Soil Physics Note

A Hysteretic Model of Hydraulic
Properties for Dual-Porosity Soils
Rudiyanto

Dep. of Civil and Environmental
Engineering
Bogor Agricultural Univ.
Bogor, Indonesia

Nobuo Toride*
Masaru Sakai

Graduate School of Bioresources
Mie Univ.
Tsu, Mie, Japan 514–8507

Jiří Šimůnek

Dep. of Environmental Sciences
Univ. of California
Riverside, CA 92521

We propose a hysteretic model of hydraulic properties for dual-porosity soils
based on the bimodal van Genuchten model and Kool and Parker (K&P) hysteresis model. Hysteresis is considered only in the first pore domain, affecting
mainly higher water contents, while a nonhysteretic behavior is assumed
in the second pore domain, affecting mainly lower water contents. The
main drying and wetting curves are described with the same set of parameters except for a1, which is different for the drying and wetting curves. The
scanning hysteresis loops in the first subregion are also described using the
K&P model. The hysteretic water retention model agrees reasonably wellwith drying and wetting retention curves and scanning loops observed forAndisols. Although the corresponding unsaturated hydraulic conductivity,
evaluated using the Mualem pore-size distribution model, as a function of thewater content, K(q), is nonhysteretic for higher water contents, unrealistic
hysteresis occurs in K(q) for lower water contents. To obtain an “almost” nonhysteretic K(q) function for the entire range of water contents, an additional
constraint on the value of the a1 parameter for K(q) is imposed, and a single
value is used for both drying and wetting curves.
Abbreviations: K&P, Kool and Parker; VG, van Genuchten.

Water retention in aggregated soils is often described as a dual-porosity(or bimodal) system consisting of inter- and intraaggregate pore domains, each having their own hydraulic properties. Durner (1992,
1994) developed a multimodal retention function as a summation of multiple van
Genuchten (VG) models (van Genuchten, 1980) to describe a stepwise water retention curve. This function, which may be reduced to a bimodal function, hasbeen used quite widely in recent years (e.g., Coppola, 2000; Peters and Durner,
2008; Schindler et al., 2010; Schelle et al., 2010, 2011; Durner and Iden, 2011;
Diamantopoulos et al., 2012).
Miyamoto et al. (2003) applied the bimodal VG model to highly aggregatedJapanese volcanic-ash soils, which exhibited a distinct stepwise water retention
curve. Similar to the capillary water retention of coarse-textured sandy soils, hysteresis may also exist for water retained in interaggregate pores, which are represented by the first part of the bimodal water retention curve in the near-saturation
range. Although remarkable hysteresis loops were observed in the retention curvenear saturation for baked ceramic aggregates (Steinberg et al., 2005), only limited
effort has been made to develop a hysteretic model for dual-porosity soils.
Kool and Parker (1987) proposed a hysteretic model based on the formulations introduced by Scott et al. (1983) for the unimodal VG model. The Kool andParker (K&P) model describes the main drying and wetting curves using different

Soil Sci. Soc. Am. J. 77:1182–1188
doi:10.2136/sssaj2012.0339n
Received 8 Oct. 2012
*Corresponding author (ntoride@bio.mie-u.ac.jp)© Soil Science Society of America, 5585 Guilford Rd., Madison WI 53711 USA
All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by
any means, electronic or mechanical, including photocopying, recording, or any information storage
and retrieval system, without permission in writing from the publisher. Permission for printing and forreprinting the material contained herein has been obtained by the publisher.

Soil Science Society of America Journal

values of the unimodal VG model parameter a. The K&P model
estimates drying and wetting scanning curves based on the scalingof the main drying and wetting curves, respectively. Because it isrelatively flexible in describing hysteretic loops, the K&P model
is widely used in numerical water flow simulation codes such as
HYDRUS (Šimůnek et al., 2008b), SWAP (Kroes et al., 2008),
and UNSAT-H (Fayer, 2000), regardless of its simple assumptions and formulations. Minor modifications of the K&P model
have been suggested by Lenhard et al. (1991) and Lenhard andParker (1989) to eliminate the so-called “pumping,” or unclosed
hysteretic loops, by keeping track of historical reversal points.
The objective of this study was to introduce hysteretic water retention into the bimodal VG model based on the K&P
model. The applicability of the proposed hysteretic model isdemonstrated using observed hysteretic water retention curves
and scanning curves for an Andisol. The unsaturated conductivity function based on the Mualem (1976) model is discussed forthe main wetting and drying retention curves. We propose anadditional constraint for a nonhysteretic hydraulic conductivity
function when expressed as a function of the water content.

MATERIAL AND METHODS

Observed Hysteretic Water Retention Curves
Because of a well-developed and stable aggregate structure
made up of noncrystalline minerals (e.g., allophane, imogolite,
and ferrihydrite), Andisols exhibit unique physical properties
such as a low bulk density, a high porosity, and a stepwise water retention curve reflecting inter- and intraaggregate pores(Miyamoto et al., 2003). Hysteretic water retention curves wereobserved for an Andisol from an upland field at the National
Institute of Vegetable and Tea Science in Mie, Japan, using a
multistep outflow–inflow experiment. The soil, sieved through a
2-mm mesh, was packed in a 5-cm-depth and 5-cm-diameter soilcolumn at a bulk density of 0.75 g cm−3. A fritted glass filter of a
6-mm thickness was used at the bottom of the soil column. The
bottom pressure head was controlled using a Marriotte bottle
during sample wetting and by decreasing the position of a drip
point during draining. A tensiometer was horizontally installedat the 2.5-cm depth. Cumulative water inflow and drainage weremonitored by regularly weighing the soil column.
After saturating the soil with a Marriotte bottle supply, thebottom pressure head was gradually decreased to −80.5 cm toobtain the main drying retention curve. After switching back to
the Marriotte bottle supply, the main wetting retention curve
was monitored by increasing the bottom pressure head to saturation. Draining and wetting processes were subsequently repeatedin the intermediate pressure head range to obtain drying andwetting scanning curves. The average water contents of the soilcolumn were determined based on the cumulative inflow anddrainage and the final water content at the end of the experiment.
Hysteretic water retention curves were obtained from the average water contents and the corresponding tensiometer readings.
Figure 1 shows the collected hysteretic water retention curves aswell as their fit using the hysteretic model described below.

Hysteretic Model
Dual-Porosity Hydraulic Functions
The dual-porosity retention function introduced by Durner(1992, 1994) is expressed as a linear superposition of the unimodal VG model for each subregion:=Sq ( h )−qr
= w 1S1 + w 2 S 2
qs − q r[1]

in which Si is given by

S=i 1+ ( a i h ) 


ni

− mi

[2]

Fig. 1. Observed and fitted hysteretic water retention curves of the Mie Andisol on (a) a logarithmic scale of the pressure head, h, and (b) a linear
scale of h. Observed drainage [qd(h)] data points are displayed as open circles and observed wetting [qw(h)] data points as solid circles. Dark bluelines represent drying curves and light blue lines represent wetting curves.
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where the subscript i represents the subregion (i = 1 or 2), S isthe effective saturation (dimensionless), h is the soil water pressure head [L], qs and qr are the saturated and residual water contents [L3 L−3], respectively, ni (dimensionless), ai [L−1], and mi(= 1 − 1/ni) are shape parameters subject to ai > 0 and ni > 1,
and w1 and w2 are weighting factors subject to 0 < wi < 1 andw1 + w2 = 1. In the following discussion, S1, subject to a1 >a2, corresponds with the first part of q(h) in the higher watercontent range, while S2 corresponds with the second part of q(h)in the lower water content range. In the case of w2 = 0, Eq. [1]is reduced to the unimodal VG model. A closed-form expressionof the unsaturated hydraulic conductivity function for the dualporosity retention function Eq. [1] based on the Mualem (1976)
model was given by Priesack and Durner (2006):
K ( q )= K s ( w1S1 + w2 S2 ) ×( w a 1−(1− S )  +w a 1−(1− S
1

1

1/m1
1

m1

2

2

( w1a 1 + w2a 2 )

2

1/m2

)  )
m2

2

[3]

2

where Ks is the saturated hydraulic conductivity [L T−1] and 
is a pore-connectivity coefficient (dimensionless). Hereafter, thedual-porosity hydraulic functions of Eq. [1–3] are referred to as
the bimodal VG model.

Hysteretic Retention Functions
When qi(h) is defined asqi ( h ) =( q s − q r ) w i S i + q ri

[4]

the total water content in Eq. [1] can be simply expressed
as a sum of three components:

q(h )=
q r +  q1 ( h ) − q r1  +  q 2 ( h ) − q r 2 

[5]Because qs1 − qr1 = (qs − qr)w1 and qs2 − qr2 = (qs − qr)w2 for h= 0 in Eq. [4], the effective saturation of the ith subregion canbe defined similarly to Eq. [1]:q ( h ) − q ri
Si = iq si − q ri

[6]

Because qr is independently defined, in addition to the watercontents of the subregions in Eq. [5], qri is assumed to be zeroin this study.
Because Fig. 1 shows that hysteresis may occur only in thefirst, high-saturation part of the dual-porosity retention function, we apply the K&P model (Kool and Parker, 1987) to q1(h)while assuming nonhysteretic q2(h). The main hysteretic loop is
described with a1d for the main drying curve, q1d(h), and a1w
for the main wetting curve, q1w(h). The remaining parameters(qs, qr, n1, w2, a2, and n2) are the same for both curves.

Scanning Retention Curves
The K&P model is also applied to q1(h) to describe the scanning curves of q(h). Drying scanning curves are scaled using the1184

main drying curve, q1d(h), and can be expressed as (Šimůnek etal., 2008a)

q1 ( h ) =q r1 + a q1  q 1d ( h ) − q dr1 [7]where aq1 is the scaling factor for the drying scanning curve passing through the latest reversal point (qD1, hD) from wetting todrying. The scaling factor aq1 is obtained by substituting (qD1,
hD) into Eq. [7] assuming qr1 = qr1d:q −q
a q1 = d D1 r1
q1 ( hD ) − q r1

[8]Note that we keep a nonzero qr1 in the above scaling Eq. [7–8]despite assuming qr1 = 0. This is because a similar scaling procedure for wetting scanning curves based on the main wetting
curve is used, as shown below. The scaling procedure results ina fictitious value of the parameter qs1*, which can be obtainedby substituting the full saturation point (qs1, 0) into Eq. [7](Šimůnek et al., 2008a; Kool and Parker, 1987):

q s1* =
q r1 + a q1 ( q s1d − q r1 )=q D1 − q r1  1− S1d ( hD ) 

[9]

S1d ( hD )Note that an identical drying scanning curve as in Eq. [7]can be derived by replacing qs1 with qs1* in the main drying curve
described with Eq. [2] and [6] (Kool and Parker, 1987). Because
the main hysteresis loop is closed at saturation, qs1d and qs1w are
equal to qs1.
Similarly, wetting scanning curves are scaled using the main
wetting curve, q1w(h), and can be described as

q1 ( h ) =q r1* +a q1  q 1w ( h ) − q r1w [10]where qr1* is the fictitious residual water content and aq1 is thescaling factor for a particular wetting scanning curve. Substituting
the reversal point (qD1, hD) and the full saturation point (qs1, 0)
into Eq. [10] leads toq D1 =
q r1* +a q1  q1w ( hD ) − q r1w [11]

q s1 =
q r1* +a q1  q s1w − q r1w [12]Subtracting Eq. [12] from Eq. [11] results in

q −q
a q1 = w D1 s1 w
q 1 ( hD ) − q s1

[13]

The fictitious residual water content, qr1*, can be obtainedby substituting Eq. [13] into Eq. [12] (Šimůnek et al., 2008a;
Kool and Parker, 1987):

q r1* =
q s1 −a q1  q s1w − q r1w 
q D1 − q s1S1w ( hD )=1− S1w ( hD )[14]

Soil Science Society of America Journal

Table 1. Parameter values of the hysteretic bimodal van Genuchten model for two
types of hypothetical soils having different n1 values, and fitted parameter values
for observed main drying and wetting retention curves for the Mie Andisol, including residual and saturated water contents (qr and qs, respectively), van Genuchten
shape parameters (ad for the drying cycle, aw for the wetting cycle, and n), and aweighting factor (w) for the two subregions (subscripts 1 and 2).
Soil
Soil with n1 = 2.56
Soil with n1 = 1.20
Mie Andisol

qr

a1d

0
0
0

0.57
0.121
0.57
0.121
0.542 0.139qs
3−3—m m —

a1w−1— cm —
0.543
0.543
0.219

n1

w2

2.56
1.2
2.8

0.52
0.52
0.559

Note again that a wetting scanning curve identical to thatderived using Eq. [10] can be also derived by replacing qr1 with
qr1* in the main wetting curve described with Eq. [2] and [6](Kool and Parker, 1987). Also note that it is possible to formulate the scanning curve model to describe unclosed hysteretic
loops, as done for the unimodal VG model by Lenhard et al.
(1991) and Lenhard and Parker (1989).

RESULTS AND DISCUSSION

Main Wetting and Drying Branches of Hysteretic
Retention Curves
Figures 1a and 1b show the observed hysteretic water retention curves on a logarithmic scale for −10−1 > h > −105 cm andon a linear scale for 0 > h > −20 cm, respectively. Drying retention data measured with a pressure plate for −100 > h > −1000
cm and a dewpoint potentiometer (WP4, Decagon Devices)for h < −1000 cm are also plotted in Fig. 1a. The wetting curve
data range between a pressure head of −76 cm and saturation.
Because the drying and wetting curves practically merged forpressure heads below h = −20 cm (Fig. 1b), we assume that theobserved retention curves can represent both the main drying

and wetting curves, confirming our assumption
made above that the distinct hysteretic behavior occurs only in the first part of the retention
curve near saturation.
To test the hysteretic model, all data fromn2
the main drying and wetting curves including
a2
data for h < −100 cm in Fig. 1a were fitted simulcm−1
taneously by minimizing the root mean square
0.000351.3
0.00001
1.3
error between the fitted and observed data sets
0.00047
1.29(RMSEq) using the Solver add-in in Microsoft
Excel. The parameters qs, a1d, a1w, n1, w2, a2,
and n2 were optimized assuming qr = 0 for the sake of simplicity(Durner, 1994). Figures 1a and 1b also show the main drying andwetting curves using optimized parameter values listed in Table 1.
Because the water retention curve exhibits a distinct air entry nearsaturation with a steep gradient of dq/dh, n1 is quite large (= 2.8);
however, the fitted drying curve underestimates and the fitted
wetting curve overestimates the experimental retention data nearsaturation because n1 is identical for both curves. While the fitted
main loop closes at around h = −100 cm, the observed loop closes
at around h = −20 cm. This disagreement comes from fitting forthe entire pressure head range (Fig. 1a). Although certain discrepancies still exist at the beginning and end of the loop, the hysteretic
water retention model can describe reasonably well both the drying and wetting retention curves as a whole for Andisols.
Figures 2a and 2b show the main hysteretic loop for two hypothetical soils having different n1 values. Drying and wetting
processes are indicated by the direction of the arrows. Parameters
for both hypothetical soils are given in Table 1. The gradient of
the water content decrease close to saturation is steeper for n1= 2.56, representing a narrower pore-size distribution. On theFig. 2. Main hysteretic loop, primary drying (qd) and subsequent wetting (qw) scanning curves of q(h) for soils with (a) the van Genuchten shapeparameter n1 = 2.56 and (b) n1 = 1.2. Dashed lines are for retention curves of the two subdomains (denoted by subscripts 1 and 2), solid lines arefor the composite retention functions, dark blue lines represent drying curves, and light blue lines represent wetting curves.
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Fig. 3. Main branches of wetting (qw) and drying (qd) retention curves as shown in Fig. 1b, as well as observed (open and solid triangles) andpredicted (dark and light blue dashed lines) scanning curves: (a) drying scanning curve departing from the main wetting curve and (b) wetting
scanning curve departing from the main drying curve.

other hand, n1 = 1.2, reflecting a wider pore-size distribution,
results in a smaller gradient.
For n1 = 2.56, hysteresis appears only in q(h) near saturation because q1(h) approaches zero for h < −100 cm (Fig. 2a).
As in the case of the unimodal VG model, the larger n1 results ina greater dq/dh gradient near saturation and a distinct air entry.
Note that the gradient of the first part of qd(h) is the same asfor qw(h) because n1 is identical for both curves. For n1 = 1.2,
hysteresis also occurs at lower pressure heads, extending even to
the second section of q(h) because q1(h) decreases gradually withdecreasing pressure head, as shown in Fig. 2b. The difference between qd(h) and qw(h) at a certain pressure head, h, is larger forthe soil with n1 = 2.56 than for the soil with n1 = 1.2.

Scanning Retention Curves
Figure 2 also shows a primary drying scanning curve anda subsequent secondary wetting scanning curve. The reversal
points (hD, qD) used in Fig. 2 are (−1.8 cm, 0.479 m3 m−3)and (−15.1 cm, 0.360 m3 m−3) for the soil with n1 = 2.56 and(−1.8 cm, 0.541 m3 m−3) and (−100.2 cm, 0.447 m3 m−3) for
the soil with n1 = 1.2.
Figure 3 presents observed and predicted scanning wetting and drying curves for Andisol. A reversal point for the
drying scanning curve in Fig. 3a is (−3.4 cm, 0.485 m3 m−3),while a reversal point for the wetting scanning curve in Fig. 3bis (−10.3 cm, 0.408 m3 m−3). A good agreement between theobserved and predicted scanning curves can be found especiallyfor the wetting scanning curve (Fig. 3b).

1186

Unsaturated Hydraulic Conductivity
Figure 4 presents the unsaturated hydraulic conductivity, K(dashed lines), as a function of the pressure head, h, and watercontent, q, derived from the main drying and wetting water retention curves using Eq. [3]. Parameter values are Ks = 1000 cm d−1
and  = 0.5, while the remaining parameters are the same as in
the water retention curves in Fig. 2. For both soils, the drying
Kd(h) function is larger than the wetting Kw(h) function for theentire range of h. On the other hand, the Kd(q) and Kw(q) functions are almost identical for high water contents (q > 0.3 for n1= 2.56 and q > 0.36 for n1 = 1.2); however, Kd(q) is larger thanKw(q) for low water contents in both soils.
For high water contents, because S2 is close to 1 and a1 issignificantly greater than a2 (Table 1), Eq. [3] can be reduced to{m1K ( q ) ≈ K s ( w 1S1 + w 2 )  1− ( 1− S11/m1 ) } [15]
2

Hence, K as a function of q is almost identical for higher watercontents, regardless of the a1 value. Conversely, S1 becomesclose to zero for lower water contents. In this case, Eq. [3] canbe reduced to{w a 1−( 1−S
K (q )≈K (w S )


s

2

2

2

2

2

1/m 2

( w 1a 1 + w 2a 2 )

) }
m2

2

2

[16]

Because a1 remains in the denominator of Eq. [16] and a1d is different from a1w, unrealistic hysteresis in K as a function of q occurs between Kd(q) and Kw(q) for lower water contents. Note thatin the case of w2 = 0 for the unimodal VG model, a1 in Eq. [3]cancels out and K is independent from a1 (van Genuchten, 1980).
Soil Science Society of America Journal

Fig. 4. The unsaturated hydraulic conductivity functions (dashed lines) for the main drying (Kd) and wetting (Kw) water retention curves shown in
Fig. 2: (a) K(h) and (b) K(q) for the soil with van Genuchten shape parameter n1 = 2.56, and (c) K(h) and (d) K(q) for the soil with n1 = 1.2. The K(h)and K(q) functions with a single value for the wetting and drying curves combined of the van Genuchten shape parameter a1K are also plotted assolid lines; dark blue lines represent drying curves and light blue lines represent wetting curves.

The unsaturated hydraulic conductivity is generally assumed to be nonhysteretic when expressed as a function of thewater content (van Genuchten, 1980; Mualem, 1986; Plagge etal., 2006). When the K&P model for the unimodal VG model isused, nonhysteretic K(q) is obtained using the same n value forboth wetting and drying curves (Kool and Parker, 1987). Because
K(q) is a function of a1 for the bimodal VG model, as shown in
Eq. [16], it is possible to obtain a nonhysteretic K(q) function if
an additional constraint in terms of a1 in Eq. [3] is imposed. In
this study, a single value of a1K is used for both Kd(q) and Kw(q),while using a1d for qd(h) and a1w for qw(h).
We demonstrate the effects of this additional constraint on
the Kd and Kw curves in Fig. 4. For example, the a1K value isdefined here as an average value of a1d and a1w on a logarithmic
scale, i.e., log a1K = (log a1d + log a1w)/2. As shown in Fig. 4aand 4c for both soils (solid lines), Kd(h) is larger than Kw(h) for
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higher pressure heads, whereas Kd(h) and Kw(h) are almost identical for lower pressure heads. This additional constraint in termsof the a1K value resulted in an almost nonhysteretic K(q) function across the entire range of water contents, as shown in Fig.
4b and 4d.
Although in Fig. 4 we demonstrated the K(q) function withan intermediate value of a1K (between a1d and a1w), in an ideal
case, a1K would be a fitting parameter obtained from an experimentally determined K(q) for low water contents; however, such
experimental data are rarely, if ever, available. Further experimental investigations will be needed to evaluate K(q) in a relatively
dry range.

CONCLUSIONSWe proposed here a hysteretic model of soil hydraulic properties for dual-porosity soils. The model is based on a bimodal1187

VG model (Durner, 1994) and the K&P hysteresis model (Kooland Parker, 1987). Hysteresis is considered only in the first
subregion, q1(h), affecting mainly higher water contents, whilea nonhysteretic behavior is assumed in the second subregion,
q2(h), affecting mainly lower water contents. The main hysteretic
loop is described with a1d for the main drying curve, q1d(h), anda1w (>a1d) for the main wetting curve, q1w(h). The remaining
parameter values are the same for the main drying and wetting
curves. For soils with a larger n1 (e.g., n1 = 2.56), hysteresis affects mainly the first part of the retention curve q(h) near saturation. On the other hand, for soils with a smaller n1 (e.g., n1 =1.2), some hysteresis extends from full saturation to lower pressure heads, also affecting the second part of the retention curve
q(h). Scanning curves of q(h) can also be successfully describedusing the K&P model in the first subregion of q1(h). The observed main hysteretic loop and scanning curves for Andisol aggregates could be reasonably well described using this proposed
hysteretic model.
The drying and wetting unsaturated hydraulic conductivity
functions Kd(q) and Kw(q), respectively, are almost identical forhigher water contents; however, Kd(q) is greater than Kw(q) forlower water contents. To overcome this unrealistic phenomenon
in the conductivity function, we use an additional constraint on
the value of the a1 parameter and use a single value a1K for bothdrying and wetting functions. As a result, an almost nonhysteretic K(q) function is obtained for the entire range of water contents. It is necessary to experimentally investigate K(q) for low
water contents to determine the a1K value.

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