the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Prediction of the absolute hydraulic conductivity function from soil water retention data

### Andre Peters

### Tobias L. Hohenbrink

### Sascha C. Iden

### Martinus Th. van Genuchten

### Wolfgang Durner

For modeling flow and transport processes in the soil–plant–atmosphere system, knowledge of the unsaturated hydraulic properties in functional form is mandatory. While much data are available for the water retention function, the hydraulic 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, *K*_{s}. In this paper we highlight the shortcomings of this approach, namely, that measured *K*_{s} 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 *K*_{s} as a scaling factor with a generally applicable effective saturated tortuosity parameter *τ*_{s} 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 *τ*_{s} was derived by fitting the new formulation to 12 data sets containing the relevant information. *τ*_{s} 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 an order 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.

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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. Classical models for the soil water retention curve (WRC) (e.g., van Genuchten, 1980; Kosugi, 1996) and the related hydraulic conductivity 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 not held by capillary forces. The non-capillary parts of the WRC and HCC become dominant when soils become dry (Iden et 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 flow ceases, and vapor flow becomes the dominant transport process. Isothermal diffusion of water vapor can be expressed in terms of an equivalent hydraulic conductivity and incorporated into an effective conductivity function (Peters, 2013). The total hydraulic conductivity can then be expressed as the sum of three components:

where *h* [m] is the suction (i.e., the absolute value of the matric head or
pressure head); *K* [m s^{−1}] is the total hydraulic conductivity; and
*K*_{c}, *K*_{nc}, and *K*_{v} [m s^{−1}] are
the hydraulic conductivity components for capillary and non-capillary flow
of liquid water and water vapor diffusion in the soil gas phase,
respectively. Under isothermal conditions, the function *K*_{v}(*h*)
can be predicted easily from the temperature-dependent diffusion coefficient
of water vapor in air and the WRC (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 conceptual model of Peters (2013)
to obtain a simple prediction scheme for the absolute non-capillary
conductivity function *K*_{nc}(*h*).

Several models have been proposed to estimate the capillary conductivity
function *K*_{c}(*h*) from conceptualizations of the pore space
involving tortuous and interconnected pore bundles, 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
distribution of 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 hydraulic conductivity function *K*_{rc}(*h*) and
scale it by fitting the function to one or more measured conductivity
points. Most commonly, the measured saturated conductivity is used for this
purpose. A comprehensive overview of these models is given by Mualem (1986).
More recently, concepts to predict the saturated hydraulic conductivity
*K*_{s} [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 *K*_{s}
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 not
possible. This is frequently the case. Second, if only measurements of
*K*_{s} are available, the unsaturated conductivity estimates will
be greatly affected by the dominant influence of structural pores on the
variability of *K*_{s}. Thirdly, even if unsaturated conductivity
data are available, the conductivity function near saturation may not be
represented well. The latter two problems are outlined below.

The problem of scaling *K*_{rc}(*h*) by *K*_{s} stems from
the influence of soil structure on the hydraulic conductivity at or near
saturation. For more than 50 years, *K*_{s} has been known to
vary over many orders of magnitude, even at the same site with a rather
homogeneous texture (Nielsen et al., 1973; Kutílek and Nielsen, 1994, p. 249). If soil structure is not properly reflected in the WRC near full
saturation, scaling *K*_{rc}(*h*) with a measured *K*_{s}
can lead to severe overestimation of conductivity in the medium moisture
range (Durner, 1992, 1994; Schaap and Leij, 2000). We exemplarily illustrate
this problem in Fig. 1, top, for a sandy soil. The average difference
between data and model in the unsaturated region in this example is about
1 order of magnitude.

A better choice is therefore to use unsaturated conductivity data to scale
the relative conductivity curve, as already proposed by Nielsen et al. (1960). However, such data are often not available, especially if the
measurements were made in the past when more recent techniques such as the
simplified evaporation method (SEM) (Schindler, 1980; Peters and Durner,
2008b, Peters et al., 2015) were not available. Moreover, the SEM typically
yields information only in a limited 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 $\mathrm{1.7}\times {\mathrm{10}}^{-\mathrm{7}}$ 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
*K*_{c}(*h*) in Eq. (1) from the WRC and thus to
circumvent a need for scaling of the relative hydraulic conductivity
function *K*_{rc}(*h*) with measured conductivity data. The paper is
organized as follows. First, we recall the basic model concept to
characterize the capillary and non-capillary pore water components of the
hydraulic conductivity in a soil. This is followed by a brief 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
*K*_{c}(*h*) from the WRC. The combination of this
model with previously developed models for predicting the complete
*K*_{nc}(*h*) and *K*_{v}(*h*) yields a soil hydraulic
conductivity function that is predicted from 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 using four
different parametrizations of the WRC and discuss the accuracy of the
conductivity estimates.

The unsaturated hydraulic conductivity function covering wet and dry
conditions can be expressed by summing up a capillary 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 *K*_{v}(*h*) can be
predicted from 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 *K*_{nc}(*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 scaled by matching measured
conductivity data. In this contribution, 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 of the pore
bundle models. To provide a clear understanding of 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 Parametrizing capillary and non-capillary pore water 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 in terms of sums of capillary and non-capillary
components. The WRC is formulated as a superposition of a capillary
saturation function *S*_{c} [–] and a non-capillary saturation
function *S*_{nc} [–] (Iden and Durner, 2014):

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*
[m^{3} m^{−3}] is the total water content, and *θ*_{s}
[m^{3} m^{−3}] and *θ*_{r} [m^{3} 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 Γ(h) is scaled by the following (Iden and Durner, 2014):

with *h*_{0} [m] being the suction head at oven dryness, which can be set to 10^{4.8} m (Schneider and Goss, 2012). Γ(h) can be 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, liquid flow 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 Weber et al. (2019).

In the original version, both the capillary and non-capillary parts of the
conductivity function needed to be scaled by matching the conductivity
function to measured conductivity data. Recently, Peters et al. (2021)
improved the model by integrating an absolute prediction of the
non-capillary liquid conductivity *K*_{nc}(*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 *K*_{sc} was required
for the capillary conductivity component in Eq. (1):

Since *K*_{sc} is orders of magnitude higher than the
non-capillary and vapor conductivity components, *K*_{sc} 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 Relative conductivity predictions using capillary bundle models

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 to a bundle of capillaries with a size
distribution that is consistent with the pore-size distribution of the
medium along with some assumptions about pore connectivity and tortuosity to
arrive at a mathematical description of the HCC. The water flux in a single
capillary under unit-gradient conditions, *Q*_{c1} [m^{3} s^{−1}], can be described with the law of Hagen–Poiseuille:

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 have a circular
cross-section. Relating *Q*_{c1} to the cross-sectional area of the
capillary yields the flux density or simply the hydraulic conductivity [m s^{−1}] assuming unit gradient conditions:

If the porous medium is regarded as a bundle of parallel capillaries of
different sizes, the hydraulic conductivity can be described as the sum of
the unit-gradient fluxes of the single water-filled capillaries, divided by
the sum of their cross-sectional areas, and corrected with the macroscopic
capillary water content, *θ*_{c} [m^{3} 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):

where *r*_{m} [m] is the maximum radius of the water-filled pores,
and *f*_{k}(r) is the pore-radius distribution. The
pore-radius distribution is related to the pore-volume distribution
*f*_{p}(*r*), reflecting volumetric fractions by ${f}_{k}\propto {f}_{\mathrm{p}}/{r}^{\mathrm{2}}$, which leads to

Since $\underset{\mathrm{0}}{\overset{{r}_{\mathrm{m}}}{\int}}{f}_{\mathrm{p}}\left(r\right)\mathrm{d}r={\mathit{\theta}}_{\mathrm{c}}$ (the capillary water content), this simplifies to

Applying the Young–Laplace relation $r=\mathrm{2}\mathit{\sigma}/\mathit{\rho}gh$, in
which *σ* [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:

where $\stackrel{\mathrm{\u0303}}{{\mathit{\theta}}_{\mathrm{c}}}$ is the dummy variable of integration.

Several factors distinguish a porous medium from a bundle of parallel tubes.
They can be accounted for mostly by implementing a tortuosity–connectivity
correction. The tortuosity describes the effect of the path length of a
single water molecule, *l*_{p} [m], being longer than a straight
line *l* [m]. The factor of path extension is then given by ${l}_{\mathrm{p}}/l\phantom{\rule{0.125em}{0ex}}$ [–]. 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 $\mathit{\tau}\phantom{\rule{0.125em}{0ex}}(\mathrm{0}<\mathit{\tau}<\mathrm{1})$ [–]:

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, the tortuosity 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 $\mathit{\beta}={\mathit{\sigma}}^{\mathrm{2}}/\left(\mathrm{2}\mathit{\eta}\mathit{\rho}g\right)$
[m^{3} s^{−1}] and considering the tortuosity correction *τ*(*θ*_{c}) leads to

Values of the physical constants used in this study are summarized in Table 1. In SI units, $\mathit{\beta}=\mathrm{3.04}\times {\mathrm{10}}^{-\mathrm{4}}$ m^{3} s^{−1} at
20 ^{∘}C.

Equation (12) is similar to the formulation by Nasta et al. (2013), who used
the same approach to predict the saturated conductivity from the WRC of
Brooks and Corey (1964). They optimized for this purpose the value of *τ* at saturation by fitting their model to measured *K*_{s} data
from 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:

in which *τ*_{s}=*τ*(*θ*_{s}). Since the degree of capillary
saturation, *S*_{c}, is given by ${S}_{\mathrm{c}}={\mathit{\theta}}_{\mathrm{c}}/({\mathit{\theta}}_{\mathrm{s}}-{\mathit{\theta}}_{\mathrm{r}})$,
${\int}_{\mathrm{0}}^{{\mathit{\theta}}_{\mathrm{c}}}{h}^{-\mathrm{2}}\mathrm{d}\stackrel{\mathrm{\u0303}}{{\mathit{\theta}}_{\mathrm{c}}}=({\mathit{\theta}}_{\mathrm{s}}-{\mathit{\theta}}_{\mathrm{r}}){\int}_{\mathrm{0}}^{{S}_{\mathrm{c}}}{h}^{-\mathrm{2}}\mathrm{d}\stackrel{\mathrm{\u0303}}{{S}_{\mathrm{c}}}$ and hence

with the relative tortuosity factor ${\mathit{\tau}}_{\mathrm{r}}=\mathit{\tau}/{\mathit{\tau}}_{\mathrm{s}}$. Note that the solution is similar for the classic (“non-PDI”) scheme, for which effective saturation is defined as ${S}_{\mathrm{e}}=\left(\mathit{\theta}-{\mathit{\theta}}_{\mathrm{r}}\right)/\left({\mathit{\theta}}_{\mathrm{s}}-{\mathit{\theta}}_{\mathrm{r}}\right)$. In this case we obtain ${\int}_{{\mathit{\theta}}_{\mathrm{r}}}^{\mathit{\theta}}{h}^{-\mathrm{2}}\mathrm{d}\stackrel{\mathrm{\u0303}}{\mathit{\theta}}=({\mathit{\theta}}_{\mathrm{s}}-{\mathit{\theta}}_{\mathrm{r}}){\int}_{\mathrm{0}}^{{S}_{\mathrm{e}}}{h}^{-\mathrm{2}}\mathrm{d}\stackrel{\mathrm{\u0303}}{{S}_{e}}$. Burdine (1953) suggested that the tortuosity (${l}_{\mathrm{p}}/l)$ is inversely related to the capillary saturation, leading to ${\mathit{\tau}}_{\mathrm{r}}\left({S}_{\mathrm{c}}\right)={S}_{\mathrm{c}}^{\mathrm{2}}$ and hence

In a more sophisticated approach, Mualem (1976a) followed the cut-and-random-rejoin model approach of Childs and Collis-George (1950) (CCG) and refined the model using the assumption 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 considering a saturation-dependent tortuosity correction ${S}_{\mathrm{c}}^{\mathit{\lambda}}$, the expression for the capillary conductivity function became (Mualem, 1976a)

with *λ* [–] as the tortuosity and connectivity factor. Applying his
model to a variety of data, Mualem found empirically that *λ*≈
0.5.

## 2.3 Absolute hydraulic conductivity prediction

For the reasons stated in the Introduction, it is preferable to predict the
absolute capillary conductivity function *K*_{c}(*h*) from the WRC
rather than calculating the relative function *K*_{rc}(*h*) and
scaling it with measured conductivity data. In this paper, we use the Mualem (1976a) model to derive the shape of the capillary conductivity function.
Our concept keeps the dependency of the relative tortuosity factor on
saturation in the original formulation of Mualem (1976a); that is, ${\mathit{\tau}}_{\mathrm{r}}=\mathit{\tau}/{\mathit{\tau}}_{\mathrm{s}}={S}_{\mathrm{c}}^{\mathit{\lambda}}$, which becomes unity at full
saturation. However, instead of following Mualem's original concept of first
normalizing the prediction integral and then scaling it with measured
conductivity values, we predict the absolute *K*_{c}(*h*) from the
WRC by introducing an absolute tortuosity coefficient, *τ*(*S*_{c}), which is given by the product of a relative and a
saturated tortuosity coefficient *τ*_{s}:

By inserting this tortuosity expression into Eq. (12), by using Mualem's integral (occurring in Eq. 16), and by applying the substitution ${\int}_{\mathrm{0}}^{{\mathit{\theta}}_{\mathrm{c}}}{h}^{-\mathrm{2}}\mathrm{d}\stackrel{\mathrm{\u0303}}{\mathit{\theta}}=({\mathit{\theta}}_{\mathrm{s}}-{\mathit{\theta}}_{\mathrm{r}}){\int}_{\mathrm{0}}^{{S}_{\mathrm{c}}}{h}^{-\mathrm{2}}\mathrm{d}\stackrel{\mathrm{\u0303}}{{S}_{\mathrm{c}}}$, we obtain the following equation for the capillary conductivity function:

Expressing the Mualem integral by ${\left[{\int}_{\mathrm{0}}^{{S}_{\mathrm{c}}}{h}^{-\mathrm{1}}\mathrm{d}{S}_{\mathrm{c}}\right]}^{\mathrm{2}}={\left[F\left(\mathrm{\Gamma}\right)-F\left({\mathrm{\Gamma}}_{\mathrm{0}}\right)\right]}^{\mathrm{2}}$,
where *F* is the solution of the indefinite integral $\int \frac{\mathrm{1}}{h\left(\mathrm{\Gamma}\right)}\mathrm{d}\mathrm{\Gamma}$ (Peters, 2014), leads to

In this model, *τ*_{s} is a new factor which scales the
capillary conductivity function. We hypothesize that *τ*_{s}
varies only moderately among different textures and that a universal value
can be determined from experimental data. If *τ*_{s} is known
and *λ* is set to Mualem's suggested value of 0.5, all three
components of the HCC given by Eq. (1) can be calculated based on the WRC
without the need for measured conductivity values.

The parameter *K*_{s} (in the classic “Non-PDI” scheme
neglecting non-capillary processes) or *K*_{sc} (within the PDI
system) of Eq. (16) is related to *τ*_{s} of Eq. (19) by

where ${\left[F\left(\mathrm{1}\right)-F\left({\mathrm{\Gamma}}_{\mathrm{0}}\right)\right]}^{\mathrm{2}}$ is the PDI formulation of the denominator in Eq. (16) (Peters, 2014).

The hydraulic tortuosity of saturated porous materials has long been
investigated using a variety of experimental and theoretical approaches. The
earliest description of hydraulic tortuosity was introduced by Carman
(1937), who modified the Kozeny (1927) equation for the saturated
permeability. Using experimental data, Carman found that ${\mathit{\tau}}_{\mathrm{s}}\approx \mathrm{2}/\mathrm{5}$ for a wide range of porosities. However, many found later that the
saturated tortuosity is variable and depends on porosity and texture. We
refer to Ghanbarian et al. (2013) for an overview of theoretical and
experimental studies about this relationship. Most of the derived values for
*τ*_{s} 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
soil-related 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 from the 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 *τ*_{s} that lumps
all these effects. The hypothesis that *τ*_{s} 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”, *τ*_{s}, for a
(virtual) porous system without structural pores.

## 2.4 Connecting the capillary conductivity function with different 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 (van Genuchten, 1980) with the usual constraint
($m=\mathrm{1}-\mathrm{1}/n$) and also in
unconstrained form (*m* independent from *n*), and the Fredlund and Xing
(1994) saturation function. The latter is the function given in the last row
of Table 2. The models are referred to as Kos-PDI, vGc-PDI, vGmn-PDI, and
FX-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 be evaluated using numerical integration. We chose these four
functions as they are the most commonly used functions in the field of
soil hydrology and geotechnics.

Although the derivation of *K*_{c}(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-capillary components. For those cases, the
expression for the integral *F*(Γ_{0}) is
zero. For the original van Genuchten–Mualem model with constraint
$m=\mathrm{1}-\frac{\mathrm{1}}{n}$, one obtains, for example,

where *S*_{e} is the effective saturation function
(${S}_{\mathrm{e}}=\left(\mathit{\theta}-{\mathit{\theta}}_{\mathrm{r}}\right)/\left({\mathit{\theta}}_{\mathrm{s}}-{\mathit{\theta}}_{\mathrm{r}}\right))$, or simply

## 3.1 Soil hydraulic models

The water retention and unsaturated hydraulic conductivity functions were
described using the PDI model system with four 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, *K*_{c}(*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), $\mathit{\beta}=\mathrm{3.04}\times {\mathrm{10}}^{-\mathrm{4}}$ m^{3} s^{−1} (Sect. 2.2, Table 1), and *τ*_{s} is the
new unknown tortuosity parameter. *K*_{nc} and *K*_{v}
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 with water retention models that
gradually approach saturation can produce a non-physical sharp decrease in
the hydraulic conductivity near saturation (e.g., Vogel et al., 2000;
Ippisch et 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, *h*_{crit}. According to Jarvis (2007), we use
*h*_{crit}= 0.06 m, corresponding to an equivalent diameter of
0.5 mm (see also Sect. 2.4).

## 3.2 Estimating the saturated tortuosity coefficient, *τ*_{s}

To obtain an estimate for *τ*_{s}, reliable data for the WRC
and in 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 of different soils. Details about the soils are given in the
original literature and are summarized in Table 3. For each of the four PDI
combinations with the capillary saturation functions given in Table 2, we
determined a value for *τ*_{s} by fitting them to the 12 data sets and
estimating the WRC parameters and *τ*_{s}. The median values of
the estimated *τ*_{s} values were used in the corresponding
prediction models.

## 3.3 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 and could 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 range of textural classes. Some of
them stemmed from soil columns taken at the same sites at which the soil
columns of 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):

where *θ*_{i} and *θ*_{mod,i} are the measured and
modeled water contents, *K*_{i} and ${\stackrel{\mathrm{\u0303}}{K}}_{i}$ are measured and modeled
hydraulic conductivities, *n*_{θ} and *n*_{K} are the respective
number of data points, *w*_{θ}=10 000 and *w*_{K}=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

*τ*

_{s}, the parameter vector contained all adjustable parameters of the water retention function plus

*τ*

_{s}. In the case of the hydraulic conductivity predictions,

*w*

_{K}was set to 0, and the estimated parameter vector contained only the WRC parameters. The performance of the different approaches was compared in terms of the root-mean-square error (RMSE) for the WRC and the HCC (common log of

*K*(

*h*)), respectively. Model comparisons were based on the Akaike information criterion, AICc, corrected for small sample sizes (Hurvich and Tsai, 1989).

## 4.1 Empirical estimate of the saturated tortuosity coefficient *τ*_{s}

Figure 2 illustrates, using 4 of the 12 calibration data sets described
in Table 3, the fitted water retention and conductivity functions for the
four basic hydraulic models listed in Table 2. A full overview on all
calibration data sets and the fitted 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 can
be hardly distinguished on the plots since they lie largely on top of each
other. Figure 3 shows the corresponding distributions of RMSE_{θ} and
RMSE_{logK}, 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 compared to those with five free
parameters (vGc-PDI and Kos-PDI). We note that fitting the *K* functions of
Peters et al. (2021), using *K*_{s} instead of *τ*_{s} as an
adjustable parameter and leaving all other settings identical, would lead
here exactly to the same results.

The distributions of the resulting values for *τ*_{s} for the four
models are shown in Fig. 4. The median values for *τ*_{s} 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 that the
two best-fitting WRC models yield almost identical estimates of *τ*_{s}.
The range of *τ*_{s} for the 12 data sets spanned less than 1.5 orders
of magnitude. We interpret this as an indication that the hypothesis of
relatively moderate overall variability in *τ*_{s} may be justified.
When fitting the classic PDI scheme (with *K*_{s} as a fitting parameter) to
these data, which do not include measured conductivity data at saturation,
the estimated *K*_{s} values varied by more than 3 orders of magnitude. In
natural soils, the measured *K*_{s} values can vary even more due to the
dominance of (texture-independent) structural pores and macropores on
*K*_{s} (e.g., Usowicz and Lipiec, 2021).

## 4.2 Tests of the absolute conductivity predictions

By using median values of *τ*_{s} 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 hydraulic conductivity functions
from the water retention functions for 23 test data sets. In Fig. 5, we show
the resulting distributions of RMSE_{θ} (fitted) and RMSE_{logK}
(predicted). Since measured conductivities were available primarily within
the range where the capillary conductivity component dominates,
RMSE_{logK} can be interpreted as an approximate error of the capillary
conductivity prediction. The medians of RMSE_{logK} for the Kos-PDI and
vGc-PDI models were 0.71 and 0.67, respectively. Combinations with the more
flexible retention models yielded median RMSE_{logK} values of 0.49 for
vGmn-PDI and 0.40 for FX-PDI. To test whether conductivity predictions were
biased, we calculated also the mean error (Fig. 6). For the FX-PDI model,
the median was close to zero, indicating an unbiased conductivity
prediction, whereas the other models tended to underestimate the
conductivity data.

Figure 7 shows fitted WRCs and predicted HCCs along with the measured conductivity data. Due to space limitations, only a subset of six randomly selected cases is shown for 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 new scheme is valuable not only for cases where no or insufficient information about the conductivity is available. It is also useful when data are available for the unsaturated hydraulic conductivity but are missing in the wet range. This is 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 in the wet moisture range; thus, an unrealistic conductivity extrapolation might result (see Fig. 1, bottom). In such cases, *λ* might be estimated, and only *τ*_{s} 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, but instead of Eq. (16), we now use Eq. (18) with *τ*_{s}=0.094. Now, the model is well able to be fitted to the data, and the hydraulic conductivity close to saturation is more reasonably predicted as in Fig. 1: predicted conductivity at saturation is $\mathrm{1.7}\times {\mathrm{10}}^{-\mathrm{7}}$ m s^{−1} (or 1.5 cm d^{−1}) for the original and $\mathrm{1.3}\times {\mathrm{10}}^{-\mathrm{5}}$ m s^{−1} (or 112 cm d^{−1}) for the new scheme. Note that the new scheme has one less adjustable parameter.

## 4.4 Considerations of the hydraulic conductivity at saturation

Because the saturated hydraulic conductivity is relatively easy to measure,
many determine *K*_{s} experimentally. As emphasized earlier, the use of
*K*_{s} for scaling the relative hydraulic conductivity function
should be avoided as much as possible. Still, *K*_{s} provides
valuable information for the hydraulic 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. Predictions of 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 *h*_{crit} = 0.06 m the “saturated matrix
conductivity” (*K*_{s,matrix}). Knowledge of
*K*_{s,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 macropore-affected range
cannot be predicted using capillary bundle models (Durner, 1994). Thus, it
is preferable to cover the region between *K*_{s,matrix} and the
measured *K*_{s} using some interpolation function such as proposed
by Schaap and van Genuchten (2006). Using the abovementioned value of 0.06 m for *h*_{crit} as a starting point for the interpolation and
assuming that the saturated conductivity (*K*_{s}) is reached at a
pore diameter of 5 mm (i.e., at *h*_{s}= 0.006 m), we can
formalize the interpolation as

As an example, we illustrate the interpolation with a simple smooth cosine interpolation function, with the log of the suction in the argument (Fig. 9). Mathematically, this interpolation is expressed as

with the transformed variables *y*=log (K) and
*x*=log(*h*) and consideration of the corresponding subscripts. We
note that the real course of the *K*(*h*) function in this moisture region
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, 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 from oven dryness to full saturation.

The unsaturated hydraulic conductivity of soils is still the most 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, such predictions 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, non-capillary, 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 is called 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 shown to vary little among different soils, and we have determined a universal value empirically from data. The new scheme 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) basic saturation function and the model of van Genuchten
with independent parameters *m* and *n* as basic function performed best.
The identified value for the saturated tortuosity coefficient *τ*_{s}
was 0.095. From a practical point of view, *τ*_{s} may simply be set to
0.1. The prediction accuracy with the new model was tested using a set of 23
soils for which measured *K* values were available. For the best-performing
model FX-PDI, the predictions matched the measured data on average with a
RMSE_{logK} of about 0.4, without a bias between the predicted 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 *K*_{s} is available and where soil
structure plays a role (which is the rule for most topsoils). In such cases,
the predicted HCC can be combined with an interpolation towards *K*_{s} to
obtain a well-estimated conductivity function over the full moisture range.
Differentiating between structural and textural effects 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 *τ*_{s} can be fitted by adjusting 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 can be implemented easily in the standard
optimization software packages.

## A1 The PDI model system

### A1.1 PDI water retention function

The capillary saturation function *S*_{c} [–] and a non-capillary
saturation function *S*_{nc} [–] may be superimposed in the following form
(Iden and Durner, 2014):

in which the first right-hand term holds for water stored in capillaries, and the
second term holds for water stored in adsorbed water films and pore corners.
*θ* [m^{3} m^{−3}] is the total water content, *h* [m] is the
suction head, and *θ*_{s} [m^{3} m^{−3}] and *θ*_{r} [m^{3} 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 Γ(h) is scaled by (Iden
and Durner, 2014)

with *h*_{0} [m] being the suction head at oven dryness, which can
be set to 10^{4.8} m following Schneider and Goss (2012). Γ(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 et al. (2021):

in which the parameter *h*_{a} [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 ${h}_{a}={\mathit{\alpha}}^{-\mathrm{1}}$ 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 ${h}_{a}={\mathit{\alpha}}^{-\mathrm{1}}$ would lead in such cases to
unrealistic retention functions with *S*_{nc} being close to unity,
whereas *S*_{c} is already close to zero. The calculation scheme
for *h*_{a} as a quantile of *S*_{c} is given in Appendix A2. The parameter *h*_{0} in Eq. (A3) is the suction head where the
water content reaches zero, which reflects the suction at oven-dry
conditions. *S*_{nc}(h) increases linearly from zero
at oven dryness to its maximum value of 1.0 at *h*_{a}, and it then
remains constant towards saturation. In order to ensure a continuously
differentiable water capacity function, *S*_{nc}(h)
must be smoothed around *h*_{a}, which is achieved by the smoothing
parameter *b* [–] (Iden and Durner, 2014), given here by

where *b*_{o}=0.1 ln (10), and ${b}_{\mathrm{1}}={\left(\frac{{\mathit{\theta}}_{\mathrm{r}}}{{\mathit{\theta}}_{\mathrm{s}}-{\mathit{\theta}}_{\mathrm{r}}}\right)}^{\mathrm{2}}$.

### A1.2 PDI hydraulic conductivity

The PDI hydraulic conductivity model is expressed as (Peters et al., 2021)

where *K*_{r,c} [–] is the relative conductivity for the capillary
component, *K*_{s,c} [m s^{−1}] is the saturated conductivity for
the capillary components, and *K*_{nc} and *K*_{v} [m s^{−1}] are the non-capillary and isothermal vapor conductivities,
respectively. *K*_{nc} is given by (Peters et al., 2021)

in which *c* is used to account for several physical and geometrical
constants and being either a free fitting parameter to scale
*K*_{nc} or $c\phantom{\rule{0.125em}{0ex}}=\mathrm{1.35}\times {\mathrm{10}}^{-\mathrm{8}}$ m${}^{\mathrm{5}/\mathrm{2}}$ s^{−1}. Parameter
*θ*_{m} [–] is the water content at *h*=10^{3} m. We refer to Saito
et al. (2006) or Peters (2013) for details regarding the formulation of
*K*_{v} as a function of the invoked WRC. Note that the capillary
liquid 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 is in this paper described using the pore bundle model of Mualem (1976a), which reads in the PDI notation (Peters, 2014) as

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

## A2 Calculation of *h*_{a} as a function of the *S*_{c} quantile

Peters (2013) proposed two methods to define the critical tension *h*_{a}
(m) for the non-capillary saturation function *S*_{nc} (–). He
decided to set ${h}_{a}={\mathit{\alpha}}^{-\mathrm{1}}$ for van Genuchten's model and
*h*_{a}=*h*_{m} for Kosugi's model (1996). His second
option was to define *h*_{a} 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

Recall that this function ensures a saturation of zero at the suction
corresponding to oven dryness, *h*_{0} (L). Iden and Durner (2014) proposed
to scale Eq. (A8) using the function

where ${\mathrm{\Gamma}}_{\mathrm{0}}=\mathrm{\Gamma}\left({h}_{\mathrm{0}}\right).$ According to Peters (2013), we define the suction *h*_{a} as

where *β* [–] represents the chosen quantile of *S*_{c}. Combining Eqs. (A8)–(A10) and solving for *h*_{a} yields

in which the constant *γ* is defined as

Applying the same approach to the capillary saturation function of Kosugi (1996), i.e.,

yields

Thirdly, for the capillary saturation of Fredlund and Xing (1994), given as

we obtain

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.

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

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.

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

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

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.

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. PE 1912/4-1).

This open-access publication was funded by Technische Universität Braunschweig.

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

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