ≤
1 1 2− ≤
⎧
⎨⎪⎪
⎩⎪⎪ [ ( )] /1 − ψ ψ ψS d
esmfor
where ψ is soil suction; ψsm is the soil suction at the startingpoint of the additional wetting scanning curve on the bound-ary drying curve; S w
e (ψ) is the effective degree of saturationalong the boundary wetting curve at a soil suction of ψ;S d
e (ψ) is the effective degree of saturation along the bound-ary drying curve at a soil suction of ψ; and S ws
e (ψsm, ψ) isthe effective degree of saturation along the additional wet-ting scanning curve at a soil suction of ψ.
Hogarth, Hopmans, Parlange, and Haverkamp (1988)model
The Hogarth et al. (1988) model is an extension of theParlange (1976, 1980) model, which uses the Néel (1942,1943) diagram to describe the hysteretic processes in soils.By applying the Brooks and Corey (1964) equation to fit theSWCC, mathematical problems near the inflection point inthe Parlange (1976, 1980) model are resolved. In addition tothe boundary drying curve, the water entry value of the soilis needed to predict the entire boundary wetting curve. Theequation for presenting the boundary drying curve can bewritten as follows:
No. Soil name Type and no. (n) of scanning curvesa Research authors
1 Adelaide dune sand D, W (11) Talsma (1970)2 Avondale clay loam D, W (0) Watson, Reginato, and Jackson (1975)3 Caribou silt loam D, W (6) Topp (1971b)4 Ceramic (No. 1) I, D, W (11) Feng (1999)5 Coarse sand D, W (1) Viaene, Vereecken, Diels, and Feyen (1994)6 Dune sand D, W (7) Gillham, Klute, and Heermann (1976)7 Glass bead (4D)b D, W (2) Nimmo and Miller (1986)8 Glass bead (20D)b D, W (2) Nimmo and Miller (1986)9 Glass bead (35D)b D, W (2) Nimmo and Miller (1986)
10 Glass bead (50D)b D, W (2) Nimmo and Miller (1986)11 Glass beads D, W (2) Bomba and Miller (1967)12 Glass beads Ballotini (mixed) D, W (8) Poulovassilis (1962)13 Mixed sand fraction D, W (11) Poulovassilis (1970b)14 Mixed sand fraction D, W (0) Poulovassilis and Ghamry (1978)15 Molongo sand D, W (8) Talsma (1970)16 Norfolk sandy loam (15D)b D, W (0) Hopmans and Dane (1986)17 Norfolk sandy loam (32D)b D, W (0) Hopmans and Dane (1986)18 Packed sand D, W (6) Vachaud and Thony (1971)19 Plainfield sand loam (20D)b D, W (2) Nimmo and Miller (1986)20 Plainfield sand loam (35D)b D, W (2) Nimmo and Miller (1986)21 Plainfield sand loam (50D)b D, W (2) Nimmo and Miller (1986)22 Plano silt loam (4D)b D, W (2) Nimmo and Miller (1986)23 Plano silt loam (35D)b D, W (2) Nimmo and Miller (1986)24 Plano silt loam (50D)b D, W (2) Nimmo and Miller (1986)25 Porous body I (sand) I, D, W (6) Poulovassilis (1970a)26 Porous body II (sand) I, D, W (6) Poulovassilis (1970a)27 Rideau clay loam D, W (6) Topp (1971b)28 Rubicon sandy loam D, W (9) Topp (1969)29 Sand I, D, W (11) Poulovassilis and Childs (1971)30 Sand (No.17) D, W (10) Perrens and Watson (1977)31 Sand (R8) D, W (11) Ayers and Watson (1977)32 Wray dune sand I, D, W (1) Mualem and Klute (1984)33 Beaver Creek sand (compacted) I, D, W (0) Pham, Fredlund, and Barbour (2003)34 Processed silt (compacted) I, D, W (0) Pham, Fredlund, and Barbour (2003)
aI, initial drying curve; D, boundary drying curve; W, boundary wetting curve; n, number of scanning curves.bSome soils are tested at different temperatures, the notations: 4D, 15D, 20D, 32D, 35D and 50D, mean the temperatures of the samples are 4, 15, 20,
32, 35, and 50 °C, respectively.
Table 2. List of the collected soils.
where ψae is a curve-fitting parameter that represents the airentry value of the soil; ψwe represents the water entry valueof the soil; and λ is a curve-fitting parameter.
It is assumed that the water entry value can be estimatedfor the soil. Two curve-fitting parameters; ψae and λ, are ob-tained from a “best-fit” of eq. [16] to the measured boundarydrying curve. The equation for the boundary wetting curvecan be expressed as follows:
[17] θ ψ
θ ψψ
λψ
ψψ
ψ − ψλ λ
w
aeae
max
ae
maxwe
( )
(
=
⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟ )
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
≥
+ −⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
for ae
aeae max
ae
max
ψ ψ
θ λ λ ψψ
λψ
ψψ
1⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
≥ ≥
≥
⎧
⎨
⎪⎪
λ
ψ − ψ ψ ψ ψ
θ ψ ψ
( )we ae we
u we
for
for
⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
where ψmax is the value of the soil suction at the meetingpoint of the two boundary curves at high suction; and θae =θu /(1 + λ – λ(ψwe/ψae).
Feng and Fredlund (1999) with enhancements by Pham,Fredlund, and Barbour (2003)
The Feng and Fredlund (1999) model is an empirical rela-tionship that utilized the boundary drying curve and twopoints on the boundary wetting curve to predict the entireboundary wetting curve. Both the boundary drying curve andthe boundary wetting curve are presented using the Feng andFredlund (1999) empirical equation (eq. [18]).
[18] ww b c
( )ψψ
ψ=
+ud
db +
where wu is the water content on the boundary drying curveat zero soil suction; and b, c, and d are curve-fitting param-eters.
The residual water content and the water content at zerosoil suction, (wu) are assumed to be the same for bothboundary curves. Once the boundary drying curve is mea-sured in the laboratory, two curve-fitting parameters for theboundary wetting curve are known (i.e., wu and c). Two ad-ditional curve-fitting parameters; bw and dw, are required topredict the boundary wetting curve. To find these parame-ters, two additional points on the boundary drying curve arerequired. The two additional points must be easy to locateand measure. Pham et al. (2003) suggested that the positionof the first point on the boundary wetting curve be definedas a point having a soil suction of ψ1 such that
[19] ψ1 ≈ ⎛⎝⎜
⎞⎠⎟
bd
10
1/
where b and d are best-fit parameters of the boundary dryingcurve (eq. [18]).
The soil suction at the second additional point, ψ2, can bedetermined from the following equation:
[20] ψ ψ2 11
1
1
12= − −−
⎡
⎣⎢
⎤
⎦⎥ −
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
b w w
w cb
d
d( )/
/u
where wu is the water content on the boundary drying curveat zero soil suction; b, c, and d are curve-fitting parametersof the boundary drying curve; ψ1 is the soil suction at thefirst additional point; and w1 is the water content at the firstadditional point.
The two parameters bw and dw can then be calculated us-ing eqs. [21] and [22]
[21] d
w c w ww w w c
w w
w w
ww
u
u=
− −− −
⎡
⎣⎢
⎤
⎦⎥log ( )( )
( )( )
log( /
1 2
1 2
2ψ ψ1w)
[22] bw c
w ww w
d
ww
u
w
= −−
( )1 1
1
ψ
Simplification of the Feng and Fredlund (1999) modelThe simplified version of the Feng and Fredlund (1999)
model assumes that the boundary wetting curve and theboundary drying curve are parallel when soil suction is plot-ted on a logarithmic scale. Consequently, only one point onthe boundary wetting curve is required to calibrate the model.
Parameter d controls the slope of the curve in eq. [18] andcan be set to the same value for both the boundary dryingand the boundary wetting curves. The curve-fitting parame-ter bw for the boundary wetting curve can be calculated asfollows:
[23] bw c
w
d
wu
= −−
( )1 1
1
ψθ
where wu is the water content on the boundary drying curveat zero soil suction; ψ1 and w1 are the soil suction and thegravimetric water content of the additional point on the bound-ary wetting curve, respectively; and c and d are the curve-fitting parameters obtained by fitting the boundary drying curve.
Predictions of the boundary wetting curve using fiveselected models
The five models were used to predict the boundary wet-ting curves for the 34 soil datasets. The Mualem (1984a) in-dependent model provides a less accurate result if anadditional wetting scanning curve starting at a lower soilsuction is used (i.e., the model requires an additional wettingscanning curve to predict the boundary wetting curve). Inthe comparison process, the required wetting scanning curvefor the Mualem (1984a) independent model was selected asthe available wetting scanning curve starting at the highestsoil suction of each soil (i.e., not the boundary wettingcurve). Of the 34 soils collected for the database, only 25soil datasets had at least one wetting scanning curve. There-fore, the Mualem (1984a) independent model could only beapplied to 25 soil datasets. The water entry value used in thecalculation of the Hogarth et al. (1988) model was estimatedfrom the measured boundary wetting curve (see Fig. 11).The predicted boundary wetting curves for the Caribou siltloam (Topp 1971b) using the five models are presented inFigs. 9–13.
The five models were ranked for each soil based on thetwo criteria (i.e., R squared and APD). Tables 3 and 4 pres-ent the ranking of the five models (i.e., rank from first tofifth) for 34 soils. The average values of the two criteria forall soils and the overall ranking for the five models are alsoshown in Tables 3 and 4. The results show that the Feng andFredlund (1999) model enhanced by Pham et al. (2003) ap-pears to be the most accurate (i.e., first position for 23 out of34 soils) followed by the Mualem (1984a) independentmodel (i.e., first position for 9 out of 34 soils). The Hogarthet al. (1988) model gave the third most accurate set of re-sults. The Mualem (1977) model and the simplified versionof the Feng and Fredlund (1999) model gave accurate resultsin some cases. The results also show good agreement be-tween the two statistical criteria (i.e., APD and R squared).
It appears that using only the boundary drying curve maynot be sufficient to predict the entire boundary wetting curve(e.g., Mualem (1977) model). The Hogarth et al. (1988) andthe simplified version of the Feng and Fredlund (1999) modelsseem to predict better for poor graded soils. The Feng andFredlund (1999) model enhanced by Pham et al. (2003) re-quires relatively simple and easily measured data for calibra-
tion. In general, it may be necessary to test a series of soilsuction decrements in the laboratory to determine the waterentry value as required for the Hogarth et al. (1988) model.
Comparison of models for predicting scanning curvesViaene et al. (1994) compared 6 models: (1) Mualem
Fig. 9. Predicted and measured boundary wetting curves for theCaribou silt loam (Topp 1971b) applying the Mualem (1977)model.
Fig. 10. Predicted and measured boundary wetting curves for theCaribou silt loam (Topp 1971b) applying the Mualem independ-ent (1984a) model.
Fig. 11. Predicted and measured boundary wetting curves for theCaribou silt loam (Topp 1971b) applying the Hogarth et al.(1988) model.
Fig. 12. Predicted and measured boundary wetting curves for theCaribou silt loam (Topp 1971b) applying the Feng and Fredlund(1999) improved by Pham et al. (2003) model.
(1974); (2) Mualem (1977); (3) Hogarth et al. (1988);(4) Mualem (1984b); (5) Hanks et al. (1969); and (6) Scottet al. (1983) using the datasets of seven soils. The authorsfound that the Mualem (1974) model-II and the Mualem(1984b) dependent model gave comparable results and pro-vided the best prediction of scanning curves.
In this section, five physically based hysteresis models forpredicting the scanning curves are compared. In addition tothe two best from Viaene et al. (1994) (i.e., Mualem (1974)and Mualem (1984b) models), three other models are se-lected for comparison: (1) Mualem and Miller (1979)model-IIIexpl; (2) Mualem (1984a) independent model; and
(3) Hogarth et al. (1988) model. The selection of the fivemodels was based on the criteria that the models must bephysically based and easy to use, requiring relatively simpledata for calibration. Of the five models, only the Hogarth etal. (1988) model uses the Néel (1942, 1943) diagram. Theremainder utilized the Mualem (1974) diagram. The fivehysteresis models apply the “similarity hypothesis” to re-duce the amount of measured data required for calibration.A brief description of the theories associated with the mod-els is given in the following sections.
Mualem (1974) model-IIMualem (1974) model-II allows the prediction of the
scanning curves from two boundary curves. The dryingscanning curve starts at a suction ψ1 on the boundary wet-ting curve and can be calculated as
[24] θ ψ ψ θ ψ θ ψ θ ψθ θ ψ
θ ψ θ ψd ww w
u wd w( , ) ( )
[ ( ) ( )][ ( )]
[ ( ) (11= + −−
− )]
where θw(ψ) is the water content on the boundary wettingcurve at suction, ψ; θd(ψ) is the water content on the boundarydrying curve at suction, ψ; and θu is the water content at themeeting point of the two boundary curves at high soil suction.
The wetting scanning curve starting at suction ψ2 on theboundary drying curve can be calculated as follows:
Fig. 13. Predicted and measured boundary wetting curves for theCaribou silt loam (Topp 1971b) applying the simplified Feng andFredlund (1999) model (presented in this paper).
Mualem(1977) model
Mualem (1984a)independent model
Hogarth et al.(1988) model
Improved Fengand Fredlund(1999) model
Simplified Fengand Fredlund(1999) model
1st position 2 9 1 20 22nd position 1 7 16 9 13rd position 6 6 10 5 74th position 8 3 6 0 175th position 17 0 1 0 7Total soils 34 25 34 34 34Average APD 17.742 6.236 6.912 4.119 10.523Standard deviation of APD 19.453 7.720 6.457 5.750 6.962Overall ranking 5 2 3 1 4
Table 3. Ranking of the five soil–water hysteresis models for predicting the boundary wetting curve based on APD.
Mualem(1977) model
Mualem (1984a)independent model
Hogarth et al.(1988) model
Improved Feng andFredlund (1999) model
Simplified Feng andFredlund (1999) model
1st position 0 9 1 23 12nd position 2 10 13 8 13rd position 7 6 10 3 84th position 7 0 7 0 205th position 18 0 3 0 4Total soils 34 25 34 34 34Average R squared 0.859 0.990 0.976 0.995 0.947Standard deviation of R squared 0.1450 0.0125 0.0221 0.0064 0.0462Overall ranking 5 2 3 1 4
Table 4. Ranking of the five soil–water hysteresis models for predicting the boundary wetting curve based on R squared.
Mualem and Miller (1979) model-IIIexplThe Mualem and Miller (1979) model-IIIexpl was meant to
be an improvement of the Mualem and Dagan (1975) model.The model is a dependent domain model with the effects ofpore blockage taken into account for the drying process.
Theoretically, the model is similar to the Mualem (1974)model. The Mualem and Miller (1979) model-IIIexpl takesinto account the effects of pore blockage by means of a newvariable (P*d). An additional drying scanning curve is re-quired to calibrate the model. The pore blockage functioncan be expressed as follows:
[26] Pd*Change of actual water content
Change of actual wa( )θ0 =
ter content (when assuming no blockage)
The drying scanning curve starting at a soil suction, ψ1,on the boundary wetting curve can be calculated as follows:
[27] θd(ψ1, ψ) = θw(ψ1) – P d*(θ0)[1 – H(ψ)][θw(ψ1)
– θw(ψ)]
The wetting scanning curve starting at suction, ψ2, on theboundary drying curve can be calculated by
[28] θw(ψ2, ψ) = θd(ψ2) + P d*(θ02)[1 – H(ψ2)][θw(ψ)
– θw(ψ2)]
where P d* is a function of water content calculated by theMualem (1974) model-II (i.e., not taking into account the ef-fect of pore blockage).
The function H(ψ) can be calculated as follows:
[29] H( )( , ) ( )
( ) ( )ψ θ ψ ψ θ ψ
θ ψ θ ψ1
1= −
−w
w w
for θ(ψ1, ψ) ≤ θ0L
where θ0 is the water content at calculated suction on theboundary drying curve (i.e., without the pore blockage ef-fect); θ02 is the water content at the suction ψ2 on the bound-ary drying curve (i.e., without the pore blockage effect); andθ0L is the limit water content that pore blockage takes effectat (i.e., P d* = 1 for θ0 > θ0L).
The model involves a complex derivation and more detailscan be found in the original paper (Mualem and Miller 1979).
Mualem (1984a) independent modelThe Mualem (1984a) independent model has the same as-
sumptions as the Mualem (1977) model. Better results canbe obtained with the aid of an additional wetting scanningcurve. To predict scanning curves, the boundary wetting curvemust first be calculated using eq. [15]. The scanning curvescan then be calculated using the equations presented for theMualem (1974) model-II.
Mualem (1984b) dependent modelThe Mualem (1984b) dependent model requires the two
boundary hysteresis curves to predict the scanning curves.Similar to Mualem and Miller (1979) model-IIIexpl, this modeltakes into account the effects of pore blockage by adding apore blockage function (Pd). In this model, Pd is a functionof the actual water content. It is assumed that the pore block-age function, Pd, is a function of actual water content on theboundary drying curve and can be expressed as follows:
[30] Pdu u d
u w
( )[ ( )]
[ ( )]θ θ θ θ ψ
θ θ ψ= −
− 2
The drying scanning curve starting at suction, ψ1, on theboundary wetting curve can be calculated as follows:
[31] θ ψ ψ θ ψ1d w( , ) ( )= 1
− − −Pd u w w w
u
( )[ ( )] [ ( ) ( )]θ θ θ ψ θ ψ θ ψθ
1
The wetting scanning curve starting at a suction, ψ2, onthe boundary drying curve can be calculated as follows:
[32] θ ψ ψ θ ψw d( , ) ( )2 2=
+ − −Pd u w w w
u
( )[ ( )] [ ( ) ( )]θ θ θ ψ θ ψ θ ψθ
2 2 2
Hogarth, Hopmans, Parlange, and Haverkamp (1988) modelThe Hogarth et al. (1988) model is an improvement on the
Parlange (1976, 1980) models. This model requires theboundary hysteresis loop to predict the scanning curves. Theboundary drying curve can be expressed using eq. [16]. Theboundary wetting curve can be expressed using eq. [17]. Thedrying scanning curve starting from a soil suction, ψ1, on theboundary wetting curve can be expressed as follows:
[33] θ ψ ψθ λ λ ψ
ψλ
ψψ
ψ1
λ
dae
max
ae
max( , ) =+ −
⎛
⎝⎜⎜
⎞
⎠⎟⎟ +
⎛
⎝⎜⎜
⎞
⎠⎟⎟1 1 ( )
( )
ψ ψ ψ ψ
θ ψ ψ ψ
1
1
−⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
>
≤
⎧
⎨⎪⎪
⎩⎪⎪
we ae
w ae
for
for
Replacing ψmax with ψ2 in eq. [17] yields an equation forcalculating the wetting scanning curve starting at a suction,ψ2, on the boundary drying curve.
Predictions of scanning curves of the five selected modelsIn this comparison, equations proposed for the boundary
hysteretic curves in the Hogarth et al. (1988) model were
best-fitted to the measured data to obtain better input param-eters for the model (i.e., ψae, λ, and ψwe). The wetting scan-ning curve starting at the highest soil suction that isavailable for each soil was used as the input wetting scan-ning for the Mualem (1984a) independent model. The dry-ing scanning curve starting at a water content approximately50% of the effective degree of saturation of each soil wasused as the input drying scanning curve for the Mualem andMiller (1979) model-IIIexpl.
The comparison considered predictions of the wettingscanning and drying scanning curves starting at relativelylow water contents (i.e., less than 50% of the effective de-gree of saturation) and relatively high water contents (i.e.,higher than 50% of the effective degree of saturation), sepa-rately. The two statistical criteria were calculated for all pre-dicted scanning curves of 28 soils using the five models. Theranking (i.e., rank from first to fifth) of the models based onthe two criteria is presented in Tables 5 and 6. Average val-ues of the two criteria for all soils and the overall rankingfor the five models are also shown in Tables 5 and 6. Thepredictions of the scanning curves for the Caribou silt loamsoil (Topp 1971b) using the five models are shown inFigs. 14–18. The results show that there is an agreement be-tween the two statistical criteria (i.e., R squared and APD).The APDs ranking appears to be more consistent with a vi-sual inspection of the goodness of fit.
The Mualem (1974) model is the most appropriate modelfor predicting the scanning curves. The Mualem (1984b) de-pendent model can predict drying scanning curves starting atrelative high water contents (i.e., higher than 50% of the ef-fective degree of saturation) better than the Mualem (1974)model. The pore blockage function was taken into account
only for the drying processes. Therefore, predictions of theboundary wetting curve for the Mualem (1974) model-II,Mualem and Miller (1979) model-IIIexpl, and Mualem(1984b) dependent model are similar. The differencesamong the calculated wetting curves for these models arecaused by the calculation procedures and the input data usedin the models. The Mualem and Miller (1979) model-IIIexplpredicts the drying scanning curve well with the aid of anadditional drying scanning curve. Some mathematical diffi-culties were encountered for some soils when using theMualem and Miller (1979) model-IIIexpl, and the pore block-age function (Pd) cannot be determined. When an additionalwetting scanning curve, starting at high suction, was usedfor calibration, the Mualem (1984a) independent model ac-curately predicted the wetting scanning curve starting at alower suction. The Hogarth et al. (1988) model was not ableto predict scanning curves well since the equations proposedfor the boundary drying and boundary wetting curves didnot fit the measured data well. The model appears to predictbetter for poor-graded soils.
A simple method for estimating the hysteretic SWCCs
The SWCCs of many soils are available in the literature,but in most cases, only the initial drying curve has beenmeasured. There are several models for predicting theboundary wetting curve from the boundary drying curve andvice versa. However, in addition to the boundary dryingcurve, these models require additional information, such asthe water contents at the two meeting points along theboundary curves or two points along the boundary wettingcurve when predicting the boundary wetting curve (Pham et
1st position 12 5 6 5 02nd position 11 14 0 3 03rd position 5 5 6 9 34th position 0 0 8 10 95th position 0 1 5 1 16Total soils 28 25 25 28 28Average APD 1.993 2.328 5.444 3.646 5.837Standard deviation of APD 2.165 2.244 6.890 3.811 4.190Overall ranking 1 2 4 3 5
Table 5. Ranking of the five hysteresis models for predicting the scanning curve based on the APD.
Mualem (1974)model-II
Mualem and Miller(1979) model-IIIexpl
Mualem (1984a)independent model
Mualem (1984b)dependent model
Hogarth et al.(1988) model
1st position 8 8 5 3 42nd position 11 8 4 3 23rd position 5 5 9 5 44th position 4 2 6 7 75th position 0 2 1 10 11Total soils 28 25 25 28 28Average R squared 0.982 0.981 0.983 0.971 0.976Standard deviation of R squared 0.0264 0.0243 0.0215 0.0436 0.02Overall ranking 1 2 3 5 4
Table 6. Ranking of the five hysteresis model for predicting the scanning curve based on R squared.
al. 2003). In this section, a statistical analysis was carriedout on the relationships among the three key SWCCs (i.e.,the initial drying curve, the boundary wetting curve, and theboundary drying curve) for the soils in the dataset usedabove. A scaling method for estimating the three keySWCCs of the soils is then presented.
Relationships among hysteretic SWCCsThere are four important parameters associated with any
SWCC; namely, the water content at zero soil suction, theair entry value (or water entry value), the slope of the curve,and the residual water content (Fig. 19). Once the informa-tion on these four parameters is available, it is possible toconstruct the entire SWCC. The residual water contents ob-tained from the initial drying curve, the boundary dryingcurve, and the boundary wetting curve are quite similarwhen the soil has been dried to a soil suction beyond the re-sidual suction. The water content at zero soil suction on theinitial drying curve is the water content at saturation, whilethe water content at zero soil suction on the boundary hys-teresis curve is approximately equal to 90% of that at satura-tion (Rogowski 1971). To study the relationship among thekey hysteretic SWCCs for a soil, the following features of
Fig. 15. Predicted and measured scanning curves for Caribou siltloam soil (Topp 1971b) applying the Mualem and Miller’s inde-pendent model (1979) with the aid of a drying scanning curvestarting at a relatively high water content (i.e., w = 35%).
Fig. 16. Predicted and measured scanning curves for Caribou siltloam soil (Topp 1971b) applying the Mualem’s independentmodel (1984a) with the aid of a wetting scanning curve startingat a relatively low water content.
Fig. 17. Predicted and measured scanning curves for Caribou siltloam soil (Topp 1971b) applying the Mualem’s dependent model(1984b).
Fig. 18. Predicted and measured scanning curves for Caribou siltloam soil (Topp 1971b) applying the Hogarth et al. (1988)model.
Fig. 14. Predicted and measured scanning curves for Caribou siltloam soil (Topp 1971b) applying the Mualem model-II (1974).
SWCCs were studied; namely, (i) the distance between theboundary drying and the boundary wetting curves, (ii) therelationship between the slopes of the boundary drying andthe boundary wetting curves, and (iii) the air entry value ofthe initial drying and boundary drying curves. The distancebetween the boundary drying curve and the boundary wet-ting curve is defined as the horizontal distance between theinflection point of the boundary drying curve and the pointhaving the same water content on the boundary wettingcurve (Fig. 19).
The Feng and Fredlund (1999) curve-fitting equation(eq. [18]) was used to investigate these relationships. Theequation has an inflection point that lies midway betweenthe saturation water content and the residual water content(i.e., at a soil suction of b1/d and water content of (wu + c)/2).The inflection points of the two boundary SWCCs have thesame water content, and the line joining the two points ishorizontal. The Feng and Fredlund (1999) curve-fittingequation expressed in terms of the degree of saturation isshown below
[34] SS b c
b
d
d( )ψ ψ
ψ= +
+u
where Su is the degree of saturation on the boundary curvesat zero soil suction; and b, c, and d are curve-fitting parame-ters.
The degree of saturation at zero soil suction on the bound-ary hysteresis curves, Su, is not 100% because of the en-trapped air. The differentiation of the Feng and Fredlund(1999) equation at a particular soil suction, ψ, can be ex-pressed as follows:
[35] SLd
du= =
+− +
+S
cd
b
S b c
bdd
d
d
dd( )
) )ψ
ψψ
ψ(ψψ
(ψψ
ψ2
where S is the degree of saturation; SL is the slope of thecurve in the algebraic soil suction coordinate system; b, c,and d are fitting parameters for the SWCC; and Su is the de-gree of saturation at zero soil suction.
The slope of the SWCC in the semilogarithmic suctioncoordinate system can be calculated as follows:
[36] SLd
d[logdd
ln 10 SL ln 10L = = =S S( )( )]
( )ψψ
ψψ
ψ ( ) ψ ( )
where SL is the slope of the curve in the algebraic soil suc-tion coordinate system.
The equation has an inflection point at a soil suction ofb1/d and a water content of (Su – c)/2. The slope of theSWCC at the inflection point on the semilogarithmic suctionscale can be calculated as follows:
[37] SL( )
ln 10Lu= −d c S
2( )
where Su is the degree of saturation on the boundary wettingcurve; and b, c, and d are curve-fitting parameters.
The ratio between the slope of the boundary drying curveand the slope of the boundary wetting curve, RSL, can be cal-culated as follows:
[38] Rd c Sd c S
SLd d ud
w w uw
( )( )
= −−
where dd, cd, Sud, and cw, dw, and Suw are curve-fitting pa-rameters for the boundary drying curve and the boundarywetting curve, respectively, corresponding to the curve-fitting parameters d, c, and Su, respectively, in eq. [34].
It is noted that if Suw = Sud and cd = cw, then eq. [38] canbe simplified as follows:
[39] Rdd
SLd
w
=
The distance between the two boundary hysteresis curvesin the semilogarithmic soil suction coordinate system, DSL,can be calculated as follows:
where ψ1, ψ1′ are soil suctions at inflection points on theboundary drying and boundary wetting curve, respectively.
A statistical analysis on the collected soil databaseA statistical analysis was carried out to investigate the re-
lationships among features on the initial drying curve, theboundary drying curve, and the boundary wetting curve forthe 34 soils (Table 2). Volume changes of the collected soilsare negligible; therefore the slope ratio, RSL, and the dis-tance, DSL, between the two boundary curves are independ-ent of the term describing the water content in the soil (i.e.,gravimetric water content, volumetric water content, or de-gree of saturation). The slopes of the two boundary curvesand the distance between the two boundary curves were cal-culated using both algebraic and semilogarithmic scales forsuction. The calculated results are shown in Table 7. Fromthe statistical analysis, some suggested values for the sloperatio and the distance between the boundary curves and theboundary wetting curve are presented in Table 8. The sloperatio depends on the extent of the soil suction applied to thesoil prior to beginning the wetting. The ratio is believed totend towards 1.0 once the applied drying suction exceeds theresidual water content. The percentage deviation betweenthe calculated values and the suggested values is also pre-sented.
Only a few soil datasets have information on the initialdrying curve. Calculations using the collected datasets showthat the volume of air entrapped in the soil, (i.e., the differ-ence in volumetric water content between the initial drying
aSome soils are tested at different temperatures, the notations: 4D, 15D, 20D, 32D, 35D and 50D, mean the temperatures of the samples are 4, 15, 20,32, 35, and 50 °C, respectively.
Table 7. Distance between the two boundary hysteresis curves and the slopes of the boundary hysteresis curves on the arithmetic soilsuction coordinate system.
Ratio of slopes in algebraiccoordinate system
Ratio of slopes in semilogarithmiccoordinate system (RSL)
Distance in semilogarithmiccoordinate system (DSL)
Table 8. Suggested value, along with percentage deviation, of the slope ratio and the distance between the two boundary curves fordifferent soil types.
curve and the boundary drying curve at zero soil suction), isabout 5% to 15% of the volume of water at saturation(Fig. 1). This information agrees with the findings ofRogowski (1971). The results also show that the air entryvalue on the boundary drying curve is approximately equalto that of the initial drying curve.
A simple scaling method for estimating hystereticSWCCs
A simple scaling method is suggested to approximate thehysteretic SWCC of a soil. The Feng and Fredlund (1999)SWCC-fitting equation can be used along with an assump-tion regarding the relationship among the key SWCCs. Thescaling method uses one of the following three curves;namely, the two boundary curves and the initial drying curveto estimate the other two curves. The boundary drying curveand the initial drying curve can be interchanged by changingthe parameter that controls the water content at zero soil suc-tion (i.e., Su and Ss). Similarly, the boundary drying curveand the boundary wetting curve can be interchanged bychanging the slope and stretching the entire curve to the leftor to the right.
If the initial drying curve is best-fit using the Feng andFredlund (1999) equation (eq. [41]) then the water contentsalong the boundary drying curve can be calculated by chang-ing the curve fitting parameter (i.e., Su = 0.9Ss) that controlswater content at zero soil suction while keeping the othercurve-fitting parameters the same (eq. [42]).
[41] SS b c
b
d
dis i i
i
i
i( )ψ ψ
ψ= +
+
[42] SS b c
b
d
ddu i i
i
i
i( )ψ ψ
ψ= +
+
where Ss is the degree of saturation at zero soil suction onthe initial drying curve (i.e., equal to 100% or 1); Su is thedegree of saturation on the boundary drying curve; and bi, ci,and di are curve-fitting parameters of the initial dryingcurve.
Let us assume that the water content along the boundarywetting curve can be described using the Feng and Fredlund(1999) equation (eq. [43]). The curve-fitting parameters ofthe boundary wetting curve (bw, cw, dw) can be calculated us-ing the curve-fitting parameters of the boundary dryingcurve, the distance, and the slope ratio between the twoboundary curves in eqs. [45], [46], and [47].
[43] SS b c
b
d
dwu w w
w
w
w( )ψ ψ
ψ= +
+
where Su is the degree of saturation at zero soil suction onthe boundary curves; and bw, cw, and dw are curve-fitting pa-rameters of the boundary wetting curve.
The curve-fitting parameters for the boundary wettingcurve can be calculated as follows:
[44] cw = cd
[45] bbD d
R
wd
SL d
SL
=⎡
⎣⎢
⎤
⎦⎥( )10
1
[46] dd
Rw
d
SL
=
where DSL and RSL are the distance and slope ratio betweentwo boundary curves on a semilogarithmic suction scale.
The Fredlund and Xing (1994) correction factor can beadded to the equation as follows:
[47] SS b c
b
d
dwu r
r
( )ln( / )
ln( / )ψ ψ
ψψ ψ
ψ= +
+− +
+⎡
⎣⎢
⎤
⎦⎥1
11 106
where ψr is the residual soil suction.Approximate values for the residual soil suction can be
defined as the soil suction of the intersection point of thestraight line that goes through the inflection point with aslope of SLL (on semilogarithmic soil suction coordinate)and the horizontal line at water content of residual watercontent (i.e., Sr). It is suggested that the value of the residualsoil suction can be calculated as follows:
[48] ψr = (2.7b)1/d
If the changes in total volume are negligible, it is possibleto apply the scaling method using one of the following termsfor the water content in the soil: the degree of saturation, thevolumetric water content, or the gravimetric water content.For a soil undergoing significant volume change during dry-ing or wetting, it is suggested that the degree of saturationbe used.
Study of the application of the scaling methodThe application of the scaling method to engineering prac-
tice can be studied by applying the suggested procedure tothree different soils. These soils are: (i) Aiken clay loam(Richards and Fireman 1943), (ii) chalks (Croney andColeman 1954), and (iii) silty sands (Croney and Coleman1954). These soils are not in the list of the collected soildatasets that were used for the statistical analysis.
The estimation for the boundary drying curve from theinitial drying curve for the Aiken clay loam is shown inFig. 20. The estimation of the boundary wetting curves fromthe boundary drying curves are presented for the chalks andthe silty sands in Figs. 21 and 22, respectively. The grain-size distribution of chalk is similar to that of the silt or siltloam soil; therefore, the SWCC for chalk is similar to that ofsilt or silt loam soil. The grain-size distribution of the siltysand is between that of sand and a silt loam soil, thereforethe SWCC for silty sand is in between that of sand and siltloam soil. Values of RSL and DSL for the soft chalks and thesilty sand (Croney and Coleman 1954) are presented in Ta-ble 9. Equations [46]–[48] were used to calculate the curve-fitting parameters for the boundary wetting curve from thecurve-fitting parameters of the boundary drying curve. Theresults are summarized in Table 9. Statistical residual valuesbetween the estimated and the actual boundary curves, RE,were calculated using the following equation:
where wes is the water content on the estimated curve; wms isthe water content on the measured (actual) curve; and n isthe number of measured points.
Figure 20 shows that the estimated boundary drying curvefor the Aiken clay loam (Richards and Fireman 1943) isquite close to the measured data. Figures 21 and 22 showthat the estimated boundary wetting curves for the chalksand silty sands appear to be reasonable. The minimum Rsquared value is 0.94, and the maximum statistical residualvalue, RE, is 1.37. It appears that the estimation of the scal-ing method for the boundary wetting curve at high soilsuctions is better than that at low soil suctions. The scalingmethod appears to be reasonable for use in geotechnical en-gineering practice.
Conclusions and recommendations
The application of unsaturated soil mechanics in geotech-nical engineering requires that consideration be given to thehysteretic effects associated with the wetting and drying pro-cesses (i.e., SWCCs). A number of conclusions regarding
hysteretic effects can be drawn from the results of the studyreported in this paper:(1) The hysteresis models for the SWCC can be categorized
as domain models and empirical models. The domaintheory has been well developed in related disciplines
Fig. 20. Estimated boundary drying curve for Aiken clay loam(Richards and Fireman 1943) obtained by scaling from the initialdrying curve at 85%, 90%, and 95% saturated water content.
Fig. 21. Estimated boundary wetting curves for the soft and hardchalks (Croney and Coleman 1954).
Fig. 22. Estimated boundary wetting curves for the loose anddense silty sands (Croney and Coleman 1954).
Soft chalks Hard chalksSilty sand (lowinitial density)
Silty sand (highinitial density)
Chosen ratio of slopes in semilogarithmic coordinate system, RSL 1.5 1 2 1
R squared between the estimated and the actual curve 0.975 0.951 0.943 0.995
Residual between the estimated and the measured curves, RE 1.37 0.744 1.07 1.097
Table 9. Chosen values for the slope ratio and the distance among the four boundary curves for chalks and silty sands and the calcu-lated results.
(i.e., soil physics) and can be used to explain the physi-cal phenomena associated with the hysteretic nature ofthe SWCC.
(2) The Feng and Fredlund (1999) model is an empiricalmodel, but it provides the closest prediction of boundarywetting curve for most soils used in this study. TheFeng and Fredlund (1999) is a simple model requiringrelatively few measured data for calibration and shouldbe used in situations where one of the two boundarycurves and two points on the other boundary curve canbe measured. The Feng and Fredlund model (1999) canalso be extended and used with other SWCC fittingequations (e.g., Fredlund and Xing 1994).
(3) The Mualem (1984a) independent model uses an addi-tional scanning curve but does not predict the most ac-curate boundary wetting curve in all cases. The Mualem(1977), Hogarth et al. (1988), and the simplified versionof the Feng and Fredlund (1999) model do not seem tobe superior to the other models. The simplified Fengand Fredlund (1999) model is simple and requires fewdata for calibration, so the model may be useful to applyin certain situations.
(4) The Mualem (1974) model appears to be the simplestand most accurate model for predicting scanning curves.The other four models are more complex and requiremore data for calibration. Taking the effect of poreblockage into account does not always produce a betterprediction of the scanning curves. It is recommendedthat the Mualem (1974) model be used in engineeringpractice where scanning curves are required.
(5) There are particular ranges of values for the slope ratioand the distance between the two boundary curves forthe soil datasets collected in this study. Values for theslope ratio between the two boundary curves on thesemilogarithmic soil suction scale for sand, sandy loam,silt loam and clay loam, and compacted silt and com-pacted sand are: 2, 2.5, 1.5, and 1, respectively. Valuesfor the distance between the two boundary curves on thelogarithmic soil suction scale for sand, sandy loam, siltloam and clay loam, and compacted silt and compactedsand are: 0.2, 0.25, 0.5 and 0.35 (log-cycles), respec-tively.
(6) Air entrapment in the soil during wetting appears to beabout 5% to 15% of the volume of the soil. A value of10% appears to be a reasonable assumption in engineer-ing practice.
(7) It is suggested that the scaling method be used to esti-mate other SWCC curves when only one of the threekey SWCCs are known.
The study in this paper is limited to the database of 34soil datasets. It is recommended that this study be extendedto a greater range of soil types.
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