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WATER RESOURCES RESEARCH, VOL. 44, W11417, doi:10.1029/2008WR007136, 2008
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A simple model for describing hydraulic conductivity
in unsaturated porous media accounting for film
and capillary flow
A. Peters1 and W. Durner1
Received 6 May 2008; revised 1 August 2008; accepted 4 September 2008; published 13 November 2008.
[1] The commonly used models for characterizing hydraulic conductivity of porous
media rely on pore bundle concepts that account for capillary flow only and neglect film
flow. Experimental evidence suggests that water flow at medium to low water contents in
unsaturated porous media can be significantly underestimated by these capillary bundle
models. We present a new model that combines a simple film flow function with the
capillary flow model of Mualem. This new model can easily be coupled to any water
retention function. Moreover, due to its mathematical simplicity, it can easily and
efficiently be implemented in existing codes for the numerical solution of unsaturated flow
problems. We investigated a set of soil water retention and conductivity data from the
literature that all reached dry conditions and were poorly described by existing capillary
bundle models. These data were well described with the new model if the model was
coupled with an appropriate retention function. Investigation of conductivity data from the
UNsaturated SOil hydraulic DAtabase (UNSODA) database showed that, in 75% of all
data sets, the new model achieved the best performance using a modified version of
Akaike’s information criterion. The numeric simulation of an evaporation scenario using
Richards’s equation showed that by neglecting film flow, the evaporation rate may be
underestimated by more than an order of magnitude.
Citation: Peters, A., and W. Durner (2008), A simple model for describing hydraulic conductivity in unsaturated porous media
accounting for film and capillary flow, Water Resour. Res., 44, W11417, doi:10.1029/2008WR007136.
1. Introduction
[2] Water transport in variably saturated porous media is
commonly described by the Richards equation. To solve this
equation, the water retention function, q(h), and the hydraulic conductivity function, K(h), must be known. For both
relationships a large variety of mathematical expressions
exist.
[3] Commonly, the retention function is described by
simple functions reflecting unimodal pore size distributions,
e.g., the frequently used models of Brooks and Corey
[1964] and van Genuchten [1980]. For soils with more
complex pore size distributions, such as structured soils,
more flexible functions of q(h) are needed. These can be
either bi- or multimodal functions obtained from simple
linear superposition of the unimodal expressions [Ross and
Smettem, 1993; Durner, 1994], or as the most flexible way,
free-form spline functions [Bitterlich et al., 2004; Iden and
Durner, 2007].
[4] The commonly used hydraulic conductivity functions
are derived from retention functions by means of pore
bundle models [Burdine, 1953; Mualem, 1976a; Alexander
and Skaggs, 1986], using the law of Hagen-Poiseuille of
capillary flow and further assumptions about tortuosity,
1
Institut für Geokölogie, Technische Universität Braunschweig,
Braunschweig, Germany.
Copyright 2008 by the American Geophysical Union.
0043-1397/08/2008WR007136$09.00
connectivity and spatial distribution of the capillaries. One
of the main advantages of this approach is the small number
of additional parameters that are required to describe K(h).
The Mualem model for instance needs only two extra
parameters, i.e., the saturated conductivity, Ks, or any other
measured conductivity at a certain pressure head to scale the
relative conductivity function, and an empirical factor, t,
accounting for tortuosity and connectivity.
[5] Experimental evidence showed already early that the
commonly used pore bundle models for K(h) often underestimate the hydraulic conductivity in the medium to dry
range [Mualem, 1976b; Pachepsky et al., 1984]. Up-to-date
this behavior is often neglected by disregarding data pairs
that do not fit the model structure [Ghezzehei et al., 2007].
Within the inverse modeling approach, where no directly
measured data for the conductivity function are obtained,
the failure of the pore bundle models is indicated by
problems in the model fit to the data at the dry range of
the experiment. These inadequacies at the laboratory scale
are supplemented by experimental evidence on the field
scale. Recently, Goss and Madliger [2007] demonstrated
that the hydraulic conductivity of a dry soil in Tanzania was
severely underestimated by the traditional concept.
[6] This shortcoming of the pore bundle models can be
attributed to neglecting film and corner flow (in this
paper we sum up both effects and refer to them as film
flow) that can be the dominant flow process in the dry range
[Lenormand, 1990; Toledo et al., 1990], where flow in fully
saturated capillaries ceases. Tuller and Or [2001] proposed
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PETERS AND DURNER: A SIMPLE MODEL FOR HYDRAULIC CONDUCTIVITY
a model of K(h) derived from hydrodynamic considerations
and specific pore geometry assumptions to account for both
capillary and film flow. Their model has received great
attention but is yet not used in standard modeling. It has the
disadvantages that it is mathematically very complex and
coupled to their retention model [Or and Tuller, 1999],
which in turn is also derived from specific pore geometry
assumptions and thermodynamic considerations.
[7] The aim of this paper is to show the shortcomings of
the common pore bundle models in dry soils and to present
a new simple model for predicting the hydraulic conductivity from saturation to dry conditions accounting for both
capillary and film flow. We will present an empirical
expression for describing film flow and couple this with a
pore bundle model, in this study the Mualem model
[Mualem, 1976a]. By testing the new model on different
data sets from the literature and own measurements we show
its usefulness for describing soil water flow in dry soils.
W11417
film thickness and the curvature radius of liquid-vapor
interfaces at the corners reduce with decreasing potential
[see Tuller and Or, 2001, equations (1) and (2)]. The
mathematical structure of this expression is similar to the
kinematic wave approach for describing film flow in
macropores [Beven and Germann, 1981] and to the
macropore flow model of Jarvis [1994]. However, our
approach deals with slow water flow at low water contents,
whereas those deal with rapid preferential water flow
processes. Various authors showed that the dependence of
hydraulic conductivity on saturation can be expressed with
power functions, especially at low water contents [e.g.,
Childs and Collis-George, 1950; Brooks and Corey, 1964;
Campbell, 1974]. Toledo et al. [1990] derived a power
function for this relationship for the case that thin films
control the water flow in fractal media.
[10] We hypothesize that the entire relative conductivity,
Kr can be described as the linear superposition of the
contributions of capillary and film flow:
2. Material and Methods
KrðSeðhÞÞ ¼ ð1 wÞKrcap þ wKrfilm ;
ð3Þ
2.1. Theory
[8] The relative unsaturated hydraulic conductivity function, Kr(h), is commonly derived from the soil water
retention characteristic, q(h), by pore bundle models. The
general form of these models can be expressed as follows
[Hoffmann-Riem et al., 1999]:
where w is the relative contribution of the film flow, subject
to w < 1. Thus we need two additional parameters, one to
separate the contributions of capillary and film flow, and
one to scale the film flow characteristic. Inserting equations (1)
and (2) into equation (3) yields:
2S
3b
Re k
h
dS
ð
h
Þ
e
6
7
6
7
KrcapðSeðhÞÞ ¼ Set 6 01
7
4R
5
hk dSeðhÞ
2S
3b
Re k
h
dS
ðhÞ
e
6
7
6
7
KrðSeðhÞÞ ¼ ð1 wÞSet 6 01
7 þ wSet 2 :
4R
5
k
h dSeðhÞ
ð1Þ
0
0
where Se = (q qr)/(qs qr) is the effective saturation,
qr [] and qs [] are the residual and saturated water contents,
respectively, and the superscript cap indicates that only
capillary flow is considered. The parameters t [], k and b
can be varied to get more specific functional expressions. For
the Burdine model [Burdine, 1953] t = 2, k = 2, and b =
1. In the Mualem model [Mualem, 1976a] t = 0.5, k = 1,
and b = 2, whereas in the model of Alexander and Skaggs
[1986] t = 1, k = 1, and b = 1. The parameter t in the
Mualem model is often treated as a free fitting parameter
[Hoffmann-Riem et al., 1999] that is frequently negative,
and thus its physical meaning is questioned [Schaap and
Leij, 2000]. These models were all derived to describe the
capillary fluid flow in porous media. Experience has shown
that they allow a sound description of the water flow in the
wet moisture range [van Genuchten, 1980]. However, they
often fail in the dry range were capillary flow becomes
negligible in comparison to film flow [Lenormand, 1990;
Toledo et al., 1990; Goss and Madliger, 2007].
[9] To overcome this shortcoming we introduce a simple
power function as an empirical approach to describe relative
film flow, Kfilm
r :
KrfilmðSeðhÞÞ ¼ Set2
ð4Þ
ð2Þ
as a function of Se, having the value 1 at full saturation and
decreasing with decreasing saturation depending on t 2,
subject to t 2 > 0. This decrease accounts for the fact that the
This general function (in this paper denoted as ‘‘unconstrained
new function’’) can be further simplified with the assumption
that t 2 = t. In this case no additional parameter is introduced to
scale the relative film flow. However, in this case the restriction
t > 0 has to be met, making the model less flexible for capillary
flow. The resulting function (in this paper denoted as
‘‘constrained new function’’) is given by:
2
2S
3b 3
Re k
6
7
6 h dSeðhÞ7
6
7
60
7
7:
KrðSeðhÞÞ ¼ Set 6
ð
1
w
Þ
þw
6
7
6
7
1
R
4
5
4
5
hk dSeðhÞ
ð5Þ
0
[11] The shape of the new function and the contributions
of the capillary and film flow are illustrated in Figure 1. It is
apparent that Kr is dominated by capillary flow in the wet
range, whereas in the dry range it is almost solely determined by film flow.
[12] With this approach the well-established uni- or multimodal van Genuchten/Mualem relations [van Genuchten,
1980; Durner, 1994] are now just extended by one or two
additional parameters for the K(h) function.
2.2. Constitutive Relationships
[13] The soil hydraulic properties in this study are
expressed by the uni- and bimodal van Genuchten expression for the retention function and either Mualem conduc-
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PETERS AND DURNER: A SIMPLE MODEL FOR HYDRAULIC CONDUCTIVITY
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Figure 1. Schematic illustrations of the contribution of capillary (Krcap) and film (Krfilm) flow to the
overall relative hydraulic conductivity (Kr) for a unimodal function (left) and a bimodal function (right).
The sum of Krcap and Krfilm gives Kr.
tivity model or the extended conductivity models (equations (4)
and (5)). The retention function is given by:
SeðhÞ ¼
k
X
ð6Þ
wi Sei ;
i¼1
where Sei are weighted subfunctions of the system, wi are the
weighting
P factors for the subfunctions, subject to 0 < wi < 1
and wi = 1. The effective saturations, Sei are given by:
SeiðhÞ ¼ ð1 þ ðai jhjÞni Þ
1=ni 1
;
constrained, and II = new model unconstrained), and the
superscript indicates whether t is fixed to the value 0.5 ()
or treated as a fitting parameter (+).
2.3. Fit of Parametric Expressions to q(h) and K(h)
Data
[16] We fit the coupled K-q-h models to the data points,
by minimizing the sum of weighted squared residuals (F(b))
between model prediction and data pairs:
FðbÞ ¼ wq
ð7Þ
r
X
k
h
i2
X
^ iðbÞ 2 ; ð9Þ
wqi qi ^qiðbÞ þ wK
wKi Ki K
i¼1
i¼1
1
where ai [cm ] and ni [] are curve-shape parameters of
the pore subsystems. For i = 1, equation (6) represents the
retention curve of van Genuchten [1980], for i = 2 the
bimodal retention function of Durner [1994].
[14] The relative unsaturated hydraulic conductivity function, Kr(h), is calculated from soil water retention characteristics according to Mualem [1976a] in the original form or
by our extended model. In combination with the q(h)
functions, the analytical solution of Priesack and Durner
[2006] for the multimodal van Genuchten functions was
extended by the film flow contribution:
2
32
k
!t P wi ai ½1 ðai jhjÞni Se k
i 7
6
X
6
7
KrðhÞ ¼ ð1 wÞ
wi Sei 6i¼1
7
k
4
5
P
i¼1
wi ai
Table 1. Summary of Hydraulic Models That Were Used to
Evaluate the Measured Data
i¼1
þw
k
X
wi Seti2 :
where r and k are the number of data pairs for the retention
function and the conductivity function, respectively, wq and
wK are the class weights of the water content data and
conductivity data, wqi and wKi are the weights of the
^ i(b) are the
individual data points, and qi, ^qi(b), Ki and K
measured and model predicted values, respectively.
[17] In equation (9), the predicted water contents, ^q, were
either calculated in a standard manner as the point water
contents at pressure head hi, (‘‘classic method’’), or in the
case that the soil column height was known as the mean
water content of the whole column (‘‘integral method’’) to
avoid a systematic bias [Peters and Durner, 2006]. The
ð8Þ
i¼1
If w = 0, equation (8) becomes to the original form of
Priesack and Durner [2006]. For w = 0 and k = 1, equation
(8) reduces to the unimodal solution of van Genuchten
[1980].
[15] These models were fitted to data points obtained
from the literature or by own measurements (see below).
Table 1 lists all model combinations and fitting cases that
were used in this study. In the codes for the cases the letter
stands for the retention model (A = van Genuchten unimodal
and B = van Genuchten bimodal), the Roman numeral for
the conductivity model (0 = Mualem, I = new model
Case
q(h) Model
K(h) Model
t
NPARAa
A0
A0+
AI
AI+
AII
AII+
B0
B0+
BI
BI+
BII
BII+
vG unimod
Mualem
fixed
fitted
fixed
fitted
fixed
fitted
fixed
fitted
fixed
fitted
fixed
fitted
5-2-3
6-3-4
6-3-4
7-4-5
7-4-5
8-5-6
8-5-6
9-6-7
9-6-7
10-7-8
10-7-8
11-8-9
new M. cons
new M. uncons
vG bimod
Mualem
new M. cons
new M. uncons
a
Number of free adjustable parameters. 1st number, soils 4 to 7; 2nd
number, soils 1 to 3; 3rd number, UNSODA K(h) data.
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PETERS AND DURNER: A SIMPLE MODEL FOR HYDRAULIC CONDUCTIVITY
latter method is favorable, even in transient evaporation
experiments as discussed by Peters and Durner [2008].
[18] To account for the different measurement frequency,
the individual weights, wqi and wKi were chosen such that
the combined data within every log10(h in cm) (pF)
increment have the same weight, i.e., the weight for a
certain data point was proportional to its distance to the
neighboring point on the pF scale. To account for the
different numerical ranges of the data types, the weights for
the data classes were calculated by wq = 1/(qmax qmin) and
wk = 1/(log10(Kmax) log10(Kmin)), where qmax, qmin, Kmax
and Kmin are the maximum and minimum values of the data
sets to be fitted.
[19] As fitting procedure for the coupled estimation of
q(h) and K(h) we used a robust combination of the shuffled
complex evolution (SCE-UA) algorithm with a LevenbergMarquardt (LM) algorithm. The SCE-UA algorithm [Duan
et al., 1992], is a global optimizer and converges within
predefined permissible parameter ranges toward the minimum (if that exists) of the objective function, not depending
on initial guesses. The LM algorithm is an efficient local
optimizer [Marquardt, 1963]. It is used to speed up the
convergence, once the close region of the global optimum is
identified by the SCE. A similar combination of a global
and a local optimizer, for the estimation of soil hydraulic
properties was introduced by Lambot et al. [2002].
2.4. Tests on Real Data
[20] To test the merit of our approach we first applied the
models to fit the q(h) and K(h) data of three soils that were
also analyzed by Tuller and Or [2001] to test their film flow
model. They showed that the K(h) data of these data sets
could not be described by the usual capillary bundle models.
The data of the sandy loam (in this study denominated as
soil 1) and the clay loam (soil 3) were digitally extracted
from their figures, whereas the data of the Gilat loam (soil 2)
where taken as numbers from the publication of Mualem
[1976b].
[21] We further analyzed data of four evaporation experiments, which were evaluated with the simplified evaporation method of Peters and Durner [2008]. The raw data
stem from an undisturbed sand described by Schindler and
Müller [2006] (soil 4), a packed sand and an undisturbed
clayey topsoil, described by Minasny et al. [2004] (soils 5
and 6), and one experiment with a packed silt soil (soil 7)
that was performed in our laboratory. Due to a relatively
high temporal resolution of the evaporation experiments
these data sets contain much more information than the data
of soils 1 to 3. The reader is referred to the original
publications for details of the experimental procedures for
soils 4 to 6.
[22] In our own experiment (soil 7), the column height
was 4.2 cm and the volume was 459 cm3. Two tensiometers
were installed at heights of 0.9 and 3.3 cm above the
bottom. The column was placed on a scale that was
manually read twice a day. The accuracy of tensiometers
and scale was sh = 0.3 cm and sw = 0.01 g, respectively.
The laboratory was air conditioned at 20° Celsius and the
potential evaporation was 0.22 cm/d.
[23] To get information about the general merit of our
approach we finally analyzed all conductivity data from the
UNSODA database [Nemes et al., 2001] that consisted of
10 or more data pairs per soil column. Thus conductivity
W11417
data of 194 soil columns could be analyzed. Only K(h) data
pairs were evaluated, because only few of the UNSODA
q(h) and K(h) data sets stem from one and the same column.
Compared to the data sets mentioned above, the UNSODA
data contain the smallest amount of information. No
preselection with respect to the suitability of the classic
pore bundle models to describe the data was carried out.
[24] The two retention models A and B were combined
with the conductivity models 0, I and II as introduced
above. Since Mualem [1976a] proposed to use for t the
value 0.5, we first ran each model on the data with a fixed
t = 0.5 to reduce the number of free fitting parameters by
one. In a second step t was treated as a free adjustable
parameter. Thus we got 6 different model combinations and
12 different fitting cases (Table 1). For the UNSODA-K(h)
data the first summand of the objective function, equation
(9), becomes zero.
2.5. Prediction of Evaporation
[25] To evaluate the effect of the different model fits on
evaporation on the field scale we simulated a steady state
evaporation scenario with the different conductivity functions fitted to the data of soil 4. To assure that differences in
the evaporation rate are only due to differences in the
conductivity function and not to differences in water capacities, we selected the retention function of case AII+ for all
simulations.
[26] The evaporation from a 5-m long vertical soil column in contact to groundwater was simulated by solving the
Richards equation with the constitutive relationships as
described by equations (6) to (8) using the finite element
code HYDRUS-1D [Simunek et al., 1998]. The soil
hydraulic property functions were fed into HYDRUS-1D
as external tables. As initial condition we chose a linear
pressure-head distribution from 0 at the bottom to 200 cm
at the top. At the lower boundary a Dirichlet boundary
condition with the value 0 was applied, simulating a
constant groundwater table at that depth. At the upper
boundary, a flux boundary condition was chosen, K(dh/dz +
1) = q0, where q0 = 0.2 cm d1. This flux was maintained until
the pressure head at the upper boundary reached a value of
3 103 cm. After the head reached that threshold value,
the boundary condition was changed toward a Dirichlet
condition. This is a common procedure to guarantee
numerical stability in the simulation. Since there were no
significant lower fluxes when choosing 1 103 cm as
threshold value we conclude that the threshold value is
justified to demonstrate the general differences between
the different models. The simulation duration was 3000 d
to assure that steady state conditions were reached. The
mass balance error in the forward simulations was in all
cases smaller than 0.5%.
2.6. Diagnostic Variables and Model Ranking
[27] We first evaluated the performance of the different
models by comparing the objective function values at their
estimated minimum, Fmin. To account for the different
number of adjustable parameters when selecting the best
model, we also calculated the Akaike Information Criterion,
AIC = 2(L + k) [Akaike, 1974], where L is the likelihood
function and k is the number of fitting parameters. If the
number of measurements, n, is small in comparison to k, the
original form of AIC should be extended by a correction
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Figure 2. q(h) and K(h) data of soils 1 to 3 and coupled fit of the unimodal van Genuchten retention
functions with the three conductivity functions with t as a free-fitting parameter.
term that accounts for small values for n/k [Hurvich and
Tsai, 1989]:
AICc ¼ n lnðFmin =nÞ þ 2k þ 2k ðk þ 1Þ=ðn k 1Þ:
ð10Þ
The first term penalizes a poor fit, the second term the
number of parameters and the third term is the
correction term for small values of n/k. If n/k becomes
large the last term becomes negligible and the AICc
converges to AIC.
[28] Since the absolute value of AICc depends on unknown scaling constants [Burnham and Anderson, 2004],
the AICc (as other model selection instruments as well)
yields only relative quantities that allow comparison of
different models. To get comparable results, Burnham and
Anderson [2004] suggest to eliminate the unknown scaling
constants and constants related to sample size that enter into
AICc by rescaling the AICc values to:
Dj ¼ AICcj AICcmin ;
ð11Þ
where the subscript j indicates the jth model and AICcmin is
the minimum AICc value of all selected models. As a rule of
thumb Burnham and Anderson [2004] suggested that
models with Dj 2 have substantial support, those with
4 Dj 7 have considerably less support, and models with
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PETERS AND DURNER: A SIMPLE MODEL FOR HYDRAULIC CONDUCTIVITY
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Table 2. Relative Fmin Values of All Soils to Which Both Hydraulic Functions Were Fitted Simultaneouslya
Model
A0
A0+
AI
AI+
AII
AII+
B0
B0+
BI
BI+
BII
BII+
Soil 1
Soil 2
Soil 3
Soil 4
Soil 5
Soil 6
Soil 7
13.74
3.40
2.18
1.79
1.13
1.00
–
–
–
–
–
–
13.18
4.08
2.39
1.05
1.25
1.00
–
–
–
–
–
–
8.27
3.27
1.86
1.07
1.47
1.00
–
–
–
–
–
–
45.48
34.81
2.36
1.03
1.00
1.00
–
–
–
–
–
–
115.65
40.39
2.40
1.77
1.89
1.00
–
–
–
–
–
–
17.90
17.86
4.48
2.11
3.93
1.00
–
–
–
–
–
–
100.39
55.26
29.97
8.17
6.31
5.36
3.45
3.45
1.98
1.08
1.81
1.00
a
For each soil, the Fmin values of the single models were divided by the Fmin value of the best fit.
a Dj > 10 have essentially no support in describing the
measured data.
3. Results
3.1. Coupled Estimation of q(h) and K(h)
[29] The q(h) and K(h) data of soils 1 to 3 and the coupled
model fit with the unimodal (A) functions listed in Table 1
are shown in Figure 2. At the moisture range dryer than h =
102 to 103, all soils show a bend in the K(h) data (right
column) on the log-log scale that is not reflected in the q(h)
data. As found already by Tuller and Or [2001], the data
cannot be described with model A0. Soil 1 is best fitted
(Table 2) by model AII+, which is ranked to be the most
adequate model by the AICc criterion (Table 3). None of the
models is able to describe both data pairs of soils 2 and 3
particularly well. This problem can also be seen in the
figures of the original publication of Tuller and Or [2001].
However, with the new model the description of the
conductivity data is significantly improved. For these soils,
the application of model AII does not improve the fit in
comparison to model AI. Thus model AI+ has the lowest
AICc (Table 3). The difference between the fmin values of
the best and the worst fit was for all soils about one order of
magnitude (Table 2).
[30] Due to the high data resolution of the evaporation
experiments, the q(h) and K(h) data of soils 4 to 7 (Figure 3)
contain more information as compared to the data of soils 1
to 3. Soils 4 to 6 can be well described by the unimodal
retention function, only the q(h) data of soil 7 require a
bimodal description. Again, the K(h) data (right column)
show a characteristic bend on the log-log scale in the range
of h = 102 to 103. Hence none of these data sets can be
fitted with conductivity model 0. According to the AICc, the
unconstrained conductivity model II is justified in each
case. The difference between the fmin values of the best and
the worst fit is for soils 4 to 6 about one to two orders of
magnitude (Table 2). Note that the evaporation experiment
comprises no information about conductivity close to
saturation, as pointed out by Tamari et al. [1993], Wendroth
et al. [1993], and Peters and Durner [2008]. The best-fit
parameters of all seven soils are listed in Table 4.
3.2. Estimation of K(h) Functions From UNSODA Data
Base
[31] Up to here only selected data sets that obviously
could not be described by unimodal retention functions in
combination with the capillary flow models were investigated. In this section we systematically analyze all K(h) data
sets from the UNSODA database that consist of more than
10 data pairs. This restriction was necessary to guarantee
that all analytical conductivity functions of Table 1, of
which the most complex case (BII) had nine free adjustable
parameters, could be fitted. 194 data sets fulfilled this
criterion and were used in the analysis.
[32] Table 5 summarizes the results of this investigation.
Model BII yielded in all cases (194 times) the best fit. This
is expected since this model has the highest flexibility and
includes actually all other models as special cases. In the
majority of the cases (102 times) the lowest AICc (Dj = 0)
was found for the constrained new model (AI and BI), the
new unconstrained (AII and BII) model was in 42 cases the
best model and in only 50 cases the Mualem model (A0 and
B0) had the lowest AICc. Thus 75% of all data sets were
Table 3. Dj Values of All Soils to Which Both Hydraulic Functions Were Fitted Simultaneously
Model
A0
A0+
AI
AI+
AII
AII+
B0
B0+
BI
BI+
BII
BII+
Soil 1
Soil 2
Soil 3
Soil 4
Soil 5
Soil 6
Soil 7
86.69
38.77
22.85
18.34
1.73
0.00
–
–
–
–
–
–
103.95
55.84
32.93
0.00
7.57
0.38
–
–
–
–
–
–
62.53
34.34
15.78
0.00
10.46
0.64
–
–
–
–
–
–
484.15
452.14
107.51
4.07
0.00
2.27
–
–
–
–
–
–
553.78
431.86
98.78
64.89
72.67
0.00
–
–
–
–
–
–
342.28
344.22
176.99
88.01
163.39
0.00
–
–
–
–
–
–
534.55
465.72
392.91
240.45
209.67
192.54
140.22
142.41
76.55
6.46
67.96
0.00
6 of 11
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PETERS AND DURNER: A SIMPLE MODEL FOR HYDRAULIC CONDUCTIVITY
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Figure 3. q(h) and K(h) data of soils 4 to 7 and coupled fit of the uni- and bimodal van Genuchten
retention functions with the three conductivity functions with t as a free-fitting parameter. A0* in soil 4:
van Genuchten/Mualem model, considering only capillary part of K(h) data (black circles) for the fit.
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PETERS AND DURNER: A SIMPLE MODEL FOR HYDRAULIC CONDUCTIVITY
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Table 4. Best-Fit Parameters of All Applied Model Combinations for Soils 1 to 7
Model
a
n
qr
qs
Ks
A0
A0+
AI
AI+
AII
AII+
0.005
0.015
0.007
0.006
0.008
0.010
1.33
1.47
1.53
1.63
1.57
1.57
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
A0
A0+
AI
AI+
AII
AII+
0.011
0.037
0.023
0.019
0.026
0.015
1.22
1.37
1.47
1.53
1.48
1.60
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
A0
A0+
AI
AI+
AII
AII+
0.001
0.003
0.002
0.002
0.002
0.001
1.29
1.31
1.35
1.38
1.35
1.39
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
A0
A0+
AI
AI+
AII
AII+
0.021
0.018
0.018
0.018
0.018
0.018
2.31
5.63
4.61
4.58
4.81
4.80
0.03
0.09
0.08
0.08
0.08
0.08
0.36
0.35
0.35
0.35
0.35
0.35
5.67
0.70
43.3
62.6
49.5
50.1
A0
A0+
AI
AI+
AII
AII+
0.273
0.027
0.029
0.028
0.028
0.027
1.55
6.90
4.64
5.38
5.22
6.71
0.00*
0.03
0.02
0.03
0.03
0.03
0.80
0.43
0.45
0.44
0.44
0.43
1163
0.03
43.7
73.5
80.8
12.6
A0
A0+
AI
AI+
AII
AII+
0.213
0.137
0.066
0.067
0.057
0.080
1.27
1.40
1.83
1.82
1.93
1.65
0.19
0.25
0.30
0.30
0.30
0.29
0.57
0.55
0.53
0.53
0.52
0.53
1233
211
131
671
87.5
25641
A0
A0+
AI
AI+
AII
AII+
B0
B0+
BI
BI+
BII
BII+
0.005
0.004
0.005
0.004
0.004
0.004
0.003
0.004
0.001
0.003
0.004
0.005
1.87
4.84
2.61
3.76
3.87
4.26
1.41
4.78
6.85
2.09
5.17
6.80
0.14
0.24
0.20
0.23
0.23
0.23
0.17
0.18
0.23
0.20
0.22
0.19
0.51
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
4.06
0.35
4.99
17.2
8.04
3.66
9.97
8.98
12.6
70.8
11.9
170
t
w
t2
a2
n2
w2
5.1
2.4
1.9
6.5
–
–
107
104
104
105
–
–
–
–
2.20
1.89
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
1.0
2.8
2.1
2.6
–
–
106
104
104
104
–
–
–
–
2.22
1.97
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
4.3
7.5
2.4
7.1
–
–
106
104
104
104
–
–
–
–
2.80
3.32
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
1.4
3.1
3.7
3.6
–
–
104
104
104
104
–
–
–
–
0.68
0.68
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
1.1
6.2
5.8
2.9
–
–
104
105
105
104
–
–
–
–
0.43
0.30
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
2.3
1.4
1.7
1.0
–
–
105
104
104
105
–
–
–
–
1.05
2.68
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
104
103
103
102
–
–
104
104
104
104
–
–
–
–
1.04
0.85
–
–
–
–
0.74
2.49
–
–
–
–
–
–
0.004
0.003
0.004
0.005
0.002
0.003
–
–
–
–
–
–
4.77
1.42
5.04
6.53
4.47
2.00
–
–
–
–
–
–
0.72
0.27
0.91
0.62
0.11
0.40
Soil 1
–
1.64
–
2.13
–
0.24
Soil 2
–
2.00a
–
2.17
–
3.97
Soil 3
–
2.00a
–
3.41
–
5.00a
Soil 4
–
1.37
–
0.73
–
0.51
Soil 5
–
1.90
–
0.41
–
0.49
Soil 6
–
0.31
–
1.85
–
5.00a
Soil 7
–
1.51
–
1.13
–
0.19
–
0.39
–
2.20
–
3.15
2.5
4.1
6.8
1.0
4.7
9.1
7.2
4.3
a
Estimated parameter reached boundary of parameter space.
best characterized by the new K(h) model, giving strong
evidence that it has a general merit in describing soil
hydraulic conductivity.
[33] For 25% of all data sets the capillary bundle model
alone was a sufficient concept. However, a lot of data sets of
the UNSODA database consist only of data in the wet range
where film flow does not play an important role (see Figure 1).
Only 24% of the data sets that were best described by the
Mualem model, but 48% of the data sets that were best
described by one of the new models contained conductivity
data at pressure heads 103 cm. Thus the relatively good
performance of the capillary bundle model in 25% of all data
sets may be just due to a lack of data in the dry range.
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Table 5. Frequencies of Best Fit (Nbest) and Greatest Merit for 12
Different Model Approaches Applied on 194 K(h) Data Sets From
the UNSODA Data Base
Model
Nbest
Dj = 0
Dj 2
Dj 7
A0
A0+
AI
AI+
AII
AII+
B0
B0+
BI
BI+
BII
BII+
0
0
0
0
0
4
0
1
0
0
5
194
20
7
32
5
16
2
20
3
51
14
11
13
26
16
46
30
31
10
32
8
67
34
37
23
40
49
81
85
86
47
85
55
116
95
96
75
[34] In contrast to the coupled fits that were discussed in
section 3.1, the cases with a fixed t were most frequently
the best describers of the data. This becomes even more
distinct if the frequencies of Dj 2 are compared. We
explain this as follows: In cases where the slope of the
retention function is small (e.g., loamy or clayey soils) the
sensitivity of the Mualem model on t is very small. In these
cases, t can be fixed to any value between 1 and 2
without a significant impact on the function’s shape [see
Schaap and Leij, 2000, Figure 1b and 1c]. Furthermore, the
conductivity model I had more than twice as often the
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lowest AICc value than model II, leading to the conclusion
that the fit for many data sets is insensitive to t. In other
words: the estimation problem of t for such soils is illposed. Hence the conductivity of fine textured soils can
often be well described by the new constrained conductivity
model I, which means that only one additional parameter is
required as compared to Mualem model.
3.3. Prediction of Evaporation
[35] In the preceding section we showed the superiority
of the new model over the classic capillary bundle model to
fit conductivity data if film flow plays a significant role. The
next question that arises is: How large is the effect of
neglecting film flow in field scale modeling? Therefore
evaporation from a 5-m long soil profile was simulated with
the properties obtained for soil 4 (Figure 3, 1st row). Soil 4
was chosen for simulation since it has a sound database in
the capillary as well as in the film flow range. Since the fit
of model A0 to the conductivity data was very poor, we
additionally followed the example of Ghezzehei et al.
[2007] and disregarded all data pairs with matric heads
<1.3 102 for the fit (case A0*), so that only data of the
capillary flow range were considered.
[36] Figure 4 shows the results of the evaporation simulations
with the different functions. The steady state flux with the
function AII is 35 times higher than with function A0* and nine
times higher than with function A0. Thus neglecting film flow
can strongly underestimate the fluxes of field scale simulations.
Figure 4. Simulation of steady state evaporation with the different hydraulic properties obtained for
soil 4. (a) Water flux through the upper boundary. Steady-state distribution of pressure head (b),
volumetric water content (c), and unsaturated hydraulic conductivity (d).
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PETERS AND DURNER: A SIMPLE MODEL FOR HYDRAULIC CONDUCTIVITY
[37] The steady state profiles of pressure head and unsaturated hydraulic conductivity are also strongly influenced
by the selected conductivity function. Goss and Madliger
[2007] found that their soil profile dried out to a greater
depth than expected by numerical simulations with the
classic hydraulic properties functions, and that this thick dry
surface layer did not act as an evaporation barrier.
Furthermore, they did not find a sharp transition zone
between the dry surface zone and the underlying wetter soil
in the upper centimeters of the soil profile as expected.
Looking at the fluxes and the matric head distribution in
Figure 4, we find that model AII yields qualitatively the
same results as found by Goss and Madliger [2007]. Thus
accounting for film flow can qualitatively describe the
phenomena found in their field study.
4. Conclusions
[38] In the medium to dry moisture range capillary flow
ceases to be the dominant flow process in porous media and
film flow becomes important, leading to failures of capillary
bundle models to describe fluid flow. A simple one-parameter power function was formulated to describe film flow in
unsaturated porous media. Coupling this function with
capillary bundle models, such as the Mualem model, yields
a conductivity function accounting for both capillary and
film flow. In the constrained form (I) only a weighting
factor for the film flow is necessary to significantly improve
model fits to data that reflect film flow phenomena. In the
unconstrained form (II) there is one additional degree of
freedom, providing a more flexible shape of K(h).
[39] Application of these new models to measured data
showed promising results. Six data sets from the literature
and one own data set were evaluated with six different
model combinations. In all cases, the new constrained or
unconstrained model performed best according to the modified information criterion of Akaike. Comparison of the
different models with all UNSODA data sets that had
sufficient measurement points showed that 75% of all data
sets could be best described by the new model.
[40] For soils having retention characteristics with a small
slope, the sensitivity of the K(h) function on t is very small.
Thus these soils can often adequately be described with
model I, whereas for soils having a steeper slope model II,
with two different parameters, may be required for an
optimal description of the two K(h) subfunctions.
[41] The negligence of film flow in modeling evaporation
scenarios significantly underestimates the evaporation rate
from the soil. Due to root water uptake the soil close to the
root surface can also reach very dry conditions, where film
flow may play the significant role. Thus considering capillary flow only may lead to a significant underestimation of
evaporation as well as transpiration rates.
[42] In the very dry range, film flow also ceases and
vapor transport becomes the dominant process for water
transport [Saito et al., 2006]. Further effort could be made
in investigating whether the new model can effectively
describe both, film and vapor transport.
[43] Acknowledgments. We thank Sascha Iden for useful discussions
and Budiman Minasny (University of Sydney) and Uwe Schindler (Leibniz
Center for Agricultural Landscape and Land Use Research, Müncheberg)
for providing data of their evaporation experiments. This study was
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supported financially by the Deutsche Forschungsgemeinschaft (DFG grant
DU 283/7-1).
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W. Durner and A. Peters, Institut für Geokölogie, Technische Universität
Braunschweig, Langer Kamp 19c, Braunschweig, Niedersachsen 38100,
Germany. ([email protected])
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