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,

We propose to reformulate the unsaturated hydraulic conductivity function by replacing the soil-specific

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.

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:

Several models have been proposed to estimate the capillary conductivity
function

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

The problem of scaling

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

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.

The objective of this study was to develop a model which predicts the
absolute capillary conductivity function

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

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

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

Since

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,

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,

Since

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,

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

Physical constants at 20

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

in which

For the reasons stated in the Introduction, it is preferable to predict the
absolute capillary conductivity function

By inserting this tortuosity expression into Eq. (12), by using Mualem's
integral (occurring in Eq. 16), and by applying the substitution

Expressing the Mualem integral by

In this model,

The parameter

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

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

Dependent on the selected WRC parametrization,

Although the derivation of

Summary of the basic water retention functions used in the PDI
scheme (see Appendix A1) as well as the analytical solutions for

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,

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,

To obtain an estimate for

To test the predictions of

Calibration data sets used for estimating water retention curves
and the saturated tortuosity coefficient

Test data sets used to test the conductivity predictions.

The models were fitted to the data by minimizing the sum of weighted squared
residuals between modeled and measured data (Peters, 2013):

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

Plots of 4 of the 12 calibration data sets and the fitted water
retention and conductivity functions used to calibrate the saturated
tortuosity coefficient

Distributions of RMSE

The distributions of the resulting values for

Distribution of fitted values of

By using median values of

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.

RMSE^{®} default settings as 1.5 times the interquartile range away from the top or bottom of the box (

Mean errors of the predicted absolute conductivity based on the soil water retention function for 23 test data sets. Black dots indicate all 23 validation data sets; red dots indicate data sets shown in Fig. 7.

Measured data (dots), fitted retention functions

Several authors (e.g., Schaap and Leij, 2000; Peters et al., 2011) have
stressed that the tortuosity parameter

Same data as in Fig. 1c and d. A new scheme with vGmn as basic
retention function and

Because the saturated hydraulic conductivity is relatively easy to measure,
many determine

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

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

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

Interpolation scheme between the predicted capillary conductivity
(red dashed line) and the measured value of

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

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

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

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

In the cases where measured unsaturated conductivity data are available (such as
from the simplified evaporation method), the proposed model with fixed

The capillary saturation function

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):

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

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

Peters (2013) proposed two methods to define the critical tension

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,

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

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:

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.