application of a soil water hysteresis model to simple water retention curves

Transport in Porous Media 44: 407–420, 2001.c© 2001 Kluwer Academic Publishers. Printed in the Netherlands. 407

Application of a Soil Water Hysteresis Model toSimple Water Retention Curves

R.D. BRADDOCK1, J.-Y. PARLANGE2, and H. LEE1

1Environmental Sciences, Griffith University, Nathan, Australia2Department of Agricultural and Biological Engineering, Cornell University, Ithaca,NY 14853, U.S.A.

(Received: 31 December 1998; in final form: 20 July 2000)

Abstract. The Parlange hysteresis model is reformulated as a pair of recurrence relations to providerelationships between wetting and drying phases to any order. The model is applied to the classicalvan Genuchten model for soil water retention used as the main wetting curve. The nonphysicalbehaviour of these retention curves is related to the existence of a point of inflection in the vanGenuchten model when it is used for the wetting boundary. Where the van Genuchten form is usedas the main drying curve, the Parlange hysteresis model provides an ordinary differential equationdescribing the main wetting curve. A number of simple analytical solutions, relating to particularvalues of the parameters of the van Genuchten model, then provide forms for the main wetting curve.In general, a numerical integration is required to generate the main wetting curve, for general valuesof the parameters of the van Genuchten model. The recurrence relations for the hysteresis cycling arestill applicable, even when the main wetting curve is only known numerically. The new main wettingcurves do not have inflection points and there is no nonphysical behaviour. The model is then appliedto the experimental data of Viaene et al. (1994)

Key words: hysteresis model, water retention curves, wetting and drying phases.

1. Introduction

The description, quantification and study of soil water flow processes leads to aconsideration of the hydraulic properties of unsaturated soils. This involves therelationship between the soil water content, θ , the hydraulic conductivity, K, andthe soil water retention function, θ(h), which relates θ to the capillary suction, h(with h positive for unsaturated soils, that is h is the negative of the soil water pres-sure). This last relationship exhibits hysteresis in that process where the functionθ(h) depends on whether the soil is wetting or drying. This hysteresis arises fromthe differences in the processes of filling or emptying of the pores in the soil. Thesolution of any unsaturated flow problem requires a knowledge of these propertiesbefore prediction or management issues can be addressed. Experimentation is usedto obtain data on these relations, and the knowledge is summarized in models,the ‘form’ of which represent the observed functions. The models also incorporateparameters to allow for least squares fitting of the data to provide sharper or more

408 R.D. BRADDOCK ET AL.

precise models. One of the larger groups of models represented in the literature,for the θ(h) relationship, is the power function model of the form(

1 − S−ae

)bS−ce = αh, (1)

where

Se = θ − θr

θs − θr,

and θr is the residual water content, θs is the saturated water content, and α is a pos-itive scaling factor. The quantities a, b and c, as well as θr, θs and α, are parametersof the model which can be used for fitting to experimental data. Vereecken (1992)discusses and tabulates various power law models occurring in the literature andrelates them to values of the six parameters, α, θr, θs, a, b and c. He also discussesthe hyperbolic cosine models of King (1965), and the exponential model of Farrelland Larson (1972). Note that (1) expresses h = h(θ), as a function of θ , whilenormally the modelling literature uses θ = θ (h).

Recently, flow models have been developed and applied to the development ofpreferential flow paths, and in particular, fingers in the soil, showing the influenceof hysteresis on the formation, persistence and recurrence of fingers through wet-ting and drying cycles (Liu et al., 1995; Ritsema et al., 1998; Di Carlo et al., 1999).There are several hysteresis models, including that of Parlange (1976), for describ-ing the relationship in the θ = θ (h) function for wetting and drying scanning curves(Jaynes, 1992). The Parlange (1976) model requires a knowledge of only one mainscanning curve (wetting or drying), whilst most other models require a knowledgeof both the wetting and drying scanning curve. Hogarth et al. (1988), Viaene et al.(1994) and Liu et al. (1995) applied the Parlange (1976) model of hysteresis toa Brooks and Corey (1964) type soil. Viaene et al. (1994) concluded that amongthe models using only one main scanning curve, the Parlange model ‘saw the bestresults’ for the 10 soils studied. Si and Kachanoski (2000) did not limit themselvesto a Brooks and Corey relationship in their application of the Parlange hysteresismodel. Like Viaene et al. (1994), Si and Kachanoski (2000) concluded that themodel of Parlange (1976) was the best one when only one main scanning curve isused. They also suggested that a combination of their Haines’ jump model with theParlange model may be an accurate way of incorporating hysteresis for infiltrationand drainage.

The power function model in Equation (1), can be inverted to obtain θ = θ (h).One such special case is the van Genuchten (1980) model written in the form

θ − θr

θs − θr=

[1

1 + (αh)n

]m, (2)

where α, n andm are parameters of the model. This expression is presently the onemostly used in numerical models because of its ability to fit data accurately. In thepresent paper, we apply the Parlange (1976) hysteresis model to a van Genuchten

APPLICATION OF A SOIL WATER HYSTERESIS MODEL 409

type soil, where Equation (2) is used to represent either the main wetting or dryingcurves and develop expressions for cyclic wetting and drying.

2. Hysteresis and the van Genuchten Model

Consider the repeated wetting and drying of a soil. There will be switch points orvalues of θ and h, for each spatial point in the soil, where a wetting phase switchesto a drying phase, and vice versa, and these switches mark the points of reversal ofthe wetting and drying phases. The main wetting and main drying curves representan envelope within which this switching occurs. The hysteresis model of Parlange(1976) for this repeated wetting and drying, can be generalized to the form

θd,p = θw,p − (h− hd,p

) d θw, p

dh(3)

and

θw,p = θw,p−1 − (h− hd,p−1

) [d θw,p−1

dh

]h=hw,p

(4)

where θw,p (h) is the pth wetting curve starting at hw,p and ending at hd,p, that is thecurve is defined for hw,p � h�hd,p; and θd,p (h) is the pth drying curve starting athd,p, and ending at hw,p+1, that is hw,p+1 � h� hd,p. The switch points are denotedby hw,p, which is the switch on the p-1 drying curve, and denotes the ending ofthe p-1th drying phase, and the start of the pth wetting phase. The switch pointdenoted by hd,p on the pth wetting curve denotes the ending of the pth wettingphase, and the start of the pth drying phase. Note that p = 0, corresponds to themain wetting and drying curves, denoted by θw,0 and θd,0 respectively. Then hd,0 isthe switch point from the main wetting to the main drying curve, and is the lowestachievable suction or capillary head.

The above is a generalization of the notation in Hogarth et al. (1988), and is bet-ter suited to describe the relationship between repeated wetting and drying cyclesof general order p. It should be noted that Equation (4) is an algebraic recurrencerelation which uses the (p-1) wetting curve and the switch points, to give the pthwetting curve. Where the main wetting curve is given, then Equation (3) is also analgebraic recurrence equation for the main drying curve (p = 0). Where the maindrying curve is known, then Equation (3) is a first order differential equation forθw,0, which is the main wetting curve. Once the differential equation is solved, allsubsequent wetting and drying curves can be found by algebraic recursion.

The expressions in Equations (1) and (2) could be simplified by taking θr = 0,however this residual water is important in practice as a curve fitting parameter. Inthe following, θ stands for (θ − θr), then θ(h) → 0 as h → ∞. Let

θvG = θs

[1

1 + (αh)n

]m(5)


be the van Genuchten form for the water content θvG (h). Also let C (h) be thespecific capacity

C(h) = −d θvG

dh,

= θsmnα(αh)n−1

[1

1 + (αh)n

]m+1

. (6)

Using the van Genuchten model as the main wetting curve, that is setting θw,0 =θvG, and substituting in Equation (3), then the corresponding main drying curve, orany drying curve issuing from hd,0 on the main wetting curve, is given by

θd,0 = θvG(h)+(h− hd,0

)C(h). (7)

For the first scanning curves, Equation (3) yields

θd,1 = θvG(h)+(h− hd,1

)C(h)+ (

hd,1 − hd,0)C

(hw,1

)(8)

and

θw,1 = θvG + (h− hd,0

)C

(hw,1

). (9)

The process can be continued to any level p of the wetting or drying phase,with an increasing number of terms in the expression for θw,p and θd,p. The generalforms for the pth wetting and drying cycles are

θw,p = θvG + (h− hd,p−1

)C

(hw,p

) +

+p−1∑k=1

(hd,k − hd,k−1

)C

(hw,k

)(10)

and

θd,p = θvG + (h− hd,p

)C(h) +

+p∑k=1

(hd,k − hd,k−1

)C

(hw,k

). (11)

The verification of these formulae is obtained using Proof by Mathematical Induc-tion. Note that for p= 1, Equations (10) and (11) reduce to θw,1 and θd,1 as givenin Equations (8) and (9). The inductive step proceeds by assuming that Equation(10) is true for some value of p=P1. Substitution into Equation (4), with p− 1replaced by P1, shows that θw,P1+1 is given by Equation (10) with p replaced byP1 + 1. The corresponding drying curve is obtained by substitution in Equation (3),and the proof by mathematical induction is complete.

An example of the use of Equations (10) and (11) is shown in Figure 1 forα= 15, n= 2, θs = 0.35, and m= 1 − 1/n. Note that the main drying curve ex-hibits a non-physical behaviour in that θd,0(h) increases above θs for small h. This


Figure 1. The hysteresis cycles of Equations (10) and (11) using the van Genuchten modelwith α= 15, n= 2, m= 1 −1/n, and θs = 0.35 as the main wetting curve. The switch pointsare at hw,1 = 0.7, hd,1 = 0.01, hw,2 = 0.6, hd,2 = 0.05, hw,3 = 0.5, and hd,3 = 0.1.

behaviour is evident for a wide range of values for α, n and m. This behaviourarises from the inflection point in the main wetting curve, and will be discussedmore fully later in the paper. The inflection point in θvG is located at

h∗ = 1

d

[n− 1

mn+ 1

]1/n

. (12)

The figure indicates that this nonphysical behaviour also occurs in θd,1, where theswitch point hd,1 is sufficiently close to zero, that is, 0 � hd,1 �h∗. In Figure 1the switch point hd,2 is sufficiently large that the curve θd,2 does not exhibit thisunphysical behaviour.

The smallest acceptable value of hd,0 is the inflection point h∗, and, on referringback to Equation (5), the saturated water content is not the parameter θs, but ratherθs/[1 + (αhd,0)

n]m, and hd,0 is the water entry value. Then the curve θd,0 and allother drying curves θd,p will not exhibit this unphysical behaviour.

In general differentiating Equation (3) with respect to h, yields

d θd,p

dh= −(h− hd,p)

d2 θw,p

dh2, (13)


where h�hd,p as hd,p is the switch point defining the low end of the range ofh for θd,p. Physical behaviour requires that the slope of θd,p be negative that is,(d θd,p/dh)� 0, and that θd,p is a monotonically decreasing function.

Then Equation (13) requires that (d2 θw,p/dh2)� 0. In Figure 1, the main wet-

ting curve θw,0 = θvG, has a point of inflection where d2 θvG/dh2 changes sign and

thus leads to the nonphysical behavior. Where the turning points hd,p are smal-ler than the corresponding h value at the inflection, the nonphysical behaviour isrepeated in the subsequent drying phase.

3. The van Genuchten Model as the Main Drying Curve

In practice, the parameter values for any θ = θ (h) model (such as Eq. (2)), areobtained by fitting either the main wetting or drying curves to experimental data.In practice, mostly drying curve data is measured and used in fitting models suchas the van Genuchten (1980) model in Equation (2).

The van Genuchten (1980) model has been used as the main drying curvefor various values of the model parameters (Ritsema et al., 1998). In this case,Equation (3), with p= 0, becomes a first order differential equation of the form

(h− hd,0)d θw,0

dh− θw,0 = −θvG. (14)

Parlange (1976) and Hogarth et al. (1988) (see their Eq. (1a)) refer to hd,0 as the‘lowest suction on a drying curve’, and hence hd,0 serves to define the start ofthe main drying curve. Whether this parameter is zero or not, is immaterial to therecurrence relations in Equations (10) and (11), since it affects only the way inwhich the first term in the summations is written. Hence, in the following, h standsfor h−hd,0, and as already stated, θ stands for θ − θr. Equation (14) is rewritten as

d θw,0

dh= (θw,0 − θvG)/h,

= F(h, θw,0) (15)

with θvG given by Equation (5). Note that the point (h= 0,θ = θs) is a singularity ofthis differential equation. For small values of αh, small compared to one, the term1/(1 + (αh)n)m can be expanded as a series in powers of h. In this series form, thedifferential equation with the series approximation, can be integrated to obtain

θw,0 ≈ C1αh+ θs + θsm

n− 1(αh)n + O

((αh)2n

), (16)

where C1 is an integration constant, which will be calculated later.Note that the expansion in Equation (16), is for general values of m, n, and α;

m, in particular, is not restricted or related to n. Generally the main wetting curvesatisfies θw,0 → 0, as h → ∞, and this is the appropriate boundary conditionfor Equation (14), for determining the main wetting curve. Since Equation (16)


is valid only for αh << 1, it cannot be used to find the value of C1. The seriessolution does indicate that the approach to the singularity is along the straight lineθ = C1αh + θs + O(αh)n, and the slope of the line depends on the value of theintegration constant C1.

For h large, then Equation (15) can be transformed and the right-hand sideexpressed as a series in terms of h−1, for h−1 small. Integration then yields

θw,0 = C2h+ θs

mn+ 1(αh)−mn + O

((αh)−mn−n

), (17)

where C2 is an integration constant. The condition θw,0 → 0, as h → ∞, impliesthat C2 = 0.Returning to Equation (14) for general α, m and n, using the change of variable

u = 1 − 1/(1 + (αh)n

), (18)

and on using the integrating factor technique, the equation can be written in theform

d

du

[θw,0

(1 − u

u

) 1n

]= −θs

n

(1 − u)m+(1/n)−1

u1+1/n. (19)

and the condition θw,0 → 0 at h → ∞, is to be applied. Integration of Equation(19) with this condition yields

θw,0

(1 − u

u

)1/n

= −θs

(m+ 1

n− 1

)B1−u

(m+ 1

n− 1, 1 − 1

n

)+

+ θs(1 − u)m+(1/n)−1

u1/n, (20)

where B1−u is the standard incomplete beta function (Prudnikov et al., 1986). Inparticular, this shows that C1 in Equation (16) is equal to −θs(m+ 1/u− 1)B(m+1/n−1, 1−1/n), where B is the complete beta function, so that in a simpler form,

C1 = −θs�(m+ 1/n)�(1 − 1/n)/�(m), (21)

where � is the standard gamma function. Note that if only the main wetting scan-ning curve was available experimentally, the parameters describing the soil shouldbe obtained by curve fitting Equation (20) to the data. Elementary functions can beobtained for special values of m, where i = m + 1/n − 1 is integer. For instance,for i = 0, m = 1 − 1/n, and i = 1, yields m = 2 − 1/n, which correspond to twoof the van Genuchten (1980) special forms. For m = 1 − 1/n, then

θw,0 = −θsαh+ θs[1 + (αh)n]1/n, (22)

For m = 2 − 1/n, then

θw,0 = θs

[(1 + (αh)n)1/n

{n− 1 + (αh)n/(1 + (αh)n)}(n− 1)

− αh

]. (23)


Obviously, other elementary relationships can be developed as i becomes largerbut still integer.

Other simple forms of the solution in Equation (20), can also be obtained. Form= 0, then, very simply, θw,0 = θs independently of n. The casem= 1 correspondsto the Taylor and Lupin scanning curve as well as the Brutsaert scanning curve(see Vereecken, 1992). In this case, that is, m= 1, Equation (19) can be directlyintegrated for specific values of n. Thus for n= 1

θw,0 = θs[1 − hln(1 + 1/αh)], (24)

while for n= 2,

θw,0 = θs

[1 + h

{arctan(αh)− π

2

}]. (25)

It should be noted that Prudnikov et al. (1986, Vol. 1, p. 25) consider differentialequations of a similar form to Equation (19), and give conditions under whichthese can be expressed in the form of finite combinations of elementary functions,that is, simpler than incomplete beta functions. In relation to Equation (19), suchelementary solutions occur when

(a) m+ 1

n− 1 is an integer, or

(b)1

nis integer, or

(c) m is integer.

Simple examples of each of these cases have been given above.The van Genuchten (1980) form for θ (h), withm= 1 − c/n, has been proposed

by Fuentes et al. (1992), where c is a parameter evaluated by fitting to data. Fuenteset al. (1992) point out that this provides a measure of independence between m andn, for the van Genuchten model, and provides more free parameters for fitting todata. The Prudnikov et al. (1986) criteria indicate that an elementary representationof θw,0(h) is possible for c= 0 and 1, but not for other values. This includes the casec= 2 considered by Vereecken (1992) and Fuentes et al. (1992), and the incompletebeta function must be used for the solution.

Once the main wetting curve is obtained for any of these models, the Parlange(1976) model of hysteresis, that is, Equations (3) and (4), is easily applied. Thisleads to the general forms in Equations (10) and (11), with θvG replaced by thechosen main wetting curve, and with C(h) given by its derivatives. Figure 2 showsthe hysteretic cycling for θw,0 as given in Equation (22), and C(h) is replaced by

C(h) = d

dh

[θs

{(1 + (αh)n)1/n − αh

}],

= θs(1 + (αh)n)(1/n)−1αnhn−1 − αθs. (26)

The main drying curve and the other drying curves, do not display the non-physical behaviour shown in Figure 1 and, indeed, Equation (21) does not have


Figure 2. The hysteresis cycles of Equations (10) and (11) using the van Genuchten model asthe main drying curve. The main wetting curve corresponds to Equation (21) and the parametervalues are α= 5, n= 2, m= 1 − 1/n, and θs = 0.35. The switch points are at hw,1 = 0.8,hd,1 = 0.01, hw,2 = 0.7, hd,2 = 0.05, hw,3 = 0.5, and hd,3 = 0.1.

an inflection point for h� 0. Thus the nonphysical behaviour where the van Ge-nuchten model is used as the main wetting curve, is avoided. Note that all of thewetting curves θw,p, in Figure 2, approach the main wetting curve as h → 0.This confirms that the integration constant C1 (see Eq. (21)), is independent of theboundary condition applied to Equation (19). Each of the θw,p(h) starts from theswitch point at (hw,p, θd,p−1 (hw,p)), which serves as the boundary value. FromEquation (14) note that

∂F (h, θw,0)

∂θw,0= −1

h, (27)

and hence the direction field of the differential equation is converging to the singu-larity. This confirms the independence of C1 to the boundary value.

4. Application

Although aware that the van Genuchten type water retention fits their data better,Viaene et al. (1994) used the Parlange (1976) model with a Brooks and Corey


representation for mathematical simplicity. Although they were pleased with theresult, it is quite pertinent to revisit their data using the van Genuchten represent-ation, to show the improvement in the fit and the mathematical simplicity of themodel.

We use the notation(θdata

wet , hdatawet

)and

(θdata

dry , hdatadry

)to represent the experimental values for the main wetting and drying curves asmeasured by Viaene et al. (1994). We select hd,0 = 5.5 and θs = 0.265 from thegraphs, and note that these parameters have well-defined physical meaning. Wecurve fit, using least squares, the experimental data, simultaneously fitting a vanGenuchten (1980) model (see Eq. (2)) to the main drying curve, and also fitting θw,0

(see Eq. (14)) to the main wetting curve. The objective function can be expressedin the form

E = λ∑main

drying data

(θdata

dry − θvG

(hdata

dry

))2 + (1 − λ)∑main

wetting data

(θdata

wet − θw0(hdata

wet

))2,

(28)

where the first summation is over the experimental main drying curve data, thesecond is over the experimental main wetting curve data, and λ is a weightingconstant. Note that

E = E(θr, α,m, n,C3),

where the van Genuchten parameters occur explicitly in the first summation term,and implicitly in the second summation term. The boundary condition

θw,0 = C3, at h = 138.1

(obtained from the Viaene et al. (1994) data) is applied to Eq. (20) or to the numer-ical solution of Eq. (14) to obtain θw,0(h). The least squares fitting was performedusing the MATLAB implementation fminsearch of the Nelder Mead algorithm.

We performed a least squares fit of the data using only the Equation (20) solu-tion fitted to the wetting curve data, that is λ= 0 in Equation (28). The fit of thesolution to the wetting curve data only, yielded the values

θr = 0.0227, m = 0.9398, n = 3.3091, α = 0.0216,

C3 = 0.0295.

The solution curve fits the wetting data extremely well (see Figure 3), but the fit ofthe corresponding drying curve with the above parameter values is notas good.


Figure 3. Solution curves of Equation (14) fitted to the wetting curve data of Viaene et al.(1994). The fitted parameters then yield a van Genuchten model for the drying curve.

We also performed a least squares fit of the data, fitting the van Genuchtenmodel to the drying curve data only, that is taking λ= 1 in Equation (28). Theresults yielded the parameter values

θr = 0.0301, m = 0.4651, n = 6.500, α = 0.0329,

C3 = 0.0310

and the data and the fitted curves are shown in Figure 4. The graph shows a verygood fit of the van Genuchten model to the drying data, but the correspondingmatch of the wetting curve is not as good. Viaene et al. (1994) did not give theerror bars on their data, but generally these experimental errors are significant. Thenumerical errors in the fitting are far smaller than the usual experimental errors.

In Figure 3, the theoretically derived drying curve displays considerable errorwhen compared with the drying curve data. In Figure 4, the theoretically derivedwetting curve, compared to the wetting curve data, displays less error.

We also performed least squares fits of Equation (28) for values of λε (0,1).For λ= 1/2, the fits are very similar to Figure 4, and the main drying curve isimportant to the fit and the least squares error. The main wetting curve seems tobecome important to the fitting, only for λ near zero.


Figure 4. Scanning curves fitted to both main wetting and drying data from Viaene et al.(1994). The solid lines are theoretical curves.

Also shown in Fig. 4 are the subsequent drying and wetting curves, calculatedusing Equations (10) and (11). The switch points selected corresponded to the datafrom Viaene et al. (1994) with hd,1 = 22.6 and hw,1 = 61.8. The secondary curveswere calculated using the same parameter values as determined by fitting the maindrying curve, and show similar fit. This provides another justification for using thevan Genuchten model to fit to the drying curve.

5. Conclusion

We have applied the Parlange (1976) hysteresis model to the van Genuchten (1980)model of the soil water characteristic function. We have shown that this leads toa pair of general formulae (Eqs (10) and (11)) for the wetting θw,p and dryingθd,p curves of order p. These relatively simple algebraic functions are readilyprogrammed into a computer to provide scanning curves to any finite order, andare ideal for use in simulation models which seek to incorporate hysteresis. Theseformulae are quite general and can be applied to a variety of scanning curve models.

We have also highlighted the nonphysical behaviour which arises when the vanGenuchten (1980) law in Equation (2) is used as the main wetting curve. This


model contains a point of inflection which can lead to nonphysical behaviour dur-ing the drying phase, and the water content may exceed θs, the saturation value.This effect may persist in subsequent wetting and drying cycles, provided that theswitch from wetting to drying occurs at values of hd,p below the inflection pointh∗.

Where the van Genuchten (1980) form θvG(h) is used to describe the main dry-ing curve, then the Parlange (1976) hysteresis model leads to a first order ordinarydifferential equation to determine the main wetting curve θw,0. We show that thisdifferential equation can be integrated to give simple closed form expressions forthe main wetting function. In the general case, the general solution can be ex-pressed in terms of an incomplete beta function. These forms do not display thenonphysical behaviour of the case where θvG is used as the main wetting curve.

A practical application of the solution using data from Viaene et al. (1994)shows its simplicity and its accuracy in predicting all scanning curves from thedrying boundary.

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application of a soil water hysteresis model to simple water retention curves

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