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RESEARCH ARTICLE
10.1002/2015WR017703
Key Points:
 Proposed soil hydraulic model
includes capillary hysteretic,
adsorptive, and ﬁlm components
 Empirical equations can predict theﬁlm conductivity using water
retention parameters Including vapor conductivity
improved conductivity predictions in
the very dry range

A complete soil hydraulic model accounting for capillary andadsorptive water retention, capillary and film conductivity, andhysteresis
Rudiyanto1, Masaru Sakai2, Martinus Th. van Genuchten3,4, A. A. Alazba5, Budi Indra Setiawan1, andBudiman Minasny6
1

Department of Civil and Environmental Engineering, Bogor Agricultural University, Bogor, Indonesia, 2Graduate School of
Bioresources, Mie University, Tsu, Japan, 3Department of Mechanical Engineering, Federal University of Rio de Janeiro, Riode Janeiro, Brazil, 4Department of Earth Sciences, Utrecht University, Utrecht, Netherlands, 5Department of AgriculturalEngineering, King Saud University, Riyadh, Saudi Arabia, 6Department of Environmental Sciences, The University ofSydney, Sydney, New South Wales, Australia

Supporting Information:
 Data Set S1
 Data Set S2
 Data Set S3
 Data Set S4
 Data Set S5
 Data Set S6
 Data Set S7
 Data Set S8
 Data Set S9
 Data Set S10
 Data Set S11
 Data Set S12
 Data Set S13
 Data Set S14
 Data Set S15
 Data Set S16
 Data Set S17
 Data Set S18

Abstract A soil hydraulic model that considers capillary hysteretic and adsorptive water retention as wellas capillary and ﬁlm conductivity covering the complete soil moisture range is presented. The model was
obtained by incorporating the capillary hysteresis model of Parker and Lenhard into the hydraulic model ofPeters-Durner-Iden (PDI) as formulated for the van Genuchten (VG) retention equation. The formulation
includes the following processes: capillary hysteresis accounting for air entrapment, closed scanning curves,
nonhysteretic sorption of water retention onto mineral surfaces, a hysteretic function for the capillary conductivity, a nonhysteretic function for the ﬁlm conductivity, and a nearly nonhysteretic function of the conductivity as a function of water content (h) for the entire range of water contents. The proposed model only
requires two additional parameters to describe hysteresis. The model was found to accurately describe
observed hysteretic water retention and conductivity data for a dune sand. Using a range of published data
sets, relationships could be established between the capillary water retention and ﬁlm conductivity parameters. Including vapor conductivity improved conductivity descriptions in the very dry range. The resulting
model allows predictions of the hydraulic conductivity from saturation until complete dryness using water
retention parameters.

Correspondence to:
Rudiyanto,
lupusae@yahoo.com

Citation:
Rudiyanto, M. Sakai, M. Th. van
Genuchten, A. A. Alazba, B. I. Setiawan,
and B. Minasny (2015), A complete soil
hydraulic model accounting forcapillary and adsorptive water
retention, capillary and ﬁlm
conductivity, and hysteresis, Water
Resour. Res., 51, doi:10.1002/
2015WR017703.
Received 16 JUN 2015
Accepted 14 OCT 2015
Accepted article online 23 OCT 2015

C 2015. American Geophysical Union.
V

All Rights Reserved.

RUDIYANTO ET AL.

1. Introduction
Estimates of the unsaturated soil hydraulic properties are essential for accurate modeling of a range of subsurface and near-surface hydrological processes, including inﬁltration, redistribution, evaporation, and solute transport in soils [e.g., Collis-George, 2012; Durner and Fl€
uhler, 2005; Saito et al., 2006; Simůnek, 2005]. The
soil hydraulic (constitutive) properties involve the water retention function deﬁning the relationshipbetween the volumetric water content, h, and the pressure head, h, and the hydraulic conductivity function
deﬁning the hydraulic conductivity, K, as a function of h or h.
Water retention in soils is governed by capillarity at relatively high water contents, and adsorption at low
water contents [Jury and Horton, 2004]. These two regions of the retention curve are often most noticeable
for relatively coarse-textured (sandy) soils [Sakai and Toride, 2007; Sakai et al., 2009; Schelle et al., 2013]. Hysteresis in the hydraulic functions, furthermore, is important in especially the capillary region at relatively
high water contents [Huang et al., 2005; Poulovassilis and Childs, 1971; Sakai et al., 2009]. Hysteresis may also
exist in the adsorption region [Arthur et al., 2015; Davis et al., 2009; Globus and Neusypina, 2006; Prunty andBell, 2007]. However, because hysteresis in the adsorb region occurs over only a small part of the retentionfunction, its effect is probably negligible for many or most practical applications [Sakai et al., 2009; Schelle
et al., 2013].
For the unsaturated hydraulic conductivity, one may similarly distinguish distinct capillary and ﬁlm ﬂow
areas at relatively high and low water contents, respectively [Lebeau and Konrad, 2010; Peters, 2013; Sakai
and Toride, 2007; Zhang, 2011]. Water in the very dry range can move furthermore in the form of vapor,while liquid ﬂow may cease there [Mehta et al., 1994; Philip and De Vries, 1957; Saito et al., 2006]. Clearly,

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comprehensive models for simulating water ﬂow in ﬁeld soils may require hydraulic functions that consider
all of these different water retention and hydraulic conductivity properties and processes from saturation all
the way to complete dryness.
Various hydraulic functions have been used over the years to describe capillary retention [Brooks and Corey,
1964; Kosugi, 1996; van Genuchten, 1980], while the capillary conductivity has been described using a rangeof statistical pore size distribution models such as those formulated by Burdine [1953] and Mualem [1976a].
The resulting classical capillary-based functions typically hold between saturation, hs, and some residual
water content, hr. Many subsequently attempted to describe the capillary and adsorption regions for thecomplete water retention curve from saturation to complete dryness [Campbell and Shiozawa, 1992; Fayer
and Simmons, 1995; Fredlund and Xing, 1994; Groenevelt and Grant, 2004; Khlosi et al., 2006; Lu et al., 2014;
Rossi and Nimmo, 1994]. Most of these studies focused only on the water retention curve, while still neglecting the contribution of ﬁlm conductivity to the overall hydraulic conductivity function. An early exception
was a study by Fayer and Simmons [1995] in which the complete water retention functions were coupledwith the capillary bundle model of Mualem [1976a] to estimate the unsaturated hydraulic conductivity forthe entire range of water contents. Their approach assumes that the capillary-based model can be usedalso at low-pressure heads where water is mostly held in place by adsorptive rather than capillary forces. An
application of their model to both a sandy and an aggregated soils was shown by Sakai and Toride [2007].
Subsequent key contributions to improved descriptions of the unsaturated hydraulic conductivity using thecapillary conductivity model of Mualem [1976a] in combination with a ﬁlm ﬂow component are by Lebeau
and Konrad [2010], Peters and Durner [2008], and Zhang [2011].
More recently, Peters [2013] introduced a set of empirical soil hydraulic models from full saturation to complete dryness. His models assumed a clear partitioning between capillary and adsorptive water retention,
but also accounted for capillary and ﬁlm conductivities. Although the models showed good agreementwith measured data, a possible shortcoming for some applications is the fact that the complete functions
are not continuously differentiable at the connection point between capillary and adsorptive water retention regions, leading to a discontinuity of the soil water capacity, dh/dh. To solve this problem, Iden andDurner [2014] modiﬁed Peters’ model by including an empirical function for adsorptive water retention to
produce a continuously differentiable soil water retention function. Following Iden and Durner [2014] andPeters [2014], we hereafter refer to this model as the Peters-Durner-Iden (PDI) model. While providing animportant contribution, the PDI model at present does not account for capillary hysteretic water retention.
Moreover, the PDI conductivity model cannot be estimated directly from water retention parameters [Peters,
2013] since it requires two extra parameters to account for the contribution of ﬁlm conductivity, as compared to the commonly used closed-form hydraulic models (e.g., those by Durner [1994], Kosugi [1996], or
van Genuchten [1980]).
Concurrently, various models have been proposed also to describe capillary hysteresis in the soil water
retention curve. Kool and Parker [1987] modiﬁed the empirical hysteresis model of Scott et al. [1983] to aform consistent with the van Genuchten [1980] function. Because its relatively simple yet realistic form, theKool and Parker [1987] hysteresis model has been widely implemented in numerical codes for water ﬂow
such as HYDRUS [Simůnek et al., 2008], SWAP [Kroes et al., 2008], and UNSAT-H [Fayer, 2000]. One drawbackof the hysteresis model of Kool and Parker [1987] is that the scanning loops are not closed and as such canlead to artiﬁcial ‘‘pumping effects’’ that can move scanning curves outside of the main wetting of drying
branches [Kool and Parker, 1987]. To avoid unclosed scanning loops, the model of Parker and Lenhard[1987], denoted here as the PL model, forces closure on the scanning curves. Although the resulting model
improved the description of scanning curves consistent with the retention equations of van Genuchten[1980], the model fails to adequately describe water retention at low water contents [Fayer and Simmons,
1995]. Moreover, the PL model uses a modiﬁed version of the capillary bundle model of Mualem [1976a] asa predictor for the unsaturated hydraulic conductivity function [Lenhard and Parker, 1987]. As shown by
Tuller and Or [2001], Mualem’s [1976a] model often fails to describe the hydraulic conductivity in themedium to dry range since it neglects ﬁlm and corner ﬂow.
In this study, we propose a model that incorporates four processes in the complete moisture range, capillary
hysteretic and adsorptive water retention and capillary and ﬁlm conductivity, by combining the PDI-based
VG and PL models. We also include a vapor model for the conductivity in very dry range. The model was

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compared with measured hysteretic water retention data of a dune sand. We further used the model todetermine capillary and ﬁlm conductivities of the same dune sand as measured using the evaporation
method. In addition, we evaluated relationships between parameters for the ﬁlm conductivity and the water
retention parameters of the PDI-based VG model using several published data sets. The relationships werevalidated further by comparison with other water retention and conductivity data sets.

2. Theoretical Development
In this section, we provide a detailed description of the soil hydraulic model between saturation and complete dryness. First, we brieﬂy review the Peters-Durner-Iden (PDI) model since it provides the kernel of ourapproach, and then include hysteresis in the formulation.
2.1. The Peters-Durner-Iden (PDI) Model
The PDI model for water retention according to Iden and Durner [2014] is given byhðhÞ5hcap ðhÞ1had ðhÞ
5ðhs 2hr ÞScap ðhÞ1hr Sad ðhÞ(1)where h is the water content, h is the pressure head, Scap(h) is relative saturation of capillary water, and Sad(h) isrelative saturation of adsorptive water. The residual water content, hr, hence serves as the maximum adsorptive
water content. As shown by Peters [2013], Scap(h) can be parameterized by a unimodal pore size distributionfunction such as the Kosugi [1996] and van Genuchten [1980] equations, or a multimodal function [Durner, 1994].
To ensure that the water content becomes zero at some large value of the pressure head, h0 (generally about2107 cm), Iden and Durner [2014] deﬁned Scap(h) by rescaling the effective capillary function, C(h), as follows
Scap ðhÞ5

CðhÞ2Cðh0 Þ
12Cðh0 Þ(2)where C(h) can be any of the standard functions [Brooks and Corey, 1964; Kosugi, 1996; van Genuchten,
1980]. For the function of van Genuchten [1980] as used further in this study, C(h) is given by

m
1
CðhÞ5
(3)
11ðajhjÞn
In which a and n are essentially empirical shape parameters, and m 5 1 – 1/n.
Iden and Durner [2014] introduced a smooth piecewise linear function to describe the relative saturation ofadsorptive water:hx 2x io1 na
Sad ðhÞ511
x2xa 1bln 11exp(4)xa 2x0bwhere x 5 log10(|h|), x0 5 log10(|h0|), xa 5 log10(|ha|), and b is a smoothing parameter given by [Iden andDurner, 2014]
(
" 
2 #)
0:2
hr
b50:11 2 12exp 2(5)hs 2hrnNote that x in equation (4) is identical to the pF value as used in many soil water retention and plant water
stress studies beginning with an early study by Schoﬁeld [1935]. We refer to Iden and Durner [2014] for amore detailed discussion of equations (4) and (5).
The hydraulic conductivity function for liquid ﬂow is given by the sum of the capillary and ﬁlm conductivities [Peters, 2013]:
K liq 5K cap 1K film
5Kscap Krcap ðScap Þ1Ksfilm Krfilm ðSad Þ(6)where the superscripts liq, cap, and ﬁlm refer to liquid, capillary, and ﬁlm contributions, respectively, Kscap
and Ksfilm are capillary and ﬁlm conductivities at saturation, Kr refers to scaled or relative hydraulic
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conductivities as a function of effective saturation of the capillary (Scap) and ﬁlm ﬂow or adsorptive (Sad)regions.
When the original van Genuchten [1980] soil hydraulic functions are used (i.e., equation (3) for the effective
capillary function), the scaled hydraulic conductivity function is given by [Peters, 2014]
Krcap 5ðScap Þ‘1212CðhÞ1=m!m !212Cðh0 Þ1=m(7)

where ‘ is an empirical pore tortuosity and/or connectivity parameter that is generally assumed to be 0.5[Mualem, 1976a]. The ﬁlm conductivity as a function of Sad is given by [Peters, 2013]:
Krfilm 5 að12Sad Þ
h0
ha

(8)

where ha is an air entry value approximated by the value of 1/a in van Genuchten’s retention model, and aan empirical parameter [Peters, 2013].
Equation (6) reﬂects the contributions of both the capillary conductivity at relatively high water contents, aswell as of the ﬁlm conductivity at low water contents, to the overall liquid conductivity. In the very dry
range, vapor transport may become more dominant than liquid ﬂow [Mehta et al., 1994, among others]. For
vapor ﬂow, one may assume that the gradient in the gravitational potential can be assumed negligible. This
allows one to add the isothermal conductivity (Kvh) to the liquid conductivity to give the total conductivity(Ktot) as follows [e.g., Peters, 2013]:
K tot 5K liq 1K vh 5K cap 1K film 1K vh(9)For completeness, details of the isothermal vapor conductivity, Kvh, are summarized in Appendix A, more orless following the development by Peters [2013]. We refer to the above PDI-based van Genuchten model as
the PvG model. Next, we include in the formulation the effects of hysteresis involving both the main water
retention branches (section 2.2) and the scanning curves (section 2.3).
2.2. Hysteresis Between the Main Retention Branches
We assumed that hysteresis occurs only in the capillary region of the water retention curve [Sakai et al.,2009; Schelle et al., 2013]. Thus, the hysteresis model of Parker and Lenhard [1987] is used here to describe
the main drying and wetting curves of the capillary region, while equation (4) is used for adsorptive water
retention without any hysteresis. The main drying and wetting curves for capillary retention in the PvG formulation are then described using:
"
#m
1
d

C ðhÞ5(10)n
11 d ajhj
and
w


m
hs 2hr1
CðhÞ5 dhs 2hr 11ðw ajhjÞn
w(11)

where C(h) is the effective capillary function given earlier by equation (3), and where the prefatory superscripts d and w denote the main drying and wetting branches, respectively. When the main branches areclosed at saturation, whs 5 dhs, and hence dC(h) 5 wC(h) 5 1 at h 5 0. On the other hand, whs < dhs when themain branches are not closed at h 5 0 due to entrapped air. Since dC(h) is assumed to be unity at h 5 0, theeffective capillary retention wC(h) for the wetting branch at h 5 0 will be less than unity since the main wetting curve is scaled by the ratio (whs – hr)/(dhs – hr) as indicated by equation (11). And hence as a result,w
C(h)  dC(h) 5 1 for nonclosing main drying/wetting branches at h 5 0.
Following Kool and Parker [1987], we assume that the hydraulic parameters of the main wetting and drying
branches are identical, except for wa < da. When the main branches are not closed at h 5 0, an additional hs
is needed since whs < dhs, leading to a total of six parameters: hr, whs < dhs, wa > da, and n. The remaining

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two water retention parameters associated with adsorptive water (ha and h0) were ﬁxed. The value of ha
was set at 1/da for both drying and wetting since adsorptive water retention is assumed to be nonhysteretic
and often measured during drying, while h0 was ﬁxed at 26.3 3 106 cm as suggested by Peters [2013]
based on Schneider and Goss [2011]. We note that the value of h0 is very much consistent with recent measurements by Cobos et al. [2014] who showed that the zero water content intercepts of 13 well-characterized
relatively coarse-textured soil occurred between 5.7 to 6.3 log kPa (or between 25 3 106 and 21.9 3 107cm).
2.3. Scanning Curves
The scanning curve approach as used by Parker and Lenhard [1987] (PL) was used for capillary water retention. The PL model enforces closure of the scanning loops in terms of the effective water content, C. The
reversal point in terms of effective water content, (DC,Dh), for reversal retention point (Dh,Dh) is obtained byﬁrst subtracting had(Dh) from the reversal water content, Dh (see equation (1)) to obtain the reversal point interms of the capillary retention function, (Dhcap,Dh). Then, since whs < dhs and 0  C  1, hs in equation (1) isequated to dhs and DC is calculated using equations (1) and (2) to give
D

D cap

C5 d

D cap
h
h
½12Cðh0 Þ1Cðh0 Þ  dhs 2hrhs 2hr(12)

Consider a drying scanning curve which begins at the reversal point (DCwd, Dhwd) from wetting to drying andproceeds to the drying end point at (DCdw, Dhdw) before reversing from drying to wetting. The main drying curveis rescaled to interpolate between the reversal points using the following equation [Parker and Lenhard, 1987]:
D
C wd 2D C dw
CðhÞ5 d DC h wd 2d C D h dwdC ðhÞ2d C D h dw 1D C dw

(13)

where dC(Dhdw) and dC(Dhwd) are the values of dC(h) (equation (10)) at h 5 Dhdw and Dhwd, respectively.
Consider similarly a wetting scanning curve starting at the reversal point (DCdw, Dhdw), from drying to wetting to the wetting end point at (DCwd, Dhwd), and then following the reversal from wetting to drying on thepreceding drying curve. The main wetting curve is scaled to interpolate between the reversal points using[Parker and Lenhard, 1987]:
D
C dw 2D C wdCðhÞ5 w DC h dw 2w C D h wdw

C ðhÞ2w C D h wd 1D C wd

(14)

where wC(Dhdw) and wC(Dhdw) are values of wC(h) function given by equation (11) with h 5 Dhdw and Dhwd,
respectively.
If there is no entrapped air, then whs 5 dhs and hence dCs 5 wCs 5 1. In contrast, if entrapped air exists, themain loop will not be closed at saturation since whs < hu < dhs or wCs < Cu < dCs. Note that hu which corresponds to Cu is deﬁned by the water content at saturation as the end point of the primary scanning curve
beginning from a reversal point at (Dhdw, Dh). Kool and Parker [1987] approximated Cu based on the Land
formulation as discussed by Aziz and Settari [1979]:
Cu 512

12D Cdw
11Rð12D Cdw Þ

(15)

1
21
12w Cs(16)

in which
R5

The primary wetting scanning curve is then calculated using equation (14) in which the end point of the primary wetting scanning curve at saturation is equal to (DCwd 5 Cu, Dhwd 5 0) with a reversal point (DCdw,
Dhdw) on the main drying curve.
2.4. Hysteresis in the Hydraulic Conductivity
The hydraulic conductivity curve is calculated using equations (6)–(8). When the main branches are closed
at h 5 0, in which case whs 5 dhs 5 hs, the saturated hydraulic conductivity will be equal to

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d cap

Ks 5 wKscap 5 Kscap as given by equation (6). Equation (11) for wC(h) will then replace C(h) in equation (7)for the wetting branch of the hydraulic conductivity, while equation (10) for dC(h) is to replace C(h) for the
drying branch. However, when the main branches are not closed at h 5 0, and hence whs < dhs or wCs < dCs,
then dKscap is used as the scale of the hydraulic conductivity, Kscap , in equation (6) for both drying and wetting. The approach is then in essence the same as the PvG model. The hydraulic conductivity curve of the
proposed model contains four parameters: Kscap 5 dKscap and ‘ for the capillary conductivity and Ksfilm and afor the ﬁlm conductivity.

3. Soil Hydraulic Data Sets
Various data sets were used to test the PvG soil hydraulic model formulations. One set of experiments carried out using Tottori dune sand from Japan was to study the ability of the extended PvG model to describe
hysteresis. We further used several published data sets from the literature to examine possible relationships
between the water retention and ﬁlm conductivity parameters.
The hysteretic water retention measurement of Tottori dune sand [Sakai and Toride, 2007] was carried out
using the hanging water column method as applied to a soil sample 10 cm in diameter and 2.5 cm inheight. Pressure heads and water contents of the soil were monitored using a tensiometer and a time
domain reﬂectrometer (TDR) connected to a data logger. After saturation, drainage was initiated by decreasing the pressure head at the bottom of the column gradually until a pressure head of 260 cm. Rewetting
was subsequently implemented using a Mariotte supply bottle by increasing the pressure head at the bottom of the column to saturation. The draining and wetting processes were repeated in the intermediatepressure head range to obtain various drying and wetting scanning curves. The observed hysteretic water
retention data were obtained from the tensiometer and TDR readings. The pressure plate method was further used to measure water retention values between 2500 and 21000 cm, while the vapor equilibrium
method was used for water retention for pressure heads below 21000 cm. Both sets of measurements
related to drying. The data were used to test the hysteresis description of the PvG formulation, and also thehydraulic conductivity between saturation and complete dryness.
In addition to the Tottori dune sand data, possible relationships between the two extra parameters in theﬁlm conductivity (i.e., Ksfilm and a) and the PvG water retention parameters were evaluated using seven published data sets. The data sets comprised Sydney Sand [Minasny and Field, 2005]; Rhinluch Sand [Schindler
and M€
uller, 2006]; Berlin Sand [Peters, 2013]; Gilat Loam, Rehovot Sand, and Pachapa Fine Sandy Clay [Mualem, 1976b]; and a Sandy Loam [Pachepsky et al., 1984]. All data sets pertained to the drying branches forboth water retention and the hydraulic conductivity. Although half of the hydraulic conductivity data weresituated between saturation and a pressure head of 21000 cm, the curves showed very distinct capillary
and ﬁlm conductivity components.

4. Results and Discussion
4.1. Hysteretic Capillary Retention
Figures 1a and 1b show the observed hysteretic water retention curves of Tottori dune sand on a logarithmic scale between pressure heads of 2101 and 2107 cm and on a linear scale between 0 and 260 cm,
respectively. Water contents decreased rapidly between 220 and 250 cm, representing the capillary water
retention region. Water contents subsequently decreased only slowly with decreasing pressure heads(h < 250 cm), typical of the adsorptive water retention region. The nonclosing nature of the main drying
and wetting curves near saturation (h 5 0) is clearly visible, indicating air entrapment during rewetting in
the capillary region. Moreover, the main wetting and drying branches merged well in the dry region forh < 250 cm. This suggests that hysteresis may be negligible in the adsorptive water retention region of thissandy soil.
To test the hysteresis model, all data from the main drying and wetting curves in Figure 1a between 0 and
260 cm, as well as the data for h < 2100 cm, were used to estimate the various model parameters in equations (1), (2), (4), (5), (10), and (11). This was done using a nonlinear least squares optimization approach thatminimized the root-mean-square error (RMSE) between the calculated and observed data. We optimized six
parameters (hr, whs, dhs, wa, da, and n), while the remaining three parameters were ﬁxed: ha at 1/da and h0 at

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Figure 1. Observed capillary hysteretic and adsorptive water retention curves (open circle, triangle, solid circle, and solid triangle), their ﬁtted main branches (solid lines) and predicted scanning curves (dash lines) for Tottori dune sand. Results are plotted versus in a (a) log10
scale and (b) linear scale.

26.3 3 106 cm as indicated earlier, and b as given by equation (5). The various parameter estimation calcuR.
lations in this study were carried out using the Solver add-in of Microsoft ExcelV
Figures 1a and 1b show excellent agreement between the observed and ﬁtted main branches using the
optimized parameter values listed in Table 1. Drying and wetting processes are indicated by the direction of
the arrows. The proposed PvG model could describe hysteretic capillary water retention very well forh > 250 cm, but also nonhysteretic retention in the adsorption region for h < 250 cm. Since the modelaccounts for air entrapment, the nonclosing main branches at and near saturation could be described
equally well. Note from Table 1 that, as expected, the ﬁtted hs and a parameters for the main drying andwetting curves differed (i.e., da < wa and whs < dhs).
Figure 1b shows that the model could describe both scanning curves very well also, with the drying scanning curve initially moving from a reversal point on the main wetting branch (h 5 217.5 cm, h 5 0.227 cm3
cm23) to some end point (h 5 236.5 cm, h 5 0.115 cm3 cm23), followed by a wetting scanning curve backfrom the end point to the initial reversal point on main wetting curve. The model produced closed scanning
curves as observed also in this experiment. We emphasize that the proposed model is relatively straightforward to apply since it requires only two additional parameters, da and dhs (or wa and whs) to describe capillary hysteretic water retention as compared to the original PDI model.
4.2. Unsaturated Hydraulic Conductivity
Figure 2a shows the unsaturated hydraulic conductivity of Tottori dune sand as a function of the pressure
head as estimated by Sakai and Toride [2007] using the evaporation method. Their parameter approach

Table 1. Soil Hydraulic Parameters for the Main Drying and Wetting Curves of the PDI-Based vG Model for Seven Soil Samples

Tottori Dune Sand
Guelph LoamIda Silt Loam (0–15 cm)
Ida Silt Loam (>15 cm)
Rubicon Sandy Loam
Rideau Clay Loam
Wray Dune SandRUDIYANTO ET AL.

hr(cm3 cm23)dhs(cm3 cm23)whs(cm3 cm23)da(cm21)wa(cm21)nKscap(cm d21)‘aKsfilm(cm d21)

0.056
0.286
0.166
0.164
0.191
0.304
0.106

0.377
0.519
0.542
0.524
0.378
0.419
0.303

0.343
0.434
0.494
0.486
0.378
0.419
0.303

0.034
0.013
0.009
0.014
0.011
0.018
0.030

0.054
0.036
0.015
0.018
0.039
0.054
0.053

7.40
2.59
1.47
1.56
5.80
3.63
8.20

97.8
31.6
491.0
1082.9
48.1
20.0
514.0

20.11
0.5
0.5
0.5
0.5
0.5
0.5

22.01021.41921.26521.27821.866
21.56522.2000.04800.00680.00150.00390.01200.00980.0140A COMPLETE SOIL HYDRAULIC MODEL

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Figure 2. Liquid conductivities (dKliq and wKliq) for the main drying and wetting branches of water retention of Tottori dune sand in Figure
1 (dark and light blue solid lines, respectively), and hydraulic conductivities (Kem) as estimated by Sakai and Toride [2007] using the evaporation method (orange dash lines). Results are plotted as a function of (a) pressure head and (b) water content.

assumed that the water retention and hydraulic conductivity curves could be described with the Fayer andSimmons [1995] and Mualem [1976a] models, respectively [Sakai and Toride, 2007]. The K(h) curve in Figure
2a consists of two subcurves indicating the presence of capillary and ﬁlm conductivities at relatively highand low-pressure heads, respectively. Figure 2a also shows the ﬁtted dK(h) drying curve using water retention parameters of the main drying curve (Table 1). Since the PvG-based K(h) model required four additionalparameters (i.e., Kscap , ‘, Ksfilm , and a), those parameters were estimated by minimizing the RMSE between
log10K(h) model predictions and log10K(h) observed data, while keeping the retention parameters ﬁxed at
the optimized values for the main drying curve as listed in Table 1. The various hydraulic parameters wereused subsequently to predict the hydraulic conductivity wK(h) for the main wetting curve. Good agreement
was obtained between dK(h) and K(h) from the evaporation method. Capillary hysteresis caused the main
K(h) branches to be hysteretic also, as well as produced unclosed K(h) curves at saturation. In accordancewith the current model formulation, the ﬁlm conductivity is assumed to be nonhysteretic.
The wetting and drying hydraulic conductivity curves as a function of water content, K(h), are presented in
Figure 2b. The curves appear almost nonhysteretic over the entire range of water contents, which is consistent with the often assumed nonhysteretic nature of K(h) [Kool and Parker, 1987; Mualem, 1986; van Genuchten, 1980]. The main difference concerns the conductivity values of the unclosed main drying andwetting branches of their end points (at h 5 0), leading to different values for dhs and whs.

4.3. Estimated Parameters of the Film Conductivity Model
One limitation of the PDI model is that it cannot be used to predict the ﬁlm conductivity from the water
retention parameters since two additional ﬁlm conductivity parameters (Ksfilm and a) are involved [Peters,
2013]. For this reason, we evaluated possible relationships between the water retention and ﬁlm conductivity parameters of Tottori dune sand as well as of six other soil samples having a broad range of n values(1.6  n  8) and a values (0.011  a  0.054) as described in Table 2.
The parameter estimation procedure was essentially the same as for the hysteretic data sets. First, we ﬁtted
the water retention parameters to the data using equations (1), (2), (3), and (4). Equations (6)–(8) were subsequently ﬁtted to the hydraulic conductivity data, leading to estimates of the four conductivity parameters(i.e., Kscap and ‘ for the capillary conductivity, and Ksfilm and a for the ﬁlm conductivity), while keeping theretention parameters at their previously ﬁtted values. Figure 3 shows a schematic of the capillary, ﬁlm, andtotal conductivities, as well as of the locations of Kscap , Ksfilm , hcf, and Kcap(hcf). We used hcf for the pressure
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Table 2. Soil Hydraulic Parameters of the PDI-Based vG Model for 11 Soil Samples

Tottori Dune Sand
Sydney Sand
Rhinluch Sand
Berlin Sand
Gilat Loam
Revoholt Sand
Pachappa Fine
Sandy Clay
Sandy Loam
Shonai Sand
Pachappa Loam
Adelanto Loamhr
(cm3 cm23)hs(cm3 cm23)

A(cm21)nKscap(cm d21)‘aKsfilm(cm d21)hcf(cm)

Kc(hcf)(cm d21)

0.056
0.041
0.094
0.030
0.156
0.020
0.106

0.375
0.427
0.347
0.305
0.437
0.402
0.324

0.034
0.027
0.018
0.054
0.017
0.045
0.011

8.05
7.30
5.15
4.44
3.42
3.22
2.70

97.4
58.7
32.8
538.3
19.4
2189.515.8

20.14720.243
0.104
0.500
1.175
1.753
0.137

22.12521.79622.43521.722
21.23821.43321.5610.07950.00490.01520.01230.00780.00160.0094255.2
295.4
2137.4266.2
2164.82114.52492.00.005760.000170.001190.000440.000400.00004
0.00055

0.047
0.058
0.118
0.237

0.418
0.419
0.441
0.426

0.011
0.041
0.007
0.004

1.64
8.05
7.30
5.15

12.8
673.0
12.0
5.0

0.002
0.5
0.5
0.5

21.16521.81321.41621.4280.00100.06410.00370.002423047.0
250.62759.621275.70.000020.034680.000290.00014head at which Kcap(hcf) is approximately equal to Kﬁlm(hcf). Please note that in reality, Ksfilm is always higher
than Kcap(hcf).
Results of the parameter analysis of the seven soils were used to correlate the ﬁlm hydraulic parameterswith the ﬁtted retention parameters. Results of the analysis are shown graphically in Figure 4. The relationship between log10(a|hcf|) and n is presented in Figure 4a. The plot suggested a hyperbolic type relationship,
which we described using the completely empirical expression:
log10 ðajhcf jÞ50:21913:811n22:187(17)

This function agreed reasonably well with the data, giving an R2 value of 0.96.
Figure 4b shows a positive and relatively strong correlation (with R2 5 0.98) between log10[Kcap(hcf)] andlog10[Ksfilm ] using the function

Figure 3. Schematic locations of Kscap , Ksfilm , Kcap(hcf), and hcf on the capillary conductivity (black dash lines), ﬁlm conductivity (black dashdot lines), and total conductivity curves (black solid lines).

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Figure 4. Relationships obtained for the water retention parameters n and a, and the ﬁlm conductivity parameters Ksfilm and a. The plots are for (a) n versus log10(a hcf); (b) log10 [Kcap(hcf)]
versus log10 [Ksfilm ], and (c) n versus a.

log10 Ks film 50:5978flog10 ½K cap ðhcf Þg20:3205(18)

Figure 4c shows the correlation between n and a, in this case, a negative correlation with R2 5 0.512. The
negative correlation suggests that a relatively steep capillary conductivity function, Kcap(h), also will lead toa relatively steep ﬁlm conductivity function, Kﬁlm(h). This is to be expected since coarser soils, and soils having a relatively narrow pore or particle-size distribution, generally have higher values of n [e.g., Carsel andParrish, 1988], leading to more rapidly declining ﬁlm conductivities when they dry out. For the data in Figure
4c, we obtained a linear function between n and a:a520:1388n21:061(19)Equation (19) indicates that a in equation (8) may not be a constant as assumed by Peters [2013] in his
model (he suggested a constant value of 21.5). According to the extended ﬁlm conductivity model of Tokunaga [2009], speciﬁc surface area inﬂuences the slope of the ﬁlm conductivity function. Since the speciﬁc
surface area in turn is also related to the value of n (i.e., coarser soils generally have higher n values), somedependency of a on n is to be expected. Another possibility of why a is not constant in or study may be
due to the assumption that ha equals 1/da in equation (8), which may have slightly affected the values of aduring the parameter estimation analysis.
4.4. Validation of the Film Conductivity Parameters
We further tested the empirical relationships for the ﬁlm ﬂow parameters (equations (17)–(19)) by predicting the unsaturated hydraulic conductivity, K(h) of three soils: Shonai Sand [Mehta et al., 1994], Adelanto
Loam, and Pachappa Loam [Jackson et al., 1965; Zhang, 2011]. The complete water retention and hydraulicconductivity curves of those three soils were available from wet to dry as shown in Figure 5. After optimizing the water retention parameters of the PvG model (hr, hs, a, n), the two extra parameters (Ksfilm and a) of
the ﬁlm conductivity model were determined. The parameter Ksfilm was calculated using equation (18)assuming ‘ 5 0.5 for Kcap(hcf) in equation (7), and with hcf determined by equation (17). The parameter a
was approximated using equation (19) as a function of n. The optimized retention parameters of the PvG
model and the estimated parameters (Ksfilm and a) in the ﬁlm conductivity model for the three soils arelisted in Table 2.
The corresponding ﬁtted water retention and unsaturated hydraulic conductivity curves are shown in Figure
5. Results show that the PvG model agrees well the observed water retention of three soils from the wet todry end. Furthermore, the model could predict the observed unsaturated hydraulic conductivity over theentire range of water contents excellently. This indicates that the capillary hydraulic conductivity model
based on the van Genuchten-Mualem equations in combination with the ﬁlm conductivity model of Peters[2013] and the empirical expressions given by equations (17)–(19) could describe the liquid conductivity
quite well, including the ﬁlm conductivity. Note that the n values in our study ranged from 1.6 to 8, andhence that the equations are valid only for this range.

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Figure 5. (left) Observed and ﬁtted water retention, h(h) and observed and predicted unsaturated hydraulic conductivity, K(h) curves of (a) Shonai Sand, (b) Pachappa Loam, and (c) Adelanto Loam. Plotted are the capillary conductivity, Kcap (pink long dash lines), ﬁlm conductivity, Kﬁlm (orange short dash lines), isothermal vapor conductivity, Kvh (light blue dot lines), andtotal conductivity, Ktot (dark blue solid lines) versus (middle) water content and (right) pressure head.

The plots in Figure 5 also show the contributions of the isothermal vapor conductivity, Kvh, to the total
conductivity, Ktot. Calculations of Kvh using the approach summarized in Appendix A assumed that thetemperature for vapor conductivity is equal to 208C, which is a common laboratory condition [Peters,
2013]. Results show that the vapor conductivity curve, Kvh, becomes more dominant than the liquid conductivity when Kliq is less than about 5 3 1028 cm d21, or for volumetric water contents less than about
0.025 for Shonai Sand (Figure 5a), 0.03 for Pachappa Loam (Figure 5b) and 0.05 cm3 cm23 for Adelanto

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Figure 6. Application of the empirical equations for the ﬁlm conductivity to estimate K(h) in the whole water content range of (a) Guelph Loam, (b) Ida Silt Loam 0–15 cm, (c) Ida Silt
Loam > 15 cm, (d) Rubicon Sandy Loam, (e) Rideau Clay Loam, and (f) Wray Dune Sand. Results are for the observed dh(h), wh(h), K(h) data (symbols: open circle, solid circle, and solid
square, respectively), the ﬁtted dh(h) and wh(h) curves (dark and light blue solid lines), the predicted dKliq and wKliq curves (dark and light blue solid lines), the predicted dKcap and wKcap
curves (dark and light blue dash lines), and the predicted Kvh curves (black dotted lines). (left) h(h), (middle) K(h), and (right) K(h).

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Loam (Figure 5c). As demonstrated especially by the curves for Shonai Sand and Pachappa Loam, the contribution of the vapor conductivity to the total conductivity becomes more important than the ﬁlm conductivity in the very dry range, ostensibly leading to an overall slightly better description of the
conductivity data.
4.5. Further Tests of the Complete Soil Hydraulic Model Formulation
We conducted several additional tests to examine the complete hysteresis and conductivity features ofthe overall soil hydraulic model. We were especially interested in the utility of the derived empirical relationships, equations (17)–(19), for estimating the two extra parameters (Ksfilm and a) of the ﬁlm conductivity model. For the tests, we used unsaturated hydraulic conductivity data of eight soils ranging from clay
loam to sand, including Guelph Loam [Elrick and Bowman, 1964], Ida Silt Loam [Green et al., 1964], Rubicon
Sandy Loam [Topp, 1969], Rideau Clay Loam [Topp, 1971], and Wray Dune Sand [Gillham et al., 1979]. Hysteretic water retention data of the eight soils were available only at relatively high-pressure heads asshown in Figure 6. The retention curves in the dry region were extrapolated to zero water contents at avalue of 26.3 3 106 cm of the pressure head, h0, as suggested by Peters [2013] based on Schneider and
Goss [2011].
To obtain the retention parameters, the observed main drying and wetting data were ﬁtted simultaneously
using the proposed hysteretic water retention model given by equations (1), (2), (4), (10), and (11). In total,
six parameters were estimated (i.e., hr, dhs, whs, da, wa, and n), while the two remaining parameters were ﬁxed
(ha 5 1/da and b). The K(h) functions were predicted with Ks equated to the measured saturated hydraulicconductivities and assuming ‘ 5 0.5 [Mualem, 1976a]. The parameters hcf, Ksfilm , and a were furthermore estimated using equations (17)–(19), respectively.
The optimized parameters for the six soils are listed in Table 1 corresponding to the main drying and wetting branches and the calculated K(h) curves in Figure 6. Although there are some disagreement for themain loop for Rubicon Sandy Loam and Rideau Clay Loam (Figures 6d and 6e), the hysteretic water retention model could describe both the drying and wetting retention curves very well. The predicted K(h) curves(solid lines) for all eight soils were essentially nonhysteric (i.e., dKliq  wKliq), and agreed well with theobserved K(h) values at the higher water content where the capillary conductivity dominates. Moreover, theliquid conductivity curves, Kliq at low water content were several orders of magnitudes higher than the capillary conductivity curves, Kcap, indicating that ﬁlm conductivities, Kﬁlm contributed signiﬁcantly to the overallliquid conductivity curves, Kliq.
The isothermal vapor conductivity Kvh components at 208C for the eight soils are also shown in Figure
6. The isothermal vapor conductivity was found to clearly dominate the liquid conductivity in the dry
end for the coarser soils with their relatively large n values (notably Wray Dune Sand and Rideau Clay
Loam in Figures 6f and 6e, respectively). Conversely, the isothermal vapor conductivities of the more
ﬁne-textured soils (Ida Silt Loam 0–15 cm and > 15 cm in Figures 6a and 6b, respectively) with theirsmaller n values (broader pore size distributions) are very similar to the liquid conductivities in thedry end. Again, this phenomenon can be explained by the extended ﬁlm conductivity model of Tokunaga [2009], which assumes that the ﬁlm conductivity increases with increasing values of speciﬁc surface area. It follows that the ﬁlm conductivities of ﬁner-grained soils (with their smaller n values) arelarger than those of coarser soils, which generally have larger n values and lower speciﬁc surface
areas at similar porosities. As a result, the ﬁlm conductivity of a ﬁner-grained soil (with its larger speciﬁc surface area) will inch toward the values of isothermal vapor conductivity encountered at lowpressure heads.

5. Conclusions
We improved the PDI model by incorporating the Parker and Lenhard (PL) hysteresis model for capillary water retention in their formulation. The resulting combined model only requires two extra parameters as compared to the PDI model. The model includes capillary hysteresis at high water contents
with either closed or unclosed main branches at full saturation due to entrapped air, closed scanning
curves, and nonhysteretic adsorptive water retention at low water contents. The model could ﬁt themain drying and wetting branches and predict the drying and wetting scanning curves of water retention very well.

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The shape of the unsaturated hydraulic conductivity curve as a function of pressure head, K(h), reﬂects
the shape of water retention, with hysteresis present in the capillary conductivity but not in the ﬁlm conductivity. The model yielded mostly nonhysteretic unsaturated hydraulic conductivity functions versus
water content, K(h), for the entire range of water contents. Inclusion of a vapor conductivity component
in the model improved predictions in the very dry range. In this paper, we concentrated on descriptionsof the water retention and unsaturated hydraulic conductivity curves. It would be interesting to implement the proposed model in numerical software for predicting various variably saturated ﬂow processes,
especially for relatively dry soil conditions (including evaporation and root water uptake processes). We
believe that the model will increase the accuracy of predictions substantially since the approach accounts
for many of the physical processes governing soil water retention and ﬂow from saturation until complete
dryness.
Water retention parameters could be related to the ﬁlm conductivity parameters by establishing empiricalmodels. The resulting combination of PvG and empirical relationships was validated using published data
sets, with results showing that the proposed model is in close agreement with the data. Thus, with thismodel, the unsaturated hydraulic conductivity for whole range of water content can be readily calculated.
Still, if both complete water retention and hydraulic conductivity data sets are available, it remains of course
preferred to ﬁt the adsorptive retention and ﬁlm water retention and hydraulic parameters directly to thedata.

Appendix A: Predicting Isothermal Vapor Hydraulic Conductivity
The isothermal vapor hydraulic conductivity, Kvh (m s21), is deﬁned as [Mehta et al., 1994; Peters, 2013; Philipand De Vries, 1957; Saito et al., 2006; Sakai et al., 2009]:
K vh 5

qvs Mg
D
Hr
qw RT

(A1)

where qvs (kg m23) is the saturated vapor density, qw (5 1000 kg m23) is the density of liquid water,
D (m2 s22) is the vapor diffusivity in soil, M is the molecular weight of water (0.018015 kg mol21), g is
the gravitational acceleration (9.81 m s22), R is the universal gas constant (58.314 J mol21 K21), T (K) isthe absolute temperature, and Hr is relative humidity. The vapor diffusivity, D, in soils can be derived asa product of the diffusivity of water vapor in air, Da (m2 s22), the tortuosity factor in the gaseous phase,sa, and the volumetric air content, ha, using
D5sa ha Da(A2)

The tortuosity factor in the gaseous phase, sa, may be calculated with the [Millington and Quirk, 1961]
relationship:sa 5

ha 7=3/2(A3)

where / [2] is the porosity, assumed to be equal to hs for simplicity, and ha 5 hs 2 h(h). The parameters Da
and qvs are functions of temperature:2TDa 52:14  1025
(A4)
273:15and
6014:79
qvs 51023 exp 31:37162
27:92495  1023 T T 21
T

(A5)

whereas Hr is derived using the Kelvin equation:


Mgjhj
Hr 5exp 2
RT(A6)

where h (m) is the pressure head.

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Acknowledgments
We would like to thank A. Peters of the€Institut f€
ur Okologie,
Technische
Universit€at Berlin, Germany, for sharing
some of the data sets used in thispaper. The published data sets for thispaper are properly cited and referred
to in the reference list: Sydney Sand
[Minasny and Field, 2005]; Rhinluch
Sand [Schindler and M€
uller, 2006];
Berlin Sand [Peters, 2013]; Gilat Loam,
Rehovot Sand, and Pachapa Fine
Sandy Clay [Mualem, 1976b]; Sandy
Loam [Pachepsky et al., 1984]; Shonai
Sand [Mehta et al., 1994], Adelanto
Loam, and Pachappa Loam [Jackson
et al., 1965; Zhang, 2011]; Guelph
Loam [Elrick and Bowman, 1964], Ida
Silt Loam [Green et al., 1964], Rubicon
Sandy Loam [Topp, 1969], Rideau Clay
Loam [Topp, 1971], and Wray Dune
Sand [Gillham et al., 1979].

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A COMPLETE SOIL HYDRAULIC MODEL

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