WATER RESOURCES RESEARCH, VOL. 46, W12554, doi:10.1029/2010WR009092, 2010
A new capillary and thin film flow model for predicting
the hydraulic conductivity of unsaturated porous media
Marc Lebeau1 and Jean‐Marie Konrad1
Received 11 January 2010; revised 21 September 2010; accepted 1 October 2010; published 31 December 2010.
[1] Most classical predictive models of unsaturated hydraulic conductivity conceptualize
the pore space as either bundles of cylindrical tubes of uniform size or assemblies of
cylindrical capillary tubes of various sizes. As such, these models have assumed that liquid
configuration is the same in both the wet and dry ranges and that a single concept can be used
to describe water transport over the entire range of matric head. Yet theoretical and
experimental findings suggest that water transport in wet media, which mostly occurs in
water saturated capillaries, is quite different from that in dry media, which occurs in thin
liquid films. Following these observations, this paper proposes a new model for predicting
the hydraulic conductivity of porous media that accounts for both capillary and thin film flow
processes. As with other predictive models, a mathematical relationship is established
between hydraulic conductivity and the water retention function. The model is
mathematically simple and can easily be integrated into existing numerical models of water
transport in unsaturated soils. In sample calculations, the model provided very good
agreement with hydraulic conductivity data over the entire range of matric head. Two other
well‐supported models, on the other hand, were unable to conform to the experimental data.
Citation: Lebeau, M., and J.‐M. Konrad (2010), A new capillary and thin film flow model for predicting the hydraulic
conductivity of unsaturated porous media, Water Resour. Res., 46, W12554, doi:10.1029/2010WR009092.
1. Introduction
[2] The description of soil‐water transport processes in
unsaturated porous media often hinges on the adequacy of
the highly nonlinear functions that represent water retention
and hydraulic conductivity. The literature abounds with
parametric models that describe both water retention and
hydraulic conductivity data. These models range from simple mathematical expressions for unstructured media, with
unimodal pore size distributions [Leij et al., 1997], to more
complex formulations for structured media with multimodal pore size distributions [Othmer et al., 1991; Ross
and Smettem, 1993; Durner, 1994]. Models have also been
proposed to describe water retention [Khlosi et al., 2008]
and hydraulic conductivity [Peters and Durner, 2008] over
the entire range of matric head. In many cases, however,
measurements of unsaturated hydraulic conductivity are
unavailable. As an alternative to direct measurements, models
have been proposed to estimate hydraulic conductivity from
water retention data, which is more easily measured. In
general, these models may be classified as those based on
(1) similarity between Darcy‐Buckingham’s law and other
viscous flow equations such as those of Navier‐Stokes and
(2) statistical representations of pore space [Brutsaert, 1967;
Mualem, 1986]. It follows that the models either conceptualize the pore space as cylindrical tubes of uniform size, with
coexisting wetting and nonwetting fluids, or as assemblies of
1
Department of Civil and Water Engineering, Faculty of Sciences
and Engineering, Laval University, Quebec City, Quebec, Canada.
Copyright 2010 by the American Geophysical Union.
0043‐1397/10/2010WR009092
cylindrical capillary tubes of various sizes, which are either
completely filled with wetting or nonwetting fluid. As such,
these models have assumed that liquid configuration is the
same in both the wet and dry ranges. Although it has generally been found that these models are successful in the wet
to moderately wet range (high values of matric head), where
water is mostly held by capillary forces, they have been
found to underestimate hydraulic conductivity under dry
conditions [Goss and Madliger, 2007; Jansik, 2009]. As
pointed out by Tuller and Or [2001], this shortcoming may
be attributed to the lack of consideration of soil‐water flow in
thin films, likely to become important at low values of matric
head.
[3] Motivated by this and other limitations, such as the
lack of consideration of adsorptive forces and unrealistic
cylindrical representation of pore space, Tuller and Or
[2001] proposed an alternative approach for the derivation
of hydraulic conductivity functions for homogeneous porous
media. Their model is based on liquid configurations in both
angular and slit‐shaped spaces, and accounts for capillary,
corner and thin film flows. Although of great scientific
interest, the model is mathematically complex and must be
used in conjunction with the water retention model of Or
and Tuller [1999], which often fails to describe experimental data in the intermediate saturation range because of
the limited flexibility of the probability distribution function
used to characterize the pore size distribution. Tokunaga
[2009] also used the principles of interfacial physical chemistry and the concept of flow in thin films to predict water
transport in dry monodisperse porous media.
[4] In this study, theoretical considerations and experimental observations are used to partition water retention into
capillary and adsorptive components. These components are
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LEBEAU AND KONRAD: A NEW CAPILLARY AND THIN FILM FLOW MODEL
associated with two different liquid configurations. The first
is an assembly of water or air filled capillary tubes of various
diameters whereas the second is a thin film of water stretched
over the surface of the solid particles. A conventional statistically based model is used to predict hydraulic conductivity
within the water filled capillaries while a theoretical model of
soil‐water flow in thin films is used to predict hydraulic
conductivity within the thin liquid films. In so doing, the
model accounts for both capillary and adsorptive forces as
well as for dual occupancy of wetting and nonwetting phases.
[5] The paper shows the effectiveness of the new model
in using water retention data to predict hydraulic conductivity over the entire range of matric head. Model performance is subsequently evaluated by comparing results with
those of other well‐supported models.
2. Proposed Model
2.1. Water Retention
[6] Most popular among the parametric models that have
been proposed to describe water retention data are the
equations of Brooks and Corey [1964] and van Genuchten
[1980]. An often problematic element in these equations is
the use of a parameter called residual water content. While
this parameter does not consistently correspond to a recognized physical entity, it is speculated to represent the largest
water content for which pore water is held primarily by
adsorptive rather than capillary forces [Corey and Brooks,
1999]. Most studies consider residual water content as a
fitting parameter and assume that liquid flow processes are
negligible at water contents smaller than a residual value.
This approach has generally been successful at large values
of matric head, where flow mostly occurs in water saturated
capillaries. Yet, it remains physically unrealistic to consider
water retention data as never becoming less than the residual
water content. In fact, there is ample experimental evidence
showing that water content ultimately approaches zero as
the soil equilibrates with moisture‐free air [Schofield, 1935;
Campbell and Shiozawa, 1992].
[7] In essence, theoretical studies of water retention have
led to believe that multilayer adsorption progressively
dominates capillary retention, and that adsorption is the
dominating force that holds water in its condensed state
when the soil is dry. Generally speaking, adsorption is a
process in which molecules of water vapor move onto a
solid or liquid surface. This process occurs until a thermodynamic equilibrium is reached between the gaseous phase
and the adsorbed layer. In very dry soils, the experimental
studies of Orchiston [1953], Amali et al. [1994], and de Seze
et al. [2000] have shown that water content is best described
with the Brunauer‐Emmett‐Teller (BET) theory, which
extends the Langmuir theory for monolayer adsorption of
gas molecules on a solid surface to multilayer adsorption
[Brunauer et al., 1938]. The BET theory is generally adequate for values of relative humidity less than approximately
35%, which translates into a matric head of −104 m when
using Kelvin’s equation at 298.15 K. For larger values of
matric head, adsorption gives rise to multilayer films with
thicknesses of the order of nanometers. These films can be
regarded as part of a thinned liquid phase [Derjaguin et al.,
1987; Churaev, 2000]. The state of this thinned liquid phase
can be described by means of the disjoining pressure isotherm, as discussed later in the paper.
[8] In response to these and other observations, new
models of water retention and physically based modifications
of existing models have been suggested by a number of
researchers [Ross et al., 1991; Campbell and Shiozawa, 1992;
Rossi and Nimmo, 1994; Fredlund and Xing, 1994; Fayer
and Simmons, 1995; Morel‐Seytoux and Nimmo, 1999;
Webb, 2000; Groenevelt and Grant, 2004; Khlosi et al., 2006;
Silva and Grifoll, 2007]. The seminal work of Campbell and
Shiozawa [1992] is of particular importance as it was the first
to note that water retention at low water content shows a linear
behavior on a semilog plot. In a later study, Fayer and
Simmons [1995] used this relationship to develop a method
of extending most parametric models to zero water content
conditions. In essence, the method consists in replacing
residual water content by the log linear form observed by
Campbell and Shiozawa [1992]. The resulting equation can
be expressed in a manner that highlights the mechanisms
(capillarity and adsorption) of water retention:
c
a
¼ c þ 1 s
ð1Þ
where is the volumetric water content, c is the volumetric
water content ascribed to capillary forces, s is the saturated
volumetric water content, and a is the volumetric water
content ascribed to adsorptive forces. It follows that the
terms on the right‐hand side of this equation are the capillary and adsorptive components of volumetric water content, respectively.
[9] Although a number of parametric models may be used
to describe the volumetric water content ascribed to capillary forces in soils with unimodal and multimodal pore
structures, it is herein represented with a simplified form of
Kosugi’s [1996] two‐parameter equation with zero residual
water content:
ln hm =hm;median
1
pffiffiffi
c ¼ s erfc
2
2
ð2Þ
where erfc( ) is the complementary error function, hm is the
matric head, hm,median is the matric head that corresponds
to the median capillary pore radius, and s is the standard
deviation of the log‐transformed capillary pore radius
distribution.
[10] The volumetric water content ascribed to adsorptive
forces is described by a form of the equation proposed by
Campbell and Shiozawa [1992]:
"
lnjhm j
a ¼ o 1 ln hm;dry #
ð3Þ
where o is the volumetric water content due to adsorption at
a matric head of −1 m, and hm,dry is the matric head at oven
dryness. The value of matric head at oven dryness depends
on prevailing laboratory conditions, i.e., temperature, pressure and relative humidity. However, experimental results
have shown that oven dryness generally corresponds to a
finite matric head of −105 m [Schofield, 1935; Russam, 1958;
Croney and Coleman, 1961; Campbell and Shiozawa, 1992].
[11] Figure 1 expresses water content in terms of degree
of saturation, and highlights the different components that
are accounted for in the water retention model: (1) a capillary component with a sigmoidal behavior in the wet range
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Figure 1. Water retention model with underlying capillary
and adsorptive components.
and (2) an adsorption component with a logarithmic
behavior in the dry range. As the mechanisms responsible
for these components of water retention are not related, the
presence of one component does not exclude or preclude the
other. Hence, in the moderately wet range, smaller pores
may be completely filled with water while a thin film of
water may cover the surface of the larger pores.
2.2. Hydraulic Conductivity
[12] According to the preceding description of water
retention, water movement in moderately wet porous media
may occur through the entire section of smaller water‐filled
pores as well as through thin films of water surrounding
larger pores. In dry media, on the other hand, liquid water
movement may only occur through the continuous water
film that covers the solid particles as the central portion of
the pores is occupied by the nonwetting phase. As a result,
the pores cannot be conceptualized solely as an assembly of
parallel tubes, which are either completely filled with water
or air. To account for the presence of a thin liquid film, it is
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herein suggested that the liquid phase be conceived as both
an assembly of water filled capillary tubes of various diameters, and a thin film of liquid stretched over the solid
particles. As illustrated in Figure 2, tubes of various radii are
used to represent capillarity. The radii of the fully saturated
tubes are inferred from the capillary component of the water
retention function using the Young‐Laplace equation. In
contrast, adsorption is represented by a thin liquid film of
varying thickness. Based on the dependence of film thickness on matric head, the thickness of the film is shown to be
greatest when the porous medium is wet and smallest when
it is dry. The extent of the film, on the other hand, is shown
to vary as a function of the capillary component of water
retention.
[13] It follows that two different models must be used
simultaneously to describe the hydraulic conductivity of the
porous media: (1) a capillary flow model and (2) a thin film
flow model. Peters and Durner [2008] recently suggested
that relative hydraulic conductivity can be described as the
weighted average of the contributions of capillary and thin
film flows. However, in the presence of a partitioned water
retention function, with both capillary and adsorptive components, it is suggested that water transport models be
combined as follows:
c
kr;a
kr ¼ kr;c þ 1 s
ð4Þ
where kr is the relative hydraulic conductivity, kr,c is the
relative hydraulic conductivity ascribed to capillary flow, and
kr,a is the relative hydraulic conductivity ascribed to thin film
flow. As formulated, the terms on the right‐hand side of this
equation represent the capillary and adsorptive (or thin film)
components of relative hydraulic conductivity, respectively.
In contrast to using a weighted average, equation (4) combines the capillary and thin film flow models without
requiring an additional fitting parameter.
2.2.1. Capillary Conductivity Model
[14] Capillary flow models conceptualize the pores as an
assembly of parallel capillary tubes, which may or may not
be sectioned and randomly rejoined. Under this assumption,
a number of fully saturated capillary tubes transmit water
while other tubes remain empty. For the sake of simplicity,
the portion of fully saturated tubes is obtained from the
Figure 2. Liquid configurations for the different components of water retention.
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LEBEAU AND KONRAD: A NEW CAPILLARY AND THIN FILM FLOW MODEL
water retention function by means of the Young‐Laplace
equation. Hydraulic conductivity at the continuum scale is
then determined by integrating the contributions of the fully
saturated tubes, as calculated with the Hagen‐Poiseuille
equation. In the literature, models have evolved from very
simple representations of pore space to more complex descriptions that account for tortuosity and variations in tube
geometry [Childs and Collis‐George, 1950; Gates and
Lietz, 1950; Fatt and Dykstra, 1951; Burdine, 1953;
Wyllie and Gardner, 1958; Mualem, 1976a; Alexander and
Skaggs, 1986]. Although many different models may be
used, Mualem’s [1976a] model has been shown to provide a
reasonable description of the relative hydraulic conductivity
due to capillary flow. Inserting equation (2) into Mualem’s
model yields the following expression of relative hydraulic
conductivity [Kosugi, 1996]:
kr;c
l
ln hm =hm;median
kc
1
pffiffiffi
¼ ¼ erfc
2
ks
2
2
ln hm =hm;median
1
pffiffiffi
þ pffiffiffi
erfc
2
2
2
described by means of the disjoining pressure isotherm
[Derjaguin et al., 1987]:
PðÞ ¼ Pg Po
ð6Þ
where P is the disjoining pressure, d is the film thickness, Pg
is the pressure in the gas phase, and Po is the pressure in the
bulk liquid phase. Based on this definition, disjoining
pressure in thin planar films (without capillary interactions)
is equal to the negative value of matric potential. It follows
that matric head is equal to −P(d)/(rog) where ro is the
density of the liquid phase and g is the gravitational acceleration constant. The form of the disjoining pressure isotherm is contingent on the type of surface forces at play. In
porous media, the major types of surface forces are ionic‐
electrostatic, molecular, structural, and adsorptive. Yet, in
the study of thin films, it is common practice to consider
only the effects of ionic‐electrostatic and molecular forces.
Hence, disjoining pressure may be defined as [Derjaguin
and Churaev, 1974]:
P ð Þ ¼ P e ð Þ þ P m ð Þ
ð5Þ
where kc is the hydraulic conductivity due to capillary flow,
ks is the saturated hydraulic conductivity, and l is a lumped
parameter that is generally assumed to account for pore
tortuosity and pore connectivity, which Mualem [1976a]
empirically set equal to 0.5 for predictive purposes. The
physical meaning of parameter l is often questioned given
that its fitted value is frequently found to be negative
[Schaap and Leij, 2000]. It is to be noted that by using
equation (5), capillary conductivity at the continuum scale is
estimated with the portion of pore water that is retained
solely by capillary forces. This is significant given that these
models are based on the assumption of capillary flow and
that there is no reason to expect them to be valid when
adsorptive forces dominate.
2.2.2. Thin Film Conductivity Model
[15] Given the similarity between Darcy‐Buckingham’s
law and other viscous flow equations (solutions of the
Navier‐Stokes equations for various geometries), fluid flow
has often been imagined to occur in the form of annuli or thin
planar films. In these cases, the wetting fluid is assumed to
be a continuous component stretched over the solid particles
whereas the nonwetting fluid is assumed to occupy the
central portion of the pores. Hence, the wetting and nonwetting fluids are assumed to coexist within the pores. In
recent years, a number of researchers have applied the
concept of flow in thin films to the study of hydraulic
conductivity in unsaturated porous media [Tuller and Or,
2001; Tokunaga, 2009].
[16] Inspired by these studies, a theoretical model of thin
film flow in porous media is obtained by solving the Navier‐
Stokes equations for steady state laminar flow through
planar liquid films. Most critical to this approach is understanding the dependence of thin film thickness on potential.
By nature, thin liquid films are composed of many transition
layers within which intensive properties and composition
may differ from those in the bulk liquid phase. The combined effect of these interfacial interactions results in larger
pressures in the thin liquid films than in the bulk liquid
phase. For planar films, the difference in potential may be
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ð7Þ
where Pe and Pm are the ionic‐electrostatic and molecular
components of disjoining pressure, respectively.
[17] In order to define the ionic‐electrostatic component
of disjoining pressure, let us consider a planar film of ionic
solution bounded by a substrate of given potential. Given its
charge, the substrate produces a much higher concentration
of ions near the surface. This concentration forces the ions to
diffuse away from the surface to equalize the concentration
throughout the solution. The charged surface and the distributed charge in the adjacent solution are termed the diffuse
double layer, which may be described with the Poisson‐
Boltzmann equation. Solutions to this equation may be
found for various cases that include truncated and extended
diffuse double layers with symmetric or asymmetric ionic
solutions. Langmuir [1938] solved the Poisson‐Boltzmann
equation for a low‐concentration symmetric ionic solution
with a high potential substrate, and found the following
expression for the ionic‐electrostatic component of disjoining pressure:
Pe ðÞ ¼
"r "o kB T 2 1
eZ
2
2
ð8Þ
where "r is the static relative permittivity of the liquid phase,
"o is the permittivity of free space, kB is the Boltzmann
constant, T is the absolute temperature, e is the electron
charge, and Z is the valence change.
[18] The molecular component of disjoining pressure
arises from long‐range fluctuating dipole bonds, or van der
Waals bonds, between condensed bodies. Once again, let us
consider the simple case of a planar film, and assume that
the film is held in place only by long‐range molecular forces. In this case, the Gibbs free energy is calculated by
adding liquid/liquid, liquid/solid, and solid/solid interactions
between two macroscopic bodies [Israelachvili, 1992]. The
disjoining pressure is subsequently calculated as the negative value of the derivative of the Gibbs free energy per unit
area, which yields
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Pm ðÞ ¼ Asvl
63
ð9Þ
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LEBEAU AND KONRAD: A NEW CAPILLARY AND THIN FILM FLOW MODEL
found the following expression for volumetric flow rate
per unit length of film:
o gB dhm
Qa ¼ 12o dx
ð11Þ
and B, a function of film thickness and temperature, is
given by
a a
B ¼ 43 5a2 a2 exp 6a2 þ a3 Ei Figure 3. Disjoining pressure isotherm for water films on
a planar substrate mainly composed of crystalline and amorphous silicates at room temperature.
where Asvl is the Hamaker constant for solid‐vapor interactions through the intervening liquid. Theoretical and
experimental values of the Hamaker constant for solid‐
liquid‐air systems range from −10−19 to −10−20 J
[Israelachvili, 1992]. In real soils, the Hamaker constant
is expected to differ slightly from these values due to the
effects of surface geometry and heterogeneity [Tuller and
Or, 2005].
[19] Equations (7) to (9) provide the basis for computing
film thickness as a function of disjoining pressure (or matric
head), and vice versa. Figure 3 illustrates the contributions
of the ionic‐electrostatic and molecular components of disjoining pressure in the case of a thin film of water on a
substrate composed primarily of crystalline and amorphous
silicates, such as the soil solid phase. Under these conditions, films in excess of 10 nm are mostly held by ionic‐
electrostatic forces whereas thinner films may be ascribed to
molecular forces.
[20] Let us now focus on the hydrodynamic behavior of
these thin liquid films. As shown in Figure 4, it is
assumed that flow is parallel to the plate in the x direction, and that other components of flow are nonexistent.
Recalling that velocity is finite at z = 0 and null at z = d,
the solution to the Navier‐Stokes equations yields a parabolic distribution of velocity. Integrating the velocity
distribution over the film thickness results in the following expression of volumetric flow rate per unit length
[Bird et al., 1960]:
Qa ¼ o g3 dhm
3o dx
ð12Þ
−7
where
R ∞ a = 1.621 · 10 /T (m), and Ei(−x) = −E1(x) =
− x [exp(−t)/t]dt is the exponential integral, which can be
evaluated using Taylor series [Gautschi and Cahill, 1964].
Note that as an alternative to equations (11) and (12), flow in
very thin films can be described by a form of equation (10)
in which the dynamic viscosity of the bulk liquid is replaced
by the average dynamic viscosity of the liquid film.
[21] To obtain the superficial, or Darcy‐Buckingham flux,
the volumetric flow rate per unit length must be multiplied
by the specific perimeter (perimeter of the solid particles per
unit cross‐sectional area). For monodisperse samples of
spherical particles, Tokunaga [2009] found the specific
perimeter to be equal to [12(1 − n)]/(pD) where n is the
porosity, and D is the diameter of the spherical particles. It
follows that the Darcy‐Buckingham flux in a monodisperse
sample of spherical particles can be expressed as
8 4o g
3 dhm
>
>
ð
1
n
Þ
dx
dhm < o D
qa ¼ ka
¼
>
dx
o gB
dhm
>
: ð1 nÞ
o D
dx
10 nm
ð13Þ
< 10 nm
where ka is the hydraulic conductivity due to film flow.
Using the experimental value of saturated hydraulic conductivity, the relative hydraulic conductivity due to film
flow can be defined as
kr;a ¼
8 1 4o g
3
>
>
10 nm
ð
1
n
Þ
> k D
<
s
o
ka
¼
>
ks > 1 o gB
>
:
ð1 nÞ
ks o D
ð14Þ
< 10 nm
[22] This equation extends the work of Tokunaga [2009]
to films thinner than 10 nm. To further generalize this work
to polydisperse samples, the diameter of the spherical particles can be replaced by an equivalent diameter, De, which is
the diameter of a spherical particle whose monodisperse
ð10Þ
where Qa is the volumetric flow rate per unit length of
film, and mo is the dynamic viscosity of the bulk liquid.
In contrast to thicker films, flow in very thin films
(thinner than 10 nm) is likely to be influenced by modified liquid viscosity close to the plate. For flow in these
films, Or and Tuller [2000] introduced an exponential
viscosity function into the Navier‐Stokes equations and
Figure 4. Adsorption on a planar surface with schematic
representation of velocity distribution.
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Table 2. Fitted Parameters for the Testing Data Sets
Fitted Parameters
Data Set
Sandy loam
Shonai sand
Masa loamy sand
Gilat loam
Pachapa loam
Adelanto loam
Pachapa fine sandy clay
Rehovot sand
Berlin medium sand 4
Poederlee loamy sand
Poederlee sand
Hupsel sand
Referencea
o
hm,median (m)
s
A
B
C
D
D
D
D
D
E
E
E
E
0.07
0.06
0.08
0.15
0.13
0.26
0.11
0.02
0.05
0.12
0.06
0.04
−2.06
−0.24
−1.18
−0.67
−2.27
−3.76
−1.14
−0.27
−0.37
−0.59
−0.63
−0.33
1.54
0.38
1.23
0.55
0.86
0.72
0.82
0.59
0.36
0.89
0.97
0.47
a
A, Pachepsky et al. [1984]; B, Mehta et al. [1994]; C, Fujimaki and
Inoue [2003a, 2003b]; D, Mualem [1976b]; E, Nemes et al. [2001].
Figure 5. Hydraulic conductivity model with underlying
capillary and thin film flow components.
sample has the same specific surface area per unit volume as
the polydisperse sample. Based on simple geometrical considerations, the specific surface area per unit volume of a
monodisperse sample of equivalent particles is found to be
equal to 6/De. The equivalent diameter can therefore be
expressed in terms of the specific surface area per unit volume
of the polydisperse sample, i.e., De = 6/Ss,polydisperse.
Although the specific surface area of the polydisperse sample
can be determined experimentally, it can also be estimated
from water retention data at low values of matric head [Tuller
and Or, 2005]. Based on this estimation, the equivalent
diameter can be expressed as
De ¼
6
1=3
m
Asvl
ð1 nÞ 6o ghm;m
ð15Þ
3. Model Evaluation (or Testing)
where m is the volumetric water content at matric head hm,m
for which capillary condensation due to surface roughness is
Table 1. Measured Soil Properties for the Testing Data Sets
Measured Soil Properties
Data Set
Sandy loam
Shonai sand
Masa loamy sand
Gilat loam
Pachapa loam
Adelanto loam
Pachapa fine sandy clay
Rehovot sand
Berlin medium sand 4
Poederlee loamy sand
Poederlee sand
Hupsel sand
Referencea
n
s
A
B
C
D
D
D
D
D
E
E
E
E
0.43
0.43
0.39
0.44
0.46
0.43
0.33
0.40
0.40
0.42
0.42
0.36
0.43
0.43
0.34
0.44
0.46
0.43
0.33
0.40
0.39
0.42
0.42
0.36
negligible [Tuller and Or, 2005]. As an approximation, this
data point can be taken as the water retention function value
closest to hm = −103 m.
[23] The preceding model builds on analyses of interfacial physical chemistry to predict relative hydraulic conductivity in the dry range without the use of any adjustable
parameters. Figure 5 highlights the contribution of this
model to relative hydraulic conductivity. As shown, relative hydraulic conductivity is dominated by thin film flow
in the dry range while capillary flow is the primary means
of water transport in the wet range. The point of crossover
separates the relative hydraulic conductivity curve into
capillary and film flow dominated regions. A smooth
transition is observed as thin film flow starts to govern
water transport at a matric head of −2.1 m. At this point,
flow is mostly ascribed to the portion of the film which is
held by ionic‐electrostatic forces. A further decrease in
matric head leads to the predominance of the portion of
film which is held by long‐range molecular forces. In
general, this reveals that thin film flow, which is often
ignored in conventional models, plays an important role in
water transport at low values of matric head.
ks (m/s)
9.26
1.09
5.90
2.00
2.00
4.50
1.40
1.33
8.01
2.79
1.90
8.16
×
×
×
×
×
×
×
×
×
×
×
×
10−7
10−4
10−6
10−6
10−6
10−7
10−6
10−4
10−5
10−5
10−5
10−5
3.1. Evaluation Data Sets
[24] A comprehensive search of relevant literature yielded
only a small number of data sets suitable to evaluate the
capabilities of the proposed model. Of these data sets, one is
reported by Pachepsky et al. [1984], one is described by
Mehta et al. [1994], another is reported by Fujimaki and
Inoue [2003a, 2003b], five are listed in Mualem’s [1976b]
catalog of unsaturated soils, and four are selected from the
UNSODA database [Nemes et al., 2001]. These data sets of
Table 3. Disjoining Pressure Parameters
Parameter
a
A, Pachepsky et al. [1984]; B, Mehta et al. [1994]; C, Fujimaki and
Inoue [2003a, 2003b]; D, Mualem [1976b]; E, Nemes et al. [2001].
6 of 15
Asvl
e
kB
T
Z
"o
"r
Definition
Unit
Hamaker constant
(J)
Electron charge
(C)
Boltzmann constant
(J/K)
Absolute temperature
(K)
Valence change
Permittivity of free space
(C2/(J m))
Static relative permittivity of water
Value
−6.00 × 10−20
1.60 × 10−19
1.38 × 10−23
298.15
1
8.85 × 10−12
78.41
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LEBEAU AND KONRAD: A NEW CAPILLARY AND THIN FILM FLOW MODEL
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Figure 6. Model results for Gilat loam. (a) Fitted water retention. (b) Predicted relative hydraulic
conductivity.
mostly undisturbed soils, with a wide range of textures and
origins, were selected because the experimental measurements covered both wet and dry regions of the water
retention and hydraulic conductivity functions. A list of the
different data sets can be found in Table 1.
3.2. Illustrative Examples
[25] The essential features of the new model are demonstrated with the various evaluation data sets, which range
from loam to sand. A summary of the model parameters
obtained by fitting the water retention model (equations (1)
through (3)) to the testing data sets is given in Table 2.
Curve fitting was conducted with a hybrid genetic‐simplex
algorithm, which combined the exploratory capacity of a
genetic algorithm with the convergence behavior of the
downhill simplex, or Nelder‐Mead, algorithm to meet
requirements of accuracy, reliability and computing time.
The set of best fit parameters obtained from the water
retention data, was then used to predict relative hydraulic
conductivity using equations (4), (5), and (14) with
equations (7), (8), and (9) defining film thickness and
equation (15) describing the equivalent diameter of the
porous media. Table 3 summarizes the disjoining pressure
parameters that were used to compute film thickness. Also
note that parameter values for the bulk liquid were taken
as those of pure water at 298.15 K (ro = 997.04 kg/m3
and mo = 8.90 × 10−4 N s/m2).
[26] Figure 6 illustrates the capabilities of the new model
for the Gilat loam data set. As shown, the water retention
model is in very good agreement with the liquid saturation
data, which ranges from saturation to a matric head of
approximately −103 m. As formulated, the water retention
model separates liquid saturation into individual contributions due to adsorptive and capillary forces. The point of
crossover of these curves separates liquid saturation into
Figure 7. Model results for Pachapa fine sandy clay. (a) Fitted water retention. (b) Predicted relative
hydraulic conductivity.
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LEBEAU AND KONRAD: A NEW CAPILLARY AND THIN FILM FLOW MODEL
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Figure 8. Model results for Sandy loam. (a) Fitted water retention. (b) Predicted relative hydraulic
conductivity.
capillary and adsorptive dominated regions. For the loam
under consideration, adsorptive forces start to dominate
water retention at a matric head of −1.0 m. At lower values
of matric head, adsorption is the predominant force that
holds water in the porous medium. The hydraulic conductivity model also shows very good agreement with the
experimental data. By considering thin film flow, the model
is able to reproduce the change in slope of the experimental
data, which occurs at a matric head of −1.5 m. It is to be
emphasized that such film contributions have generally been
overlooked in conventional predictive models of hydraulic
conductivity. Figure 7 shows somewhat similar results for
the Pachapa fine sandy clay data set. In this case, however,
the hydraulic conductivity model slightly underestimates the
experimental data in both wet and dry ranges.
[27] In contrast to the finer‐grained soils, the sandy loam
data set presents a much smaller quantity of adsorbed water.
Yet, as shown in Figure 8, the water retention model shows
very good agreement with the experimental data over the
entire range of matric head. The hydraulic conductivity
model also predicts the experimental data quite well. For
this soil, the point of transition from capillary‐dominated
flow to film‐dominated flow occurs at a matric head of
−10.1 m. The model also shows close agreement with the
experimental data for other coarse‐grained soils. As shown
in Figures 9, 10, and 11, the model captures the distinctive
features of both the liquid saturation and relative hydraulic
conductivity data.
[28] Figures 12 and 13 show experimental data for Poederlee loamy sand and Poederlee sand, respectively. In both
cases, the water retention model is in close agreement with
the data while the hydraulic conductivity model shows
significant differences with the experimental data in the dry
range. For these data sets, the hydraulic conductivity data in
the dry range was determined with the hot air method,
which has been shown to overestimate hydraulic conduc-
Figure 9. Model results for Rehovot sand. (a) Fitted water retention. (b) Predicted relative hydraulic
conductivity.
8 of 15
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LEBEAU AND KONRAD: A NEW CAPILLARY AND THIN FILM FLOW MODEL
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Figure 10. Model results for Berlin medium sand 4. (a) Fitted water retention. (b) Predicted relative
hydraulic conductivity.
tivity [Stolte et al., 1994]. The experimental error can be
ascribed to thermal water (liquid and vapor) transport as
well as redistribution and evaporation loss during sampling. According to van Grinsven et al. [1985], the effects
of water redistribution can be expected to be less pronounced in finer‐grained soils, such as the Poederlee
loamy sand.
[29] Figure 14 illustrates the features of the new model
for four (4) data sets in which hydraulic conductivity is
expressed as a function of liquid saturation. As shown,
the water retention model is in good agreement with the
measured data, capturing the behavior in both wet and
dry regions. For most of the data sets, the hydraulic
conductivity model is also in close agreement with the
measurements. The discrepancy observed in some of the
data sets can be attributed to the fact that film thickness
is computed as a function of matric head, and not liquid
saturation. Hence, hydraulic conductivity in the dry range
is computed in terms of matric head, and subsequently
expressed in terms of liquid saturation using the water
retention function. Model errors are therefore compounded
by small inaccuracies in the water retention function.
[30] On the whole, these examples show that the new
theoretically based model, which uses simplified conceptualizations of pore space, has a great deal of explanatory
power, and that it is generally consistent with that found by
Tuller and Or [2001, 2002].
3.3. Comparison With Other Models
[31] In the following paragraphs, inferences on model
performance are drawn from model predictions for the
evaluation data sets. In adhering to the method of multiple
working hypotheses, three (3) well‐supported models are
Figure 11. Model results for Hupsel sand. (a) Fitted water retention. (b) Predicted relative hydraulic
conductivity.
9 of 15
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LEBEAU AND KONRAD: A NEW CAPILLARY AND THIN FILM FLOW MODEL
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Figure 12. Model results for Poederlee loamy sand. (a) Fitted water retention. (b) Predicted relative
hydraulic conductivity.
considered: (1) Kosugi‐Mualem, K1‐M, (2) extended
Kosugi‐Mualem, K2‐M, and (3) extended Kosugi‐Lebeau
and Konrad, K2‐LK. These models are formed with one
of two parametric models of water retention. The first is
the well‐documented parametric model of Kosugi [1996],
which is herein referred to as the K1 model, and the
second is the extended form of the Kosugi model, which is
referred to as the K2 model. In terms of hydraulic conductivity, two different models are used. The first is the
conventional capillary model of Mualem [1976a], and the
second is the capillary and thin film flow model presented
earlier in this paper. These models are referred to as the M
and LK models, respectively. For the sake of completeness, a detailed description of the K1‐M and K2‐M models
is given in Appendix A.
[32] Figure 15 presents a visual comparison of the various
models for the Gilat loam data set. As shown, the K1‐M
model is essentially unable to describe the experimental
data in the dry range. Moreover, the model adjusts residual
water content to an unrealistically large finite value, which
is in no way related to the quantity of adsorbed water. A
significant improvement is observed when using the
K2‐M model, which describes water retention from
saturation to oven dryness. Although this model adequately describes the liquid saturation data and imparts
a change in the slope of the relative hydraulic conductivity
curve, it is unable to conform to the experimental hydraulic
conductivity data in the dry range. This shortcoming is
ascribed to the fact that conventional models of hydraulic
conductivity omit thin film flow. This is highlighted by the
fact that the K2‐LK model, which accounts for thin film
Figure 13. Model results for Poederlee sand. (a) Fitted water retention. (b) Predicted relative hydraulic
conductivity.
10 of 15
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LEBEAU AND KONRAD: A NEW CAPILLARY AND THIN FILM FLOW MODEL
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Figure 14. Model results for various soils.
flow, shows much closer agreement with the experimental
data.
[33] Such visual observations can be translated into
quantifiable terms by means of the mean squared error,
MSE, which is an indicator of the overall magnitude of the
residuals (differences between the observed and predicted
data). In this study, the data is log‐transformed to ensure that
the complete range of relative hydraulic conductivity is well
Figure 15. Comparison of model results with those of other well‐supported models for Gilat loam.
(a) Fitted water retention. (b) Predicted relative hydraulic conductivity.
11 of 15
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LEBEAU AND KONRAD: A NEW CAPILLARY AND THIN FILM FLOW MODEL
Table 4. Model Prediction Statistics
Mean Squared Error
Data Set
Sandy loam
Shonai sand
Masa loamy sand
Gilat loam
Pachapa loam
Adelanto loam
Pachapa fine sandy clay
Rehovot sand
Berlin medium sand 4
Poederlee loamy sand
Poederlee sand
Hupsel sand
b
Referencea
K1‐Mc
K2‐Md
K2‐LKe
A
B
C
D
D
D
D
D
E
E
E
E
60.33
19.95
0.99
165.72
3.67
18.60
44.84
25.86
121.55
21.20
34.01
30.47
45.60
12.36
6.17
0.16
3.53
3.42
1.95
1.06
3.31
1.56
0.96
3.97
2.32
3.40
2.03
2.02
0.79
0.29
0.67
1.92
1.01
2.55
0.89
2.84
2.85
3.59
1.79
a
A, Pachepsky et al. [1984]; B, Mehta et al. [1994]; C, Fujimaki and
Inoue [2003a, 2003b]; D, Mualem [1976b]; E, Nemes et al. [2001].
b
Lowest values of MSE are highlighted in bold.
c
K1‐M: Kosugi‐Mualem.
d
K2‐M: extended Kosugi‐Mualem.
e
K2‐LK: extended Kosugi‐Lebeau and Konrad.
represented, and the mean squared error is computed as
follows:
MSE ¼
N h i2
1X
ln kr;i ln ^kr;i
N i¼1
ð16Þ
where N is the number of observations in the data set, kr,i is
the ith observed relative hydraulic conductivity, and ^k r,i is
the ith predicted relative hydraulic conductivity. Table 4
summarizes the MSE of the various models for each of the
data sets. For the sake of interpretation, the lowest values of
MSE are highlighted in bold. As shown, the K2‐LK model is
the best of the candidate models for nine (9) out of the twelve
(12) evaluation data sets, including the Gilat loam data set.
This overall performance is reflected in the mean value of the
mean squared error, MMSE, which is equal to 1.79. As such,
the MMSE is 25 times smaller than that for the K1‐M model
and 2 times smaller than that for the K2‐M model. These
significant differences suggest that omitting thin film flow
entails a large loss in predictive capability. This is clearly
illustrated in the scatterplots of observed versus predicted
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values shown in Figure 16. The predicted values of the K2‐M
model show a general tendency to underestimate hydraulic
conductivity in the dry range. In contrast, the predicted values
of the K2‐LK model are close to the observed values, and the
data generally falls along a straight line with a slope of one.
4. Conclusion
[34] Most classical predictive models of unsaturated
hydraulic conductivity have been derived by conceptualizing the pore space as either bundles of cylindrical tubes of
uniform size or assemblies of cylindrical capillary tubes of
various sizes. As such, these models have assumed that a
single concept can be used to describe water transport in
both the wet and dry ranges. In contrast, theoretical and
experimental findings have shown that thin film flow most
likely prevails over capillary flow in dry porous media.
Motivated by this shortcoming, a new approach for modeling unsaturated hydraulic conductivity in both wet and dry
ranges was proposed. As a starting point, water retention
was partitioned into capillary and adsorptive components.
These components were then associated with two different
liquid configurations: (1) an assembly of water or air filled
capillary tubes of various diameters and (2) a thin film of
adsorbed water stretched over the solid particles. A conventional statistically based model was subsequently used to
determine the conductivity of the water filled capillaries
whereas a theoretical model of soil‐water flow in thin films
was used to describe the conductivity of the adsorbed water.
This is significant given that there is no reason to expect
conventional statistical models to be valid when adsorptive
forces dominate. Finally, the components of conductivity
were coupled to form a hydraulic conductivity function that
accounts for both capillary and thin film flows. In this
manner, conductivity at the continuum scale was determined
with both the portions of pore water that are retained by
capillary and adsorptive forces.
[35] Sample calculations for a number of testing data sets
showed the prevalence of thin film flow in the dry range. In
general, the proposed predictive model provided very good
agreement with hydraulic conductivity data over the entire
range of matric head. Moreover, comparison with two other
well‐supported models revealed significant advantages of
Figure 16. Scatterplot of observed versus predicted values for the various models.
12 of 15
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LEBEAU AND KONRAD: A NEW CAPILLARY AND THIN FILM FLOW MODEL
the proposed model. Overall, the new model shows promise,
and warrants further comparison to experimental data as it
becomes available. At present, model testing is limited by
the paucity of combined measurements of water retention
and hydraulic conductivity in the dry range.
Appendix A
[36] Numerous capillary flow models have been proposed
to predict unsaturated hydraulic conductivity from the water
retention function. Among the most popular, the model put
forward by Mualem [1976a] may be written as follows:
2
R
1
32
jhm ð X Þj dX 7
6
k
6
7
kr ¼ ¼ Q0:5 6 X ¼0
7
4 Rs
5
ks
1
jhm ð X Þj dX
X ¼0
where kr is the relative hydraulic conductivity, k is the
unsaturated hydraulic conductivity, ks is the saturated
hydraulic conductivity, Q = ( − r)/(s − r) is the normalized
volumetric water content, is the volumetric water content, r
is the residual volumetric water content, s is the saturated
volumetric water content, X is a dummy variable of integration representing volumetric water content, and hm is the
matric head. As with other capillary flow models, the model
expressed by equation (A1) can be used with any parametric
model of water retention. It is nonetheless more convenient to
perform the integration along the logarithm of the absolute
value of the matric head axis when water retention is
described from saturation to oven dryness. In such a case, the
following variant of equation (A1) is preferred:
2
32
dðeY Þ
dY 7
6
6Y ¼lnjhm;dry j dhm
7
k
7
0:5 6
kr ¼ ¼ Q 6 ∞
7
R
6
ks
dðeY Þ 7
4
dY 5
dhm
Y ¼lnjhm;dry j
radius distribution. By integrating this model into equation
(A1), the following closed form expression for relative
hydraulic conductivity may be derived:
0:5
ln hm =hm;median
k
1
pffiffiffi
kr ¼ ¼ erfc
ks
2
2
2
ln hm =hm;median
1
pffiffiffi
erfc
þ pffiffiffi
2
2
2
lnR
jhm j
ðA2Þ
ln hm =hm;median
1
pffiffiffi
þ a
¼ ðs a Þ erfc
2
2
ðA5Þ
where a = o [1 − lnjhmj/lnjhm,dryj] is the volumetric water
content ascribed to adsorptive forces, and o is the volumetric water content due to adsorptive forces at a matric
head of −1 m. Using this water retention model in conjunction with equation (A2) does not result in a closed form
expression for relative hydraulic conductivity. For this reason, relative hydraulic conductivity must be computed by
numerical integration. It is to be noted that the combined use
of equations (A2) and (A5) is herein referred to as the K2‐M
model.
Notation
Kosugi Retention Model (Nonextended Form)
[38] The equation of Kosugi [1996] is among the most
widely used parametric models of water retention, and is
generally expressed as
ln hm =hm;median
1
pffiffiffi
¼ ðs r Þ erfc
þ r
2
2
Kosugi Retention Model (Extended Form)
[39] Fayer and Simmons [1995] presented a method of
extending most parametric models to zero water content
conditions by replacing residual water content with an
equation proposed by Campbell and Shiozawa [1992]. By
applying this method to the Kosugi model, the following
expression is found [Khlosi et al., 2006]:
where Q = /s is the normalized volumetric water content,
Y is a dummy variable of integration representing the logarithm of the absolute value of matric head, and hm,dry is the
matric head at oven dryness.
[37] The following paragraphs present the equations
obtained when Mualem’s [1976a] capillary flow model is
used in conjunction with the extended and nonextended
forms of the Kosugi [1996] water retention model.
A1.
ðA4Þ
The combination of the water retention and hydraulic conductivity models expressed by equations (A3) and (A4) is
herein referred to as the K1‐M model.
A2.
ðA1Þ
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ðA3Þ
where erfc( ) is the complementary error function, hm,median is
the matric head that corresponds to the median pore radius,
and s is the standard deviation of the log‐transformed pore
13 of 15
a Viscosity constant (L).
Asvl Hamaker constant for solid‐vapor interactions
through the intervening liquid (ML2 T−2).
B Thin film variable viscosity function (L3).
D Diameter of the spherical particles (L).
De Equivalent diameter (L).
e Electron charge (TI).
g Gravitational acceleration constant (LT−2).
hm Matric head (L).
hm,dry Matric head at oven dryness (L).
hm,m Matric head for which capillary condensation
due to surface roughness is negligible (L).
hm,median Matric head that corresponds to the median
capillary pore radius (L).
k Hydraulic conductivity (LT−1).
ka Hydraulic conductivity ascribed to film flow
(LT−1).
kB Boltzmann constant (ML2 T−2 Q−1).
kc Hydraulic conductivity ascribed to capillary
flow (LT−1).
kr Relative hydraulic conductivity.
kr,a Relative hydraulic conductivity ascribed to
thin film flow.
kr,c Relative hydraulic conductivity ascribed to
capillary flow.
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LEBEAU AND KONRAD: A NEW CAPILLARY AND THIN FILM FLOW MODEL
kr,i
^
k r,i
ks
l
MSE
MMSE
n
N
Pg
Po
qa
Qa
Sr
Ss,polydisperse
T
vf
x,z
Z
d
"o
"r
a
c
m
o
r
s
Q
mo
P
Pe
Pm
ro
s
ith observed relative hydraulic conductivity.
ith predicted relative hydraulic conductivity.
Saturated hydraulic conductivity (LT−1).
Capillary model parameter.
Mean squared error.
Mean value of the mean squared error.
Porosity.
Number of observations in the data set.
Pressure in the gas phase (ML−1 T−2).
Pressure in the bulk liquid phase (ML−1 T−2).
Superficial volumetric flux due to film flow
(LT−1).
Volumetric flow rate per unit length of film
(L2 T−1).
Degree of saturation.
Specific surface area per unit volume of the
polydisperse sample (L−1).
Absolute temperature (Q).
Film velocity (LT−1).
Spatial coordinates (L).
Valence change.
Film thickness (L).
Permittivity of free space (M−1 L−3 T4 I2).
Static relative permittivity of the liquid phase.
Volumetric water content.
Volumetric water content ascribed to adsorptive forces.
Volumetric water content ascribed to capillary
forces.
Volumetric water content at matric head hm,m.
Volumetric water content due to adsorption at
a matric head of −1 m.
Residual volumetric water content.
Saturated volumetric water content.
Normalized volumetric water content.
Dynamic viscosity of the bulk liquid (ML−1 T−1).
Disjoining pressure (ML−1 T−2).
Ionic‐electrostatic component of disjoining
pressure (ML−1 T−2).
Molecular component of disjoining pressure
(ML−1 T−2).
Density of the liquid phase (ML−3).
Standard deviation of the log‐transformed
capillary pore radius distribution.
[40] Acknowledgments. The authors wish to acknowledge the
financial participation of Natural Sciences and Engineering Research
Council of Canada (NSERC) Industrial Research Chair in the Operation
of Infrastructures Exposed to Freezing. The authors also extend their
appreciation to Nicolas Venkovic for his help in programming the
hybrid genetic‐simplex algorithm. Finally, the authors thank John R.
Nimmo, Andre Peters, and the anonymous reviewer for their very
insightful comments.
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J.‐M. Konrad and M. Lebeau, Department of Civil and Water
Engineering, Faculty of Sciences and Engineering, Laval University,
Quebec City, QC G1V 0A6, Canada. ([email protected])
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