Hydrol. Earth Syst. Sci., 27, 1565–1582, 2023
https://doi.org/10.5194/hess-27-1565-2023© Author(s) 2023. This work is distributed under
the Creative Commons Attribution 4.0 License.

Prediction of the absolute hydraulic conductivity
function from soil water retention data
Andre Peters1 , Tobias L. Hohenbrink1,a , Sascha C. Iden1 , Martinus Th. van Genuchten2,3 , and Wolfgang Durner1
1 Division of Soil Science and Soil Physics, Institute of Geoecology,

Technische Universität Braunschweig, Braunschweig, Germany
2 Department of Earth Sciences, Utrecht University, Utrecht, the Netherlands
3 Department of Nuclear Engineering, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil
a now at: Deutscher Wetterdienst (DWD), Agrometeorological Research Center, Braunschweig, Germany
Correspondence: Andre Peters (a.peters@tu-braunschweig.de)Received: 21 December 2022 – Discussion started: 13 January 2023
Revised: 20 March 2023 – Accepted: 21 March 2023 – Published: 17 April 2023

Abstract. For modeling flow and transport processes in thesoil–plant–atmosphere system, knowledge of the unsaturated
hydraulic properties in functional form is mandatory. While
much data are available for the water retention function, thehydraulic conductivity function often needs to be predicted.
The classical approach is to predict the relative conductivity from the retention function and scale it with the measured saturated conductivity, Ks . In this paper we highlightthe shortcomings of this approach, namely, that measured Ks
values are often highly uncertain and biased, resulting in poor
predictions of the unsaturated conductivity function.
We propose to reformulate the unsaturated hydraulic conductivity function by replacing the soil-specific Ks as a scaling factor with a generally applicable effective saturated tortuosity parameter τs and predicting total conductivity using
only the water retention curve. Using four different unimodal
expressions for the water retention curve, a soil-independent
general value for τs was derived by fitting the new formulation to 12 data sets containing the relevant information. τs
was found to be approximately 0.1.
Testing of the new prediction scheme with independent
data showed a mean error between the fully predicted conductivity functions and measured data of less than half anorder of magnitude. The new scheme can be used when insufficient or no conductivity data are available. The model
also helps to predict the saturated conductivity of the soil
matrix alone and thus to distinguish between the macropore
conductivity and the soil matrix conductivity.

1

Introduction

Accurate representations of the soil hydraulic properties(SHPs) in functional form are essential for simulations of water, energy, and solute transport in the vadose zone. Classicalmodels for the soil water retention curve (WRC) (e.g., vanGenuchten, 1980; Kosugi, 1996) and the related hydraulicconductivity curve (HCC) derived using pore-bundle concepts (e.g., Burdine, 1953; Mualem, 1976a) account for water storage and flow in completely filled capillaries but neglect adsorption of water and water flow in films and corners. We will refer to the latter processes as “non-capillary”
as opposed to “capillary” in the remainder of this article.
In this paper, the term “non-capillary” is used only for water held by adsorption, although water in very large pores(i.e., larger than 0.3 mm in diameter; Jarvis, 2007) is also notheld by capillary forces. The non-capillary parts of the WRC
and HCC become dominant when soils become dry (Idenet al., 2021a, b). Therefore, improved models of the SHPs
have been proposed that extend models that were established
for the wet range towards the dry range (e.g., Tuller and Or,
2001; Peters and Durner, 2008a, Lebeau and Konrad, 2010;
Zhang, 2011; Peters, 2013). In the very dry range, liquid flowceases, and vapor flow becomes the dominant transport process. Isothermal diffusion of water vapor can be expressed interms of an equivalent hydraulic conductivity and incorporated into an effective conductivity function (Peters, 2013).
The total hydraulic conductivity can then be expressed as thesum of three components:

Published by Copernicus Publications on behalf of the European Geosciences Union.

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A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

K(h) = Kc (h) + Knc (h) + Kv (h),(1)

where h [m] is the suction (i.e., the absolute value of thematric head or pressure head); K [m s−1 ] is the total hydraulic conductivity; and Kc , Knc , and Kv [m s−1 ] are thehydraulic conductivity components for capillary and noncapillary flow of liquid water and water vapor diffusion in thesoil gas phase, respectively. Under isothermal conditions, thefunction Kv (h) can be predicted easily from the temperaturedependent diffusion coefficient of water vapor in air and theWRC (Saito, 2006; Peters and Durner, 2010). Recently, Peters et al. (2021) combined the mechanistic models of Lebeau
and Konrad (2010) and Tokunaga (2009) with the conceptualmodel of Peters (2013) to obtain a simple prediction scheme
for the absolute non-capillary conductivity function Knc (h).
Several models have been proposed to estimate the capillary conductivity function Kc (h) from conceptualizationsof the pore space involving tortuous and interconnected porebundles, most of which go back to the seminal studies by
Burdine (1953) and Childs and Collis-George (1950) (CCG).
Today, the capillary bundle model of Mualem (1976a), who
refined the assumptions of the CCG model, is most frequently used (see Assouline and Or, 2013, for a critical review of this and similar models). The pore-size distributionof a porous medium is derived from the WRC, while the
HCC is predicted using Poiseuille’s law and some assumptions about the connectivity and tortuosity of the pore network. Attempts to predict the absolute capillary conductivity
based on these theories (e.g., Millington and Quirk, 1961;
Kunze et al., 1968) were not very satisfying because of large
deviations with measured conductivities. However, the general shape of the HCC could be described well. Therefore,
the models used in practice today predict a relative hydraulicconductivity function Krc (h) and scale it by fitting the function to one or more measured conductivity points. Most commonly, the measured saturated conductivity is used for thispurpose. A comprehensive overview of these models is given
by Mualem (1986). More recently, concepts to predict thesaturated hydraulic conductivity Ks [m s−1 ] from the WRC
were derived by Guarracino (2007), who used a fractal approach, and Mishra and Parker (1990) and Nasta et al. (2013),
who used capillary bundle models to estimate Ks as a function of the WRC.
When predicting the hydraulic conductivity using a relative conductivity function that needs to be scaled by matching it to measured data, one faces three types of problems.
First and most obviously, if no conductivity data are available for matching, scaling the relative conductivity is notpossible. This is frequently the case. Second, if only measurements of Ks are available, the unsaturated conductivity
estimates will be greatly affected by the dominant influenceof structural pores on the variability of Ks . Thirdly, even if
unsaturated conductivity data are available, the conductivity
Hydrol. Earth Syst. Sci., 27, 1565–1582, 2023

function near saturation may not be represented well. The
latter two problems are outlined below.
The problem of scaling Krc (h) by Ks stems from the influence of soil structure on the hydraulic conductivity at ornear saturation. For more than 50 years, Ks has been known
to vary over many orders of magnitude, even at the samesite with a rather homogeneous texture (Nielsen et al., 1973;
Kutílek and Nielsen, 1994, p. 249). If soil structure is notproperly reflected in the WRC near full saturation, scaling
Krc (h) with a measured Ks can lead to severe overestimationof conductivity in the medium moisture range (Durner, 1992,
1994; Schaap and Leij, 2000). We exemplarily illustrate thisproblem in Fig. 1, top, for a sandy soil. The average difference between data and model in the unsaturated region in thisexample is about 1 order of magnitude.
A better choice is therefore to use unsaturated conductivity data to scale the relative conductivity curve, as alreadyproposed by Nielsen et al. (1960). However, such data are often not available, especially if the measurements were madein the past when more recent techniques such as the simplified evaporation method (SEM) (Schindler, 1980; Petersand Durner, 2008b, Peters et al., 2015) were not available.
Moreover, the SEM typically yields information only in alimited suction range, typically between h ≈ 0.6 to h ≈ 8 m,
because of the limited measurement range of tensiometers
and the fact that the highest measurable conductivity by SEM
is of the same order of magnitude as the evaporation rate(i.e., between 10−8 and 10−7 m s−1 depending on the laboratory conditions). This is particularly problematic with coarse
materials for which the conductivity close to saturation is
many orders of magnitude larger. We illustrate this problem
in Fig. 1, bottom, which shows data for a well-graded sand,together with the fitted water retention and hydraulic functions. Whereas the match between model and the available
data appears almost perfect, the conductivity curve near saturation is unreliable, and the model-predicted saturated conductivity of 1.7 × 10−7 m s−1 (or 1.5 cm d−1 ) is at least 2 orders of magnitude too low for such a soil.
The objective of this study was to develop a model which
predicts the absolute capillary conductivity function Kc (h)in Eq. (1) from the WRC and thus to circumvent a need for
scaling of the relative hydraulic conductivity function Krc (h)with measured conductivity data. The paper is organized asfollows. First, we recall the basic model concept to characterize the capillary and non-capillary pore water componentsof the hydraulic conductivity in a soil. This is followed by abrief review of the essentials of conductivity estimation using
pore-bundle models, which is required to understand our approach. We then develop a model to predict Kc (h) from theWRC. The combination of this model with previously developed models for predicting the complete Knc (h) and Kv (h)yields a soil hydraulic conductivity function that is predictedfrom the WRC and covers the dry (vapor-dominated), the dry
to medium wet (film-dominated), and the medium wet to wet(capillary-dominated) ranges. We apply the obtained scheme
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A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

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tion, we extend the conductivity predictions further towards
capillary pores, which will lead to an absolute prediction of
all terms in Eq. (1), without the need for any measured conductivity data. Our concept is based on classic concepts ofthe pore bundle models. To provide a clear understandingof our approach, we first outline below the PDI model concepts since the PDI parameterization differentiates between
the capillary, non-capillary, and vapor-flow components of
the SHPs.
2.1

Figure 1. Example of conductivity predictions for two soils as obtained by scaling the capillary conductivity with measured conductivity data. Plots on the left side show fitted retention functions;plots on the right side show the corresponding predicted conductivity functions. (a, b) K prediction by matching the relative K
function to the saturated conductivity, Ks . (c, d) K prediction by
matching the relative K function to unsaturated conductivity data,obtained using the simplified evaporation method. The retentionfunctions and corresponding predicted hydraulic conductivity functions were parameterized using the Peters–Durner–Iden model system with the basic van Genuchten saturation function (Peters et al.,2021), as described in Appendix A. Data source: Peters et al. (2019).

using four different parametrizations of the WRC and discuss
the accuracy of the conductivity estimates.

2

TheoryThe unsaturated hydraulic conductivity function covering
wet and dry conditions can be expressed by summing up acapillary component, a film flow component, and a contribution of isothermal vapor diffusion, as given by Eq. (1).
This conceptualization is reflected in the PDI model system (Peters, 2013, 2014; Iden and Durner, 2014), where water retention and the liquid hydraulic conductivity are parameterized as sums of capillary and non-capillary components in a relatively simple, yet consistent, manner. Under
isothermal conditions, the function Kv (h) can be predictedfrom the WRC (Saito, 2006; Peters and Durner, 2010). Using the mechanistic models of Lebeau and Konrad (2010)
and Tokunaga (2009), the absolute non-capillary conductivity function Knc (h) can also be predicted from the WRC (Peters et al., 2021). But still, the capillary part of the conductivity function of the PDI model system needs to be scaledby matching measured conductivity data. In this contribuhttps://doi.org/10.5194/hess-27-1565-2023

Parametrizing capillary and non-capillary porewater components in the PDI model

The PDI model system (Peters, 2013, 2014; Iden and Durner,
2014) describes in a relatively simple, yet consistent, manner the water retention and liquid hydraulic conductivity interms of sums of capillary and non-capillary components.
The WRC is formulated as a superposition of a capillary saturation function Sc [–] and a non-capillary saturation function
Snc [–] (Iden and Durner, 2014):
θ (h) = θc + θnc = (θs − θr )Sc + θr Snc ,(2)

in which the first term considers water stored in saturated
capillaries, and the second term considers water stored in adsorbed films and pore corners. θ [m3 m−3 ] is the total water content, and θs [m3 m−3 ] and θr [m3 m−3 ] are the saturated and maximum adsorbed water contents, respectively.
To meet the physical requirement that the capillary saturation function reaches zero at oven dryness, a basic saturation
function 0 (h) is scaled by the following (Iden and Durner,
2014):Sc (h) =0 (h) − 0 (h0 ),1 − 0 (h0 )(3)with h0 [m] being the suction head at oven dryness, which
can be set to 104.8 m (Schneider and Goss, 2012). 0 (h) canbe any unimodal or multimodal saturation function, such as
the unimodal functions used by van Genuchten (1980), Kosugi (1996), or Fredlund and Xing (1994), or their bimodal
counterparts or combinations (Durner, 1994; Romano et al.,2011).
The total effective hydraulic conductivity function in the
PDI model system is given by Eq. (1). It accounts for liquid water flow in completely filled capillary pores, liquidflow in partly filled pores such as in films on grain surfaces and in pore edges, and the isothermal vapor conductivity. Again, any capillary conductivity model (e.g., Burdine,
1953; Mualem, 1976a) can be used in the PDI system, as outlined by Peters (2013), Peters and Durner (2015), and Weberet al. (2019).
In the original version, both the capillary and non-capillaryparts of the conductivity function needed to be scaled bymatching the conductivity function to measured conductivity
data. Recently, Peters et al. (2021) improved the model byHydrol. Earth Syst. Sci., 27, 1565–1582, 2023

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A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

integrating an absolute prediction of the non-capillary liquid
conductivity Knc (h) as based on the WRC. This decreased
the number of model parameters to the same number as for
traditional models, which do not consider non-capillary storage and conductivity. Nevertheless, a scaling factor Ksc was
required for the capillary conductivity component in Eq. (1):Kc (h) = Ksc Krc (h).

(4)

Since Ksc is orders of magnitude higher than the noncapillary and vapor conductivity components, Ksc can be interpreted as being equal to the total saturated conductivity.
A detailed description of the PDI model system is given in
Appendix A1.
2.2

Capillary bundle models use information about the effective
pore-size distribution of a porous medium as contained in
the WRC. Generally, the Hagen–Poiseuille law is applied toa bundle of capillaries with a size distribution that is consistent with the pore-size distribution of the medium along withsome assumptions about pore connectivity and tortuosity toarrive at a mathematical description of the HCC. The water
flux in a single capillary under unit-gradient conditions, Qc1[m3 s−1 ], can be described with the law of Hagen–Poiseuille:
Qc1 =

8η

,

(5)

where ρ [kg m−3 ] is the fluid density, g [m s−2 ] is gravitational acceleration, η [N s m−2 ] is dynamic viscosity, and r[m] is the radius of the capillary, which is assumed to havea circular cross-section. Relating Qc1 to the cross-sectional
area of the capillary yields the flux density or simply the hydraulic conductivity [m s−1 ] assuming unit gradient conditions:
Qc1
ρgr 2
Kc1 =
=
.
8η
πr2

Hydrol. Earth Syst. Sci., 27, 1565–1582, 2023

Since

(6)

Rrm

fp (r)dr = θc (the capillary water content), thissimplifies toZρg rm 2r fp (r)dr.
8η 0

(9)

Applying the Young–Laplace relation r = 2σ/ρgh, in whichσ [N m−2 ] is the surface tension between the fluid and gas
phases and h [m] the suction, leads to the following expression for a bundle of parallel capillaries:

σ2
Kc =
2ηρg

Zθc

h−2 dθ˜c ,

(10)

0

where θ˜c is the dummy variable of integration.
Several factors distinguish a porous medium from a bundle of parallel tubes. They can be accounted for mostly byimplementing a tortuosity–connectivity correction. The tortuosity describes the effect of the path length of a single watermolecule, lp [m], being longer than a straight line l [m]. The
factor of path extension is then given by lp / l [–]. This causes
both a reduction in the local conductivity and the local hydraulic gradient (Bear, 1972), leading to lower effective hydraulic conductivity by a tortuosity factor τ (0 < τ < 1) [–]:
 2
l
.
lp

(11)

Note that deviations from flow in straight capillary bundles are not only affected by tortuosity in the strict sense but
also by additional effects which will be discussed in Sect. 2.3
within the context of model development. Furthermore, thetortuosity factor is not a constant but a function of the capillary water content since the path length increases with decreasing water contents. Lumping the physical parameters of
Eq. (10) into β = σ 2 / (2ηρg) [m3 s−1 ] and considering the
tortuosity correction τ (θc ) leads to
Zθc

(7)

(8)

0

τ=

If the porous medium is regarded as a bundle of parallel
capillaries of different sizes, the hydraulic conductivity canbe described as the sum of the unit-gradient fluxes of the single water-filled capillaries, divided by the sum of their crosssectional areas, and corrected with the macroscopic capillary water content, θc [m3 m−3 ]. The latter is necessary because air-filled pores and the soil matrix do not contribute to
the macroscopic conductivity. This yields then the following(Flühler and Roth, 2004):
Rrρg 0 m r 4 fk (r)drR
Kc = θc,8η 0rm r 2 fk (r)drRr
ρg 0 m r 2 fp (r)drR.
Kc = θc8η 0rm fp (r)drKc =Relative conductivity predictions using capillarybundle models

πρgr 4

where rm [m] is the maximum radius of the water-filled
pores, and fk (r) is the pore-radius distribution. The poreradius distribution is related to the pore-volume distribution
fp (r), reflecting volumetric fractions by fk ∝ fp /r 2 , whichleads toKc (θc ) = βτ (θc )h−2 dθ˜c .

(12)

0

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A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data
Values of the physical constants used in this study are summarized in Table 1. In SI units, β = 3.04 × 10−4 m3 s−1 at
20 ◦ C.
Equation (12) is similar to the formulation by Nastaet al. (2013), who used the same approach to predict
the saturated conductivity from the WRC of Brooks andCorey (1964). They optimized for this purpose the value ofτ at saturation by fitting their model to measured Ks datafrom the GRIZZLY database (Haverkamp et al., 1997). As
mentioned in the Introduction, Eq. (12) has proven to be insufficient to describe the unsaturated conductivity function
K(h). Burdine (1953) for this reason normalized the expression by the corresponding integral over all capillary pores,
which leads to the following relative conductivity function:Rθβτ 0 c h−2 dθ˜c
,
Krc =
Rθ
βτs 0 s h−2 dθ˜c

(13)

in which τs = τ (θs ). Since the degreeR of capillary satuθ
ration, Sc , is given by Sc = θc /(θs − θr ), 0 c h−2 dθ˜c = (θs −R Sc −2θr ) 0 h dS˜c and hence
R Sc −2h dS˜c,h−2 dS˜c

Krc = τr (Sc ) R01

(14)

0

with the relative tortuosity factor τr = τ/τs . Note thatthe solution is similar for the classic (“non-PDI”)
scheme, for which effective saturation isRdefined as Se =θ(θ − θr ) / (θs − θr ). In this case we obtain θr h−2 dθ̃ = (θs −R Se −2θr ) 0 h dS˜e . Burdine (1953) suggested that the tortuosity(lp / l) is inversely related to the capillary saturation, leading
to τr (Sc ) = Sc2 and hence
R Sc −2˜
2 0 h dSc.
K c = K s Sc R 1−2 ˜
0 h d Sc(15)

In a more sophisticated approach, Mualem (1976a) followed
the cut-and-random-rejoin model approach of Childs andCollis-George (1950) (CCG) and refined the model using theassumption that the length of a pore is directly proportional
to its radius. Normalizing his integral expression by the corresponding integral over all capillary pores and consideringa saturation-dependent tortuosity correction Scλ , the expression for the capillary conductivity function became (Mualem,
1976a)"R S
Kc = Ks Scλc −1 ˜0 h d ScR1
−1 ˜
0 h dSc#2,

Absolute hydraulic conductivity prediction

For the reasons stated in the Introduction, it is preferable topredict the absolute capillary conductivity function Kc (h)from the WRC rather than calculating the relative function
Krc (h) and scaling it with measured conductivity data. In
this paper, we use the Mualem (1976a) model to derive theshape of the capillary conductivity function. Our concept
keeps the dependency of the relative tortuosity factor on saturation in the original formulation of Mualem (1976a); thatis, τr = τ/τs = Scλ , which becomes unity at full saturation.
However, instead of following Mualem’s original concept offirst normalizing the prediction integral and then scaling itwith measured conductivity values, we predict the absolute
Kc (h) from the WRC by introducing an absolute tortuosity
coefficient, τ (Sc ), which is given by the product of a relative
and a saturated tortuosity coefficient τs :
τ (Sc ) = τs τr (Sc ) = τs Scλ .

(17)

By inserting this tortuosity expression into Eq. (12), by
using Mualem’s integralR θ(occurring in Eq. 16),
R S and by applying the substitution 0 c h−2 dθ̃ = (θs − θr ) 0 c h−2 dS˜c , weobtain the following equation for the capillary conductivity
function:

Kc (h) = βτs Scλ (θs − θr )2Z Sc

h−1 dSc

2
(18)

.

0

Expressing the Mualem integral by

hR

2
Sc −1
=
0 h dSci[F (0) −RF (00 )]2 , where F is the solution of the indefinite
1d0 (Peters, 2014), leads tointegral h(0)Kc (h) = βτs Scλ (θs − θr )2 [F (0) − F (00 )]2 .

(19)

In this model, τs is a new factor which scales the capillary conductivity function. We hypothesize that τs varies
only moderately among different textures and that a universal value can be determined from experimental data. If τs isknown and λ is set to Mualem’s suggested value of 0.5, allthree components of the HCC given by Eq. (1) can be calculated based on the WRC without the need for measured
conductivity values.
The parameter Ks (in the classic “Non-PDI” scheme neglecting non-capillary processes) or Ksc (within the PDI system) of Eq. (16) is related to τs of Eq. (19) by(16)

with λ [–] as the tortuosity and connectivity factor. Applying
his model to a variety of data, Mualem found empirically thatλ ≈ 0.5.
https://doi.org/10.5194/hess-27-1565-2023

2.3

1569

Ksc = βτs (θ s − θr )2 [F (1) − F (00 )]2 ,(20)

where [F (1) − F (00 )]2 is the PDI formulation of the denominator in Eq. (16) (Peters, 2014).

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A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

Table 1. Physical constants at 20 ◦ C used in this study.
ParameterσηρgDefinition

Unit

Surface tension between fluid and gas phases
Dynamic viscosity of the bulk liquid
Density of pure water at 298.15 K
Gravitational acceleration constant

N m−1
N s m−2
kg m−3m s−2

Value
0.07278.90 × 10−4
997.04
9.81

The hydraulic tortuosity of saturated porous materials haslong been investigated using a variety of experimental andtheoretical approaches. The earliest description of hydraulictortuosity was introduced by Carman (1937), who modified
the Kozeny (1927) equation for the saturated permeability.
Using experimental data, Carman found that τs ≈ 2/5 for awide range of porosities. However, many found later that thesaturated tortuosity is variable and depends on porosity andtexture. We refer to Ghanbarian et al. (2013) for an overview
of theoretical and experimental studies about this relationship. Most of the derived values for τs are between approximately 0.7 and 0.2.
Importantly, current schemes for the tortuosity generally
only account for pathway elongation due to tortuous flow
paths according to Eq. (11). In real soils, however, the deviation from flow in straight capillary bundles is not only affected by tortuosity in the strict sense but also by other soilrelated factors such as the surface roughness of pore walls,
non-circular capillaries, and dead-end pores. Additionally,
not only the geometry of the pore space may differ fromthe ideal case but also such fluid properties as surface tension and viscosity likely will be different from those of pure
free water. Finally, capillary bundle models will not represent
the pore distribution and connectivity in an ideal way. Therefore, we seek in this contribution an empirical value of τsthat lumps all these effects. The hypothesis that τs varies only
moderately among different textures will be tested by fitting
predicted K functions to test data. In doing so, conductivity
data at or very close to saturation are not considered in the fitting, since the actual saturated tortuosity depends strongly on
the nature of macropores (e.g., inter-aggregate space, wormholes, decayed plant roots). Therefore, we use the term “saturated tortuosity coefficient”, τs , for a (virtual) porous system
without structural pores.

Genuchten, 1980) with the usual constraint (m = 1 − 1/n)
and also in unconstrained form (m independent from n), andthe Fredlund and Xing (1994) saturation function. The latteris the function given in the last row of Table 2. The models are referred to as Kos-PDI, vGc-PDI, vGmn-PDI, andFX-PDI. The saturation functions and the solutions for the
integral F are given in Table 2. For the Kos and vGc saturation functions, F is given in analytical form. For the unconstrained vGmn and FX saturation functions, F needs to beevaluated using numerical integration. We chose these fourfunctions as they are the most commonly used functions in
the field of soil hydrology and geotechnics.
Although the derivation of Kc (h) is presented here for the
PDI model, we note that the model concept is not limited to
PDI-type soil hydraulic functions and that closed-form expressions can also be derived easily for “classical” models
that use a residual water content and neglect the non-capillarycomponents. For those cases, the expression for the integral F (00 ) is zero. For the original van Genuchten–Mualem
model with constraint m = 1 − n1 , one obtains, for example,

2.4

The water retention and unsaturated hydraulic conductivity
functions were described using the PDI model system withfour different unimodal basis functions for capillary water(Table 2), combined with the Mualem (1976a) capillary bundle model to predict the shape of the capillary conductivity
function, Kc (h). This function is given by Eq. (19) and included in the total conductivity function given by Eq. (1).
The relative tortuosity parameter λ was set to 0.5 following Mualem (1976a), β = 3.04 × 10−4 m3 s−1 (Sect. 2.2, Table 1), and τs is the new unknown tortuosity parameter. Knc

Connecting the capillary conductivity function withdifferent WRC parametrizations

Dependent on the selected WRC parametrization, F in
Eq. (19) can be expressed in closed form or must be calculated numerically. For this study, we used four unimodal
models to describe the WRC and correspondingly to predict
K(h). All four models are used within the PDI system. The
basic capillary saturation functions are given by the function of Kosugi (1996), the van Genuchten functions (vanHydrol. Earth Syst. Sci., 27, 1565–1582, 2023

2
 
1
K(h) = βτs Seλ (θs − θr )2 α 1 − (1 − Sem )m,(21)

where Se is the effective saturation function (Se =(θ − θr ) / (θs − θr )), or simply
Ks = βτs (θ s − θr )2 α 2 .

3Materials and methods

3.1

Soil hydraulic models(22)

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A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

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Table 2. Summary of the basic water retention functions used in the PDI scheme (see Appendix A1) as well as the analytical solutions for F(Eq. 19) as given in Peters (2014). The parameters α, n, m, σkos , and hm are shape parameters, and e is the Euler number. In the case of FX
and vGmn, no analytical solution for the integral in Eq. (19) is known, and F must be evaluated numerically.
Name

Basic function (0 (h))"  h #Expression for F (0)Kosugi (Kos)

1 erfc2−1−1/n11+(αh)nm−α 1 − 0 1/m

Van Genuchten with mn independent (vGmn)



m
1
n
1+(αh)

R1h(0) d0

Fredlund–Xing (FX)m1ln[e+(αh)n ]R1h(0) d0

Estimating the saturated tortuosity coefficient, τs

To obtain an estimate for τs , reliable data for the WRC andin particular for the HCC in the wet range (but not at saturation) are needed. We used six data sets used by Peters (2013)
and six additional data sets from Sarkar et al. (2019), which
fulfill the abovementioned requirements. Soil textures varied
from pure sand to loamy clay, representing a wide variety ofdifferent soils. Details about the soils are given in the original literature and are summarized in Table 3. For each of thefour PDI combinations with the capillary saturation functions
given in Table 2, we determined a value for τs by fitting themto the 12 data sets and estimating the WRC parameters andτs . The median values of the estimated τs values were usedin the corresponding prediction models.
3.3


h
i
2 /2
exp σkos−1 (20) + σ√kos
erf
erfc
2hm
2

Van Genuchten with m = 1 − 1/n (vGc)

and Kv were predicted from the WRC (Peters et al., 2021;
see Appendix A1).
For soils with a wide pore-size distribution, Mualem’s
model (as all capillary bundle models) in combination withwater retention models that gradually approach saturation
can produce a non-physical sharp decrease in the hydraulicconductivity near saturation (e.g., Vogel et al., 2000; Ippischet al., 2006). Madi et al. (2018) developed a mathematical
criterion to test individual WRC parameterizations for physical plausibility. To avoid this model artifact, we used the“hclip approach” of Iden at al. (2015) in all cases. The hclip
approach limits the pore size in the conductivity prediction
integral to a maximum value, which is equivalent to a minimum suction, hcrit . According to Jarvis (2007), we use hcrit =0.06 m, corresponding to an equivalent diameter of 0.5 mm(see also Sect. 2.4).
3.2

ln hm
√
2σkos

Validation of the absolute K predictions

To test the predictions of K(h) from the WRC, we selected
data sets that cover a relatively wide moisture range andcould be described well using unimodal WRCs. We used
23 data sets, which were obtained at Technische Universität
Braunschweig, Germany. The data comprised again a broad
https://doi.org/10.5194/hess-27-1565-2023

range of textural classes. Some of them stemmed from soilcolumns taken at the same sites at which the soil columnsof Sarkar et al. (2019) were taken (locations JKI, GG, and
SAU). However, we used independent data from different
soil samples taken in different years. All data sets except one(Test 4) are from undisturbed samples. Details about the validation data are given in Table 4.
3.4

Parameter estimation and diagnostics

The models were fitted to the data by minimizing the sum of
weighted squared residuals between modeled and measured
data (Peters, 2013):
Xnθ 
2
φ (b) = wθ
θ
−
θ
(b)
i
mod,i
i=1
i2
XnK h
+ wK
log
(K
)
−
log
(
K̃
(b))
,
(23)
i
i
10
10
i=1
where θi and θmod,i are the measured and modeled water contents, Ki and K̃i are measured and modeled hydraulic conductivities, nθ and nK are the respective number of data points, wθ = 10 000 and wK = 16 are weights
for the two data groups (Peters, 2011), and b is the vector of
unknown model parameters. The shuffled-complex-evolution
algorithm SCE-UA (Duan et al., 1992) was used to minimize
the objective function given by Eq. (23). In the case of estimating the general value of τs , the parameter vector contained all adjustable parameters of the water retention function plus τs . In the case of the hydraulic conductivity predictions, wK was set to 0, and the estimated parameter vector contained only the WRC parameters. The performance ofthe different approaches was compared in terms of the rootmean-square error (RMSE) for the WRC and the HCC (common log of K(h)), respectively. Model comparisons werebased on the Akaike information criterion, AICc, corrected
for small sample sizes (Hurvich and Tsai, 1989).

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1572

A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

Table 3. Calibration data sets used for estimating water retention curves and the saturated tortuosity coefficient τs .
Data set ID

Data set name in original publication

SourceTexture class

Cal 1
Cal 2
Cal 3

Rehovot sandGilat loam
Pachappa fine sandy clay (PFSC)

Mualem (1976b)

Sand
Loam
Sandy Clay

Cal 4
Cal 5
Cal 6

–
–
–

Pachepsky et al. (1984)

Sandy Loam
Silt Loam
Clay Loam

Cal 7
Cal 8
Cal 9
Cal 10
Cal 11
Cal 12

GG first sample
GG second sample
JKI first sample
JKI second sample
SAU first sample
SAU second sample

Sarkar et al. (2019)

Silt Loam
Silt Loam
Loamy Sand
Loamy Sand
Sand
Sand

Table 4. Test data sets used to test the conductivity predictions.
Data set ID

Data set name inoriginal publication

Test 1a
Test 2a
Test 3aTest 4b
Test 5
Test 6
Test 7
Test 8
Test 9Test 10
Test 11
Test 12

–
–
–
–
–
–
–
–
–
–
–
–

Test 13
Test 14
Test 15
Test 16a
Test 17a
Test 18asand 1silt loam 1
sand 2asilt loam 2
sand 2bsilt

Test 19a
Test 20a
Test 21a
Test 22
Test 23

GG
JKI
SAU
HEB
SEL

SourceTexture classOwn data(not published)Silt Loam
Sandy Loam
Sand
Sand
Sandy Loam
Loamy Sand
Loamy Sand
Sand
Sand
Loamy Sand
Loamy Sand
Sandy Loam

Schelle et al. (2013)

Kirste et al. (2019)

Sand
Silt Loam
Sand
Silt Loam
Sand
Silt
Silt Loam
Sandy Loam
Sand
Silt Loam
Silty Clay Loama samples taken at same sites but different years as some of the calibration data (Cal 7 to Cal 12).
b disturbed sample.

4
4.1

Results and discussion
Empirical estimate of the saturated tortuosity
coefficient τsFigure 2 illustrates, using 4 of the 12 calibration data sets
described in Table 3, the fitted water retention and conducHydrol. Earth Syst. Sci., 27, 1565–1582, 2023

tivity functions for the four basic hydraulic models listed inTable 2. A full overview on all calibration data sets and thefitted models is given in the Supplement. All estimated parameters are given in Table A1. In general, all four models
are well suited to describe the data. Actually, the models canbe hardly distinguished on the plots since they lie largely on
top of each other. Figure 3 shows the corresponding distribu-

https://doi.org/10.5194/hess-27-1565-2023

A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

1573

indication that the hypothesis of relatively moderate overallvariability in τs may be justified. When fitting the classic PDI
scheme (with Ks as a fitting parameter) to these data, which
do not include measured conductivity data at saturation, theestimated Ks values varied by more than 3 orders of magnitude. In natural soils, the measured Ks values can vary evenmore due to the dominance of (texture-independent) structural pores and macropores on Ks (e.g., Usowicz and Lipiec,2021).
4.2

Figure 2. Plots of 4 of the 12 calibration data sets and the fitted
water retention and conductivity functions used to calibrate the saturated tortuosity coefficient τs in Eq. (19). Parameter λ was set toa value of 0.5 according to Mualem (1976a). Parameter τs and theretention parameters were allowed to vary. Numbers in the subplots
indicate RMSEθ and RMSElogK values for the various model combinations.

tions of RMSEθ and RMSElogK , which allow one to better
differentiate the fitting performance. Results confirm the visual impression from Fig. 2 that all four models fit similarly,
with a slightly better performance of the models having six
free parameters (i.e., FX-PDI and vGmn-PDI) as comparedto those with five free parameters (vGc-PDI and Kos-PDI).
We note that fitting the K functions of Peters et al. (2021),
using Ks instead of τs as an adjustable parameter and leaving all other settings identical, would lead here exactly to thesame results.
The distributions of the resulting values for τs for the fourmodels are shown in Fig. 4. The median values for τs were
0.062 for the constrained van Genuchten function (vGc),
0.084 for the Kosugi function (Kos), and 0.094 and 0.095
for the unconstrained van Genuchten (vGmn) and Fredlund–
Xing (FX) functions, respectively. It appears noteworthy thatthe two best-fitting WRC models yield almost identical estimates of τs . The range of τs for the 12 data sets spanned
less than 1.5 orders of magnitude. We interpret this as an
https://doi.org/10.5194/hess-27-1565-2023

Tests of the absolute conductivity predictions

By using median values of τs for the different models (0.084,
0.062, 0.094, and 0.095 for Kos-PDI, vGc-PDI, vGmn-PDI,and FX-PDI, respectively; Fig. 4), we predicted the hydraulicconductivity functions from the water retention functions for
23 test data sets. In Fig. 5, we show the resulting distributions of RMSEθ (fitted) and RMSElogK (predicted). Since
measured conductivities were available primarily within therange where the capillary conductivity component dominates, RMSElogK can be interpreted as an approximate error of the capillary conductivity prediction. The medians of
RMSElogK for the Kos-PDI and vGc-PDI models were 0.71
and 0.67, respectively. Combinations with the more flexible
retention models yielded median RMSElogK values of 0.49
for vGmn-PDI and 0.40 for FX-PDI. To test whether conductivity predictions were biased, we calculated also the meanerror (Fig. 6). For the FX-PDI model, the median was close tozero, indicating an unbiased conductivity prediction, whereas
the other models tended to underestimate the conductivity
data.
Figure 7 shows fitted WRCs and predicted HCCs alongwith the measured conductivity data. Due to space limitations, only a subset of six randomly selected cases is shownfor the FX-PDI combination. The WRC fits and HCC predictions for all 23 test soils and all four models are listed in the
Supplement.
4.3

Improved estimation of K functions when K data
are available

Several authors (e.g., Schaap and Leij, 2000; Peters et al.,2011) have stressed that the tortuosity parameter λ might
differ greatly from the value suggested by Mualem (1976a),since the change in tortuosity with respect to capillary saturation can obviously be different for different soils. The newscheme is valuable not only for cases where no or insufficient information about the conductivity is available. It isalso useful when data are available for the unsaturated hydraulic conductivity but are missing in the wet range. Thisis the case, for example, with the commonly used evaporation method (Schindler, 1980; Peters and Durner, 2008b;
Peters et al., 2015). Then, there is often high uncertainty inthe wet moisture range; thus, an unrealistic conductivity extrapolation might result (see Fig. 1, bottom). In such cases,
Hydrol. Earth Syst. Sci., 27, 1565–1582, 2023

1574

A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

Figure 3. Distributions of RMSEθ and RMSElogK of the fitted retention and conductivity models for the 12 data sets. Black dots indicate
single realizations.

Figure 4. Distribution of fitted values of τs for 4 different retentionfunctions.

λ might be estimated, and only τs might be fixed. We illustrate this in Fig. 8 for the data set shown in Fig. 1 (bottom). Again, the vGmn-PDI retention model is used, butinstead of Eq. (16), we now use Eq. (18) with τs = 0.094.
Now, the model is well able to be fitted to the data, andthe hydraulic conductivity close to saturation is more reasonably predicted as in Fig. 1: predicted conductivity at saturation is 1.7 × 10−7 m s−1 (or 1.5 cm d−1 ) for the original and1.3 × 10−5 m s−1 (or 112 cm d−1 ) for the new scheme. Notethat the new scheme has one less adjustable parameter.
4.4

Considerations of the hydraulic conductivity atsaturation

Because the saturated hydraulic conductivity is relatively
easy to measure, many determine Ks experimentally. As emphasized earlier, the use of Ks for scaling the relative hydraulic conductivity function should be avoided as much as
possible. Still, Ks provides valuable information for the hyHydrol. Earth Syst. Sci., 27, 1565–1582, 2023

draulic behavior of soils at and close to saturation, which cannot be derived from the WRC. Within the context of modeling macropore flow, Nimmo (2021) identified a need for approaches to determine the properties of the matrix only while
excluding the remainder of the porous medium. Predictionsof a capillary conductivity function may help to fill this research gap.
Our approach predicts the capillary hydraulic conductivity in the matrix domain up to a minimum suction. Following Jarvis (2007), we may choose for this a suction of about
0.06 m (pore diameter approximately 0.5 mm) up to which
the macropore conductivity can be neglected. Accordingly,
we call the conductivity at hcrit = 0.06 m the “saturated matrix conductivity” (Ks,matrix ). Knowledge of Ks,matrix could
substantially improve the parameterization of simulation
models that explicitly distinguish between matrix and macropore flow (e.g., Reck et al., 2018; van Schaik et al., 2010).
The shape of the conductivity function in the macroporeaffected range cannot be predicted using capillary bundle models (Durner, 1994). Thus, it is preferable to cover
the region between Ks,matrix and the measured Ks using
some interpolation function such as proposed by Schaap andvan Genuchten (2006). Using the abovementioned value of
0.06 m for hcrit as a starting point for the interpolation andassuming that the saturated conductivity (Ks ) is reached at apore diameter of 5 mm (i.e., at hs = 0.006 m), we can formalize the interpolation as Ks,matrix
Kint
K=

Ks

for
for
for

h = hcrit ,hcrit > h > hs ,h ≤ hs .

(24)

As an example, we illustrate the interpolation with a simple smooth cosine interpolation function, with the log of thehttps://doi.org/10.5194/hess-27-1565-2023

A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

1575

Figure 5. RMSEθ and RMSElogK for prediction of the absolute conductivity from the soil water retention function of 23 test data sets. Black
dots indicate the validation data sets; red dots indicate the data sets used to estimate a general value of the saturated tortuosity coefficient τs .
The red cross indicates an outlier, defined by the MATLAB® default settings as 1.5 times the interquartile range away from the top or bottomof the box (https://de.mathworks.com/help/matlab/ref/boxchart.html, last access: 13 April 2023).

while K(h) was predicted from the WRC from dryness to
h = 0.06 m. From h = 0.06 m to h = 0.006 m, the smooth interpolation scheme (Eq. 24) was applied. With this scheme,
we obtained a description of hydraulic conductivity fromoven dryness to full saturation.
5

Figure 6. Mean errors of the predicted absolute conductivity basedon the soil water retention function for 23 test data sets. Black dots
indicate all 23 validation data sets; red dots indicate data sets shownin Fig. 7.

suction in the argument (Fig. 9). Mathematically, this interpolation is expressed as (x − xs )+ 1 (ycrit − ys ) , (25)yint = ycrit +0.5 cos π(xcrit − xs )with the transformed variables y = log (K) and x = log(h)and consideration of the corresponding subscripts. We notethat the real course of the K(h) function in this moistureregion probably will be different; hence, other interpolation
schemes could be used. Still, any interpolation will probably
improve the performance of numerical models if such conditions close to full saturation are encountered.
Figure 10 shows the practical application of the above interpolation scheme for the data given in Fig. 1 (top). The
PDI water retention function was fitted to the retention data,
https://doi.org/10.5194/hess-27-1565-2023

Summary and conclusions

The unsaturated hydraulic conductivity of soils is still themost difficult hydraulic property to directly measure. The
availability of commercial systems that allow one to determine SHPs using the simplified evaporation method has improved the situation somewhat; still, available conductivity
data generally are restricted to a relatively limited soil moisture range so that predictive models for the hydraulic conductivity curve continue to play a critical role. To date, suchpredictions mostly use pore-bundle models that require measured conductivity data to scale the predicted curves. However, the HCC outside the range for which measured data
are available is highly uncertain. In this contribution we presented a prediction scheme for the hydraulic conductivity
covering the moisture range from very dry conditions to almost full saturation. The PDI modeling framework predicts
three components of the conductivity, namely, vapor, noncapillary, and capillary liquid conductivity as absolute values
from the water retention function.
Pore-bundle models do not in themselves account for important characteristics such as path elongation due to tortuosity, surface roughness of pore walls, non-circular capillaries,
dead-end pores, physical properties of the liquid phase, etc.
These effects can be accounted for with a parameter that iscalled tortuosity coefficient. We divide this parameter into
two factors: a saturated tortuosity factor and a relative tortuosity function that takes the dependence of tortuosity on
water content into account. The saturated tortuosity factor is
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1576

A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

Figure 8. Same data as in Fig. 1c and d. A new scheme with vGmn
as basic retention function and τs fixed at 0.094 was fitted to the data(new fit). For comparison, the original fit using Ks as adjustableparameter is given by dashed lines. Note that the new scheme has
one adjustable parameter less.

Figure 9. Interpolation scheme between the predicted capillary conductivity (red dashed line) and the measured value of Ks (gray dot).

Figure 7. Measured data (dots), fitted retention functions (a, c) andpredicted conductivity functions (b, d). Shown are for six randomly
selected soils out of 23 validation data sets. Numbers in the subplots
indicate RMSEθ and RMSElogK values Note that the conductivity
curves are not fits to the data.

shown to vary little among different soils, and we have determined a universal value empirically from data. The newscheme using a saturated tortuosity factor with an assigned
universal value can be used to predict the hydraulic conductivity curve from the water retention curve when insufficient
or no conductivity data are available.
The proposed general prediction scheme was tested by
combining it with four parametric water retention models. Of
these, the PDI model with the Fredlund and Xing (1994) baHydrol. Earth Syst. Sci., 27, 1565–1582, 2023

sic saturation function and the model of van Genuchten withindependent parameters m and n as basic function performed
best. The identified value for the saturated tortuosity coefficient τs was 0.095. From a practical point of view, τs may
simply be set to 0.1. The prediction accuracy with the newmodel was tested using a set of 23 soils for which measured
K values were available. For the best-performing model FXPDI, the predictions matched the measured data on averagewith a RMSElogK of about 0.4, without a bias between thepredicted functions and measured data.
The conductivity estimation using our approach involves
the conductivity of the soil matrix only and as such excludes
the effects of the soil structure. The scheme is applicable
only if retention data are available, and it opens new possibilities to use existing retention data collections (e.g., Gupta
et al., 2022). The approach will also be helpful for situations
where a measured value of the saturated conductivity Ks isavailable and where soil structure plays a role (which is therule for most topsoils). In such cases, the predicted HCC canbe combined with an interpolation towards Ks to obtain awell-estimated conductivity function over the full moisture
range. Differentiating between structural and textural effects
https://doi.org/10.5194/hess-27-1565-2023

A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

Figure 10. Application of the interpolation scheme given by
Eq. (24) to the data set shown in Fig. 1a and b. The FX-PDI model
was fitted to the water retention data and K(h) was predicted fromdryness to h = 0.06 m.

enables a physically more consistent use of measured SHP
information.
In the cases where measured unsaturated conductivity
data are available (such as from the simplified evaporation
method), the proposed model with fixed τs can be fitted byadjusting a soil-specific relative tortuosity coefficient. This
leads to a more reliable description of the conductivity function in the wet range, where no data are available, relative
to current model approaches. Our new scheme can therefore
improve the fitting of SHP models to measurements and canbe implemented easily in the standard optimization software
packages.
Appendix A
A1
A1.1

The PDI model system
PDI water retention function

The capillary saturation function Sc [–] and a non-capillary
saturation function Snc [–] may be superimposed in the following form (Iden and Durner, 2014):
θ (h) = (θs − θr )Sc + θr Snc ,(A1)

in which the first right-hand term holds for water stored incapillaries, and the second term holds for water stored in adsorbed water films and pore corners. θ [m3 m−3 ] is the total
water content, h [m] is the suction head, and θs [m3 m−3 ] andθr [m3 m−3 ] are the saturated and maximum adsorbed watercontents, respectively. To meet the physical requirement thatthe capillary saturation function reaches zero at oven dryness, a basic saturation function 0 (h) is scaled by (Iden andDurner, 2014)
Sc (h) =0 (h) − 0 (h0 ),1 − 0 (h0 )(A2)with h0 [m] being the suction head at oven dryness, which
can be set to 104.8 m following Schneider and Goss (2012).
https://doi.org/10.5194/hess-27-1565-2023

1577

0 (h) can be any unimodal or multimodal saturation function
such as the unimodal functions of van Genuchten (1980) and
Kosugi (1996) or their bimodal versions (Durner, 1994; Romano et al., 2011).
The saturation function for non-capillary water is given
by a smoothed piecewise linear function (Iden and Durner,
2014), which is here given in the notation of Peters etal. (2021): 
h i1/b 
ln hh0 − bln 1 + hha
 
,
(A3)
Snc (h) =ln hha0in which the parameter ha [m] reflects the suction head where
non-capillary water reaches its saturation (fixed in our study
to the suction at which capillary saturation reaches 0.75). We
note that in earlier publications, we set ha = α −1 for the constrained van Genuchten function. For the vGmn and the FX
models, however, α −1 may be very high although the capillary saturation decreases already at low suctions. Setting
ha = α −1 would lead in such cases to unrealistic retentionfunctions with Snc being close to unity, whereas Sc is alreadyclose to zero. The calculation scheme for ha as a quantile of
Sc is given in Appendix A2. The parameter h0 in Eq. (A3) isthe suction head where the water content reaches zero, whichreflects the suction at oven-dry conditions. Snc (h) increases
linearly from zero at oven dryness to its maximum value of
1.0 at ha , and it then remains constant towards saturation.
In order to ensure a continuously differentiable water capacity function, Snc (h) must be smoothed around ha , which isachieved by the smoothing parameter b [–] (Iden and Durner,
2014), given here by
1 − e−b1,(A4)
b = bo 1 + 2
n2

2
r
where bo = 0.1 ln(10), and b1 = θs θ−θ
.
r
A1.2

PDI hydraulic conductivity

The PDI hydraulic conductivity model is expressed as (Peters
et al., 2021)
K (h) = Ks,c Kr,c + Knc + Kv ,(A5)

where Kr,c [–] is the relative conductivity for the capillarycomponent, Ks,c [m s−1 ] is the saturated conductivity for thecapillary components, and Knc and Kv [m s−1 ] are the noncapillary and isothermal vapor conductivities, respectively.
Knc is given by (Peters et al., 2021)
 −1.5(1−Snc )−1.5 h0
Knc = c θm ha
,
(A6)
ha
in which c is used to account for several physical and geometrical constants and being either a free fitting parameter toHydrol. Earth Syst. Sci., 27, 1565–1582, 2023

1578

A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention datascale Knc or c = 1.35 × 10−8 m5/2 s−1 . Parameter θm [–] isthe water content at h = 103 m. We refer to Saito et al. (2006)
or Peters (2013) for details regarding the formulation of Kv
as a function of the invoked WRC. Note that the capillaryliquid conductivity is formulated as a relative conductivity,
which has to be scaled with a measured value, whereas the
non-capillary conductivity and the isothermal vapor conductivity are formulated as absolute conductivities.
The relative conductivity for water flow in capillaries isin this paper described using the pore bundle model of
Mualem (1976a), which reads in the PDI notation (Peters,2014) as"R 0

−1
0 h dX

Kr,c (Sc ) = (Sc )λ R 10

−100 h dX

0 (h) − 00,
1 − 00(A9)

where 00 = 0 (h0 ) . According to Peters (2013), we define
the suction ha as(A10)

Sc (ha ) = β,where β [–] represents the chosen quantile of Sc . Combining
Eqs. (A8)–(A10) and solving for ha yields
i1/n
h 1
,
ha = α −1 γ − m − 1

(A11)

#2
,

(A7)

where λ [–] is the tortuosity and connectivity parameter, and
X is a dummy variable of integration.
A2

Sc (h) =Calculation of ha as a function of the Sc quantile

Peters (2013) proposed two methods to define the critical tension ha (m) for the non-capillary saturation function Snc (–).
He decided to set ha = α −1 for van Genuchten’s model andha = hm for Kosugi’s model (1996). His second option was
to define ha as a quantile of the capillary saturation function,
while suggesting the value of 0.5 as a potential choice. For
completeness, we repeat here the relevant equations.
The capillary saturation function of van Genuchten is
given by
−m0 (h) = 1 + (αh)n.
(A8)
Recall that this function ensures a saturation of zero at
the suction corresponding to oven dryness, h0 (L). Iden andDurner (2014) proposed to scale Eq. (A8) using the function

Hydrol. Earth Syst. Sci., 27, 1565–1582, 2023

in which the constant γ is defined as(A12)

γ = β (1 − 00 ) + 00 .

Applying the same approach to the capillary saturation function of Kosugi (1996), i.e., log hhm
,
0 (h) = 0.5 erfc  √(A13)
2σyields√−1 (2γ )ha = hm e 2 σ erfc

.

(A14)

Thirdly, for the capillary saturation of Fredlund andXing (1994), given as−m
0 (h) = ln e + (αh)n,(A15)
we obtain11nha = α −1 exp(γ − m ) − e .

(A16)

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A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

1579

Appendix B: Additional information

Table B1. Estimated parameter values for all four model combinations for the calibration data set.

Kos-PDIvGc-PDIvGm-PDIFX-PDI

Cal1
Cal2
Cal3
Cal4
Cal5
Cal6
Cal7
Cal8
Cal9
Cal10
Cal11
Cal12

Cal1
Cal2
Cal3
Cal4
Cal5
Cal6
Cal7
Cal8
Cal9
Cal10
Cal11
Cal12

Cal1
Cal2
Cal3
Cal4
Cal5
Cal6
Cal7
Cal8
Cal9
Cal10
Cal11
Cal12

Cal1
Cal2
Cal3
Cal4
Cal5
Cal6
Cal7
Cal8
Cal9
Cal10
Cal11
Cal12

https://doi.org/10.5194/hess-27-1565-2023–

θs [–]

θr [–]

hm [m]

σkos [–]

τs [–]

0.404
0.437
0.322
0.417
0.515
0.496
0.432
0.442
0.321
0.362
0.363
0.382

0.019
0.168
0.110
0.078
0.187
0.293
0.000
0.267
0.113
0.031
0.090
0.128

0.256
0.669
1.301
2.518
5.595
7.367
14.635
4.900
2.358
2.879
0.468
0.436

0.628
0.514
0.790
1.515
1.638
2.213
2.823
1.561
1.230
1.923
0.388
0.381

θs [–]

θr [–]

α [m−1 ]

n [–]

0.399
0.437
0.321
0.413
0.511
0.493
0.424
0.445
0.319
0.359
0.364
0.381

0.021
0.164
0.106
0.054
0.134
0.175
0.000
0.105
0.102
0.000
0.087
0.126

4.612
1.711
1.018
0.940
0.496
0.832
1.706
0.671
0.844
1.706
2.321
2.469

3.264
3.566
2.627
1.661
1.534
1.244
1.155
1.279
1.808
1.377
4.649
4.735

θs [–]

θr [–]

α [m−1 ]

n [–]

m [–]

τs [-]

0.409
0.434
0.317
0.429
0.524
0.500
0.423
0.443
0.321
0.360
0.350
0.378

0.020
0.159
0.103
0.088
0.209
0.317
0.000
0.098
0.104
0.000
0.080
0.124

2.588
2.063
1.362
0.022
0.001
0.012
2.123
1.487
0.644
1.585
2.893
2.730

2.121
4.784
5.371
0.756
0.725
0.677
1.364
2.225
1.540
1.311
11.231
5.799

1.978
0.397
0.230
6.825
29.350
4.016
0.099
0.067
0.630
0.298
0.202
0.524

0.167
0.022
0.174
0.055
0.031
0.121
0.015
0.024
0.127
0.067
0.313
0.306

θs [–]

θr [–]

α [m−1 ]

n [–]

m [–]

τs [–]

0.411
0.435
0.318
0.429
0.524
0.500
0.427
0.444
0.321
0.361
0.354
0.378

0.019
0.147
0.083
0.088
0.209
0.317
0.000
0.128
0.092
0.000
0.068
0.120

2.351
1.887
1.165
0.031
0.002
0.017
1.643
1.027
0.683
1.380
2.596
2.532

1.980
4.059
4.135
0.756
0.727
0.675
0.942
1.454
1.511
1.153
7.112
5.052

5.799
1.261
0.815
14.09948.102
8.692
0.010
0.010
1.384
0.724
1.033
1.743

0.169
0.022
0.159
0.055
0.031
0.120
0.014
0.023
0.126
0.070
0.252
0.302

0.162
0.022
0.153
0.050
0.029
0.110
0.013
0.037
0.128
0.058
0.354
0.349
–

τs [–]
0.148
0.023
0.158
0.048
0.024
0.055
0.014
0.019
0.138
0.068
0.342
0.326

Hydrol. Earth Syst. Sci., 27, 1565–1582, 2023

1580

A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

Data availability. The 12 data sets used in this paper for model calibration are collected from the published literature and are available
as follows. Cal 1 to Cal 3: Mualem (1976b); Cal 4 and Cal 6 (originally published in Pachepsky et al., 1984): Tuller and Or (2001);
Cal 5: (originally published in Pachepsky et al., 1984): Zhang(2010); Cal 7 to Cal 12: Sarkar et al. (2019). The test data sets
Test 1 to Test 23 can be obtained from the corresponding author
upon request.

Supplement. The supplement related to this article is available online at: https://doi.org/10.5194/hess-27-1565-2023-supplement.

Author contributions. Conceptualization: AP; model implementation and analysis: AP and TLH; draft preparation and discussions: AP, TLH, SCI, MTvG and WD; All authors read and approved the final paper.

Competing interests. The contact author has declared that none of
the authors has any competing interests.

Disclaimer. Publisher’s note: Copernicus Publications remains
neutral with regard to jurisdictional claims in published maps andinstitutional affiliations.

Acknowledgements. This study was supported by the Deutsche
Forschungsgemeinschaft (DFG grant PE 1912/4-1). We thank Gerrit de Rooij and John Nimmo as reviewers for their constructive
comments, as well as Erwin Zehe for handling the manuscript.

Financial support. This research has been supported by theDeutsche Forschungsgemeinschaft (grant no. PE 1912/4-1).
This open-access publication was funded
by Technische Universität Braunschweig.

Review statement. This paper was edited by Erwin Zehe and reviewed by John R. Nimmo and Gerrit H. de Rooij.

References
Assouline, S. and Or, D.: Conceptual and parametric representationof soil hydraulic properties: A review, Vadose Zone J., 12, 1–20,
https://doi.org/10.2136/vzj2013.07.0121, 2013.
Bear, J.: Dynamics of Fluids in Porous Media, Elsevier, New York,ISBN 0486131807, 1972.
Brooks, R. H. and Corey, A. T.: Hydraulic properties of porous
media. Hydrology Paper No. 3. Civil Engineering Department, Colorado State University, Fort Collins, CO, https://hess.
copernicus.org/articles/27/385/2023/hess-27-385-2023.pdf (lastaccess: 14 April 2023), 1964.

Hydrol. Earth Syst. Sci., 27, 1565–1582, 2023

Burdine, N.: Relative permeability calculations from pore size distribution data, J. Petrol. Technol., 5, 71–78, 1953.
Carman, P. C.: Fluid flow through granular beds, Transactions, Institution of Chemical Engineers, London, 15, 150–166, 1937.
Childs, E. C. and Collis-George, N.: The permeability of porous
materials, Proc. R. Soc. Lon. Ser.-A, 201, 392–405, 1950.
Duan, Q., Sorooshian, S., and Gupta, V.: Effective and efficient
global optimization for conceptual rainfall-runoff models, Water
Resour. Res., 28, 1015–1031, 1992.
Durner, W.: Predicting the unsaturated hydraulic conductivity using multi-porosity water retention curves, in: Proceedings ofthe International Workshop, Indirect Methods for Estimating
the Hydraulic Properties of Unsaturated Soils, edited by: Van
Genuchten, M. Th., Leij, F. J., and Lund, L. J., Univ. of California, Riverside, 185–202, 1992.
Durner, W.: Hydraulic conductivity estimation for soils with heterogeneous pore structure, Water Resour. Res., 30, 211–222,
https://doi.org/10.1029/93WR02676, 1994.
Flühler, H. and Roth, K.: Physik der ungesättigten Zone. Lecturenotes, Institute of Terrestrial Ecology, Swiss Federal Institute ofTechnology Zurich, Switzerland, 2004.
Fredlund, D. G. and Xing, A. Q.: Equations for the soilwater characteristic curve, Can. Geotech. J., 31, 521–532,
https://doi.org/10.1139/t94-061, 1994.
Ghanbarian, B., Hunt, A. G., Ewing, R. P., and Sahimi, M.: Tortuosity in porous media: a critical review, Soil Sci. Soc. Am. J., 77,
1461–1477, 2013.
Guarracino, L.: Estimation of saturated hydraulic conductivity Ks
from the van Genuchten shape parameter α, Water Resour. Res.,
43, W11502, https://doi.org/10.1029/2006WR005766, 2007.
Gupta, S., Papritz, A., Lehmann, P., Hengl, T., Bonetti, S., and Or,
D.: Global Soil Hydraulic Properties dataset based on legacy site
observations and robust parameterization, Scientific Data, 9, 1–
15, 2022.
Haverkamp, R., Zammit, C., Boubkraoui, F., Rajkai, K., Arrue, J.
L., and Heckmann, N.: GRIZZLY, Grenoble soil catalogue: Soilsurvey of field data and description of particle-size, soil water
retention and hydraulic conductivity functions, Lab. d’Etude desTransferts en Hydrol. et en Environ., Grenoble, France, 1997.
Hurvich, C. and Tsai, C.: Regression and time series model
selection in small samples, Biometrika, 76, 297–307,
https://doi.org/10.1093/biomet/76.2.297, 1989.
Iden, S. and Durner, W.: Comment on “Simple consistent models
for water retention and hydraulic conductivity in the complete
moisture range” by A. Peters, Water Resour. Res., 50, 7530–
7534, https://doi.org/10.1002/2014WR015937, 2014.
Iden, S. C., Peters, A., and Durner, W.: Improving prediction of
hydraulic conductivity by constraining capillary bundle models
to a maximum pore size, Adv. Water Resour., 85, 86–92, 2015.
Iden, S. C., Blöcher, J., Diamantopoulos, E., and Durner, W.: Capillary, film, and vapor flow in transient bare soil evaporation (1):
Identifiability analysis of hydraulic conductivity in the medium
to dry moisture range, Water Resour. Res., 57, e2020WR028513,
https://doi.org/10.1029/2020WR028513, 2021a.
Iden, S. C., Diamantopoulos, E., and Durner, W.: Capillary, film,
and vapor flow in transient bare soil evaporation (1): Experimental identification of hydraulic conductivity in the medium to
dry moisture range, Water Resour. Res., 57, e2020WR028514,
https://doi.org/10.1029/2020WR028514, 2021b.

https://doi.org/10.5194/hess-27-1565-2023

A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data
Ippisch, O., Vogel, H.-J., and Bastian, P.: Validity limits for the van
Genuchten–Mualem model and implications for parameter estimation and numerical simulation, Adv. Water Resour., 29, 1780–
1789, 2006.
Jarvis, N. J.: A review of non-equilibrium water flow and solutetransport in soil macropores: Principles, controlling factors andconsequences for water quality, Eur. J. Soil. Sci., 58, 523–546,
2007.
Kirste, B., Iden, S. C., and Durner, W.: Determination of the soil
water retention curve around the wilting point: Optimized protocol for the dewpoint method, Soil Sci. Soc. Am. J., 83, 288–299,
2019.
Kosugi, K.: Lognormal distribution model for unsaturated soil hydraulic properties, Water Resour. Res., 32, 2697–2703, 1996.
Kozeny, J.: Über kapillare Leitung des Wassers im Boden, Sitzungsberichte Wiener Akademie, 136, 271–306, 1927.
Kunze, R. J., Uehara, G., and Graham, K.: Factors important in thecalculation of hydraulic conductivity, Soil Sci. Soc. Am. J., 32,
760–765, 1968.
Kutílek, M. and Nielsen, D. R.: Soil hydrology: texbook for students of soil science, agriculture, forestry, geoecology, hydrology, geomorphology and other related disciplines, Catena Verlag,
ISBN 9783923381265, 1994.
Lebeau, M. and Konrad, J.-M.: A new capillary and thin film
flow model for predicting the hydraulic conductivity of unsaturated porous media, Water Resour. Res., 46, W12554,
https://doi.org/10.1029/2010WR009092, 2010.
Madi, R., de Rooij, G. H., Mielenz, H., and Mai, J.: Parametric soil
water retention models: a critical evaluation of expressions forthe full moisture range, Hydrol. Earth Syst. Sci., 22, 1193–1219,
https://doi.org/10.5194/hess-22-1193-2018, 2018.
Millington, R. J. and Quirk, J. P.: Permeability of porous solids, T.
Faraday Soc., 57, 1200–1207, 1961.
Mishra, S. and Parker, J. C.: On the relation between saturated
conductivity and capillary retention characteristics, Groundwater, 28, 775–777, 1990.
Mualem, Y.: A new model for predicting the hydraulic conductivityof unsaturated porous media, Water Resour. Res., 12, 513–522,
1976a.
Mualem, Y.: A catalog of the hydraulic properties of unsaturated
soils (Tech. Rep), Technion – Israel Institute of Technology,
1976b.
Mualem, Y.: Hydraulic conductivity of unsaturated soils:
Prediction and formulas, Methods of Soil Analysis:
Part 1 Physical and Mineralogical Methods, 5, 799–822,
https://doi.org/10.2136/sssabookser5.1.2ed.c31, 1986.
Nasta, P., Vrugt, J. A., and Romano, N.: Prediction of the saturated
hydraulic conductivity from Brooks and Corey’s water retention
parameters, Water Resour. Res., 49, 2918–2925, 2013.
Nielsen, D. R., Kirkham, D., and Perrier, E. R.: Soil capillary conductivity: Comparison of measured and calculated values, Soil
Sci. Soc. Am. J., 24, 157–160, 1960.
Nielson, D. R., Biggar, J. W., and Erh, K. T.: Spatial variabilityof field-measured soil-water properties, Hilgardia, 42, 215–259,
1973.
Nimmo, J. R.: The processes of preferential flow in the unsaturated
zone, Soil Sci. Soc. Am. J., 85, 1–27, 2021.

https://doi.org/10.5194/hess-27-1565-2023

1581

Pachepsky, Y., Scherbakov, R., Varallyay, G., and Rajkai, K.: On
obtaining soil hydraulic conductivity curves from water retention
curves, Pochvovedenie, 10, 60–72, 1984 (in Russian).
Peters, A.: Simple consistent models for water retention and hydraulic conductivity in the complete moisture range, Water Resour. Res., 49, 6765–6780, https://doi.org/10.1002/wrcr.20548,
2013.
Peters, A.: Reply to comment by S. Iden and W. Durner on “Simple
consistent models for water retention and hydraulic conductivity
in the complete moisture range”, Water Resour. Res., 50, 7535–
7539, https://doi.org/10.1002/2014WR016107, 2014.
Peters, A. and Durner, W.: A simple model for describing hydraulic conductivity in unsaturated porous media accounting
for film and capillary flow, Water Resour. Res., 44,W11417,
https://doi.org/10.1029/2008WR007136, 2008a.
Peters, A. and Durner, W.: Simplified evaporation method for determining soil hydraulic properties, J. Hydrol., 356, 147–162,
https://doi.org/10.1016/j.jhydrol.2008.04.016, 2008b.
Peters, A. and Durner, W.: Reply to comment by N. Shokri
and D. Or on “A simple model for describing hydraulicconductivity in unsaturated porous media accounting forfilm and capillary flow”, Water Resour. Res., 46, W06802,
https://doi.org/10.1029/2010WR009181, 2010.
Peters, A. and Durner, W.: SHYPFIT 2.0 User’s Manual. Research
Report. Institut für Ökologie, Technische Universität Berlin, Germany, 2015.
Peters, A., Durner, W., and Wessolek, G.: Consistent parameter
constraints for soil hydraulic functions, Adv. Water Resour., 34,
1352–1365, 2011.
Peters, A., Iden, S. C., and Durner, W.: Revisiting the simplified
evaporation method: Identification of hydraulic functions considering vapor, film and corner flow, J. Hydrol., 527, 531–542,
https://doi.org/10.1016/j.jhydrol.2015.05.020, 2015.
Peters, A., Iden, S. C., and Durner, W.: Local Solute Sinks andSources Cause Erroneous Dispersion Fluxes in Transport Simulations with the Convection–Dispersion Equation, Vadose Zone
J., 18, 190064, https://doi.org/10.2136/vzj2019.06.0064, 2019.
Peters, A., Hohenbrink, T. L., Iden, S. C., and Durner, W.: A
simple model to predict hydraulic conductivity in medium todry soil from the water retention curve, Water Resour. Res.,57, e2020WR029211, https://doi.org/10.1029/2020WR029211,
2021.
Reck, A., Jackisch, C., Hohenbrink, T. L., Schröder, B.,
Zangerlé, A., and van Schaik, L.: Impact of Temporal
Macropore Dynamics on Infiltration: Field Experiments
and Model Simulations, Vadose Zone J., 17, 170147,
https://doi.org/10.2136/vzj2017.08.0147, 2018.
Romano, N., Nasta, P., Severino, G., and Hopmans, J. W.:
Using Bimodal Lognormal Functions to Describe Soil Hydraulic Properties, Soil Sci. Soc. Am. J., 75, 468–480,
https://doi.org/10.2136/sssaj2010.0084, 2011.
Saito, H., Šimůnek, J., and Mohanty, B. P.: Numerical analysis ofcoupled water, vapor, and heat transport in the vadose zone, Vadose Zone J., 5, 784–800, 2006.
Sarkar, S., Germer, K., Maity, R., and Durner, W.: Measuring nearsaturated hydraulic conductivity of soils by quasi unit-gradientpercolation – 2. Application of the methodology, J. Plant Nutr.
Soil Sc., 182, 535–540, https://doi.org/10.1002/jpln.201800383,
2019.

Hydrol. Earth Syst. Sci., 27, 1565–1582, 2023

1582

A. Peters et al.: Prediction of the absolute hydraulic conductivity function from soil water retention data

Schaap, M. G. and Leij, F. J.: Improved prediction of unsaturated
hydraulic conductivity with the Mualem-van Genuchten model,
Soil Sci. Soc. Am. J., 64, 843–851, 2000.
Schaap, M. G. and Van Genuchten, M. Th.: A modified Mualem–van Genuchten formulation for improved description of the hydraulic conductivity near saturation, Vadose Zone J., 5, 27–34,
2006.
Schelle, H., Heise, L., Jänicke, K., and Durner, W.: Water retention
characteristics of soils over the whole moisture range: A comparison of laboratory methods, Eur. J. Soil. Sci., 64, 814–821,
2013.
Schindler, U.: Ein Schnellverfahren zur Messung der Wasserleitfähigkeit im teilgesättigten Boden an Stechzylinderproben, Arch.
Acker-u. Pflanzenbau u. Bodenkd. Berlin, 24, 1–7, 1980.
Schneider, M. and Goss, K.-U.: Prediction of the water sorption isotherm in air dry soils, Geoderma, 170, 64–69,
https://doi.org/10.1016/j.geoderma.2011.10.008, 2012.
Tuller, M. and Or, D.: Hydraulic conductivity of variably
saturated porous media: Film and corner flow in angular pore space, Water Resour. Res., 37, 1257–1276,
https://doi.org/10.1029/2000WR900328, 2001.
Tokunaga, T. K.: Hydraulic properties of adsorbed water films in
unsaturated porous media, Water Resour. Res., 45, W06415,
https://doi.org/10.1029/2009WR007734, 2009.

Hydrol. Earth Syst. Sci., 27, 1565–1582, 2023

Usowicz, B. and Lipiec, J.: Spatial variability of saturated hydraulicconductivity and its links with other soil properties at the regional
scale, Sci. Rep., 11, 1–12, 2021.
van Genuchten, M. Th.: A closed-form equation for predicting thehydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J.,
44, 892–898, 1980.
van Schaik, L., Hendriks, R. F. A., and Jvan Dam, J.: Parameterization of macropore flow using dye-tracer infiltration
patterns in the SWAP model, Vadose Zone J., 9, 95–106,
https://doi.org/10.2136/vzj2009.0031, 2010.
Vogel, T., Van Genuchten, M. T., and Cislerova, M.: Effect of the
shape of the soil hydraulic functions near saturation on variablysaturated flow predictions, Adv. Water Resour., 24, 133–144,
2000.
Weber, T. K., Durner, W., Streck, T., and Diamantopoulos, E.:
A modular framework for modeling unsaturated soil hydraulicproperties over the full moisture range, Water Resour. Res., 55,
4994–5011, 2019.
Zhang, Z. F.: Soil water retention and relative permeability for conditions from oven-dry to full saturation, Vadose Zone J., 10,
1299–1308, https://doi.org/10.2136/vzj2011.0019, 2011.

https://doi.org/10.5194/hess-27-1565-2023