Few parametric expressions for the soil water retention curve are suitable for dry conditions. Furthermore, expressions for the soil hydraulic conductivity curves associated with parametric retention functions can behave unrealistically near saturation. We developed a general criterion for water retention parameterizations that ensures physically plausible conductivity curves. Only 3 of the 18 tested parameterizations met this criterion without restrictions on the parameters of a popular conductivity curve parameterization. A fourth required one parameter to be fixed.

We estimated parameters by shuffled complex evolution (SCE) with the objective function tailored to various observation methods used to obtain retention curve data. We fitted the four parameterizations with physically plausible conductivities as well as the most widely used parameterization. The performance of the resulting 12 combinations of retention and conductivity curves was assessed in a numerical study with 751 days of semiarid atmospheric forcing applied to unvegetated, uniform, 1 m freely draining columns for four textures.

Choosing different parameterizations had a minor effect on evaporation, but cumulative bottom fluxes varied by up to an order of magnitude between them. This highlights the need for a careful selection of the soil hydraulic parameterization that ideally does not only rely on goodness of fit to static soil water retention data but also on hydraulic conductivity measurements.

Parameter fits for 21 soils showed that extrapolations into the dry range of the retention curve often became physically more realistic when the parameterization had a logarithmic dry branch, particularly in fine-textured soils where high residual water contents would otherwise be fitted.

The pore architecture of the soil influences its hydraulic behavior, typically described by two curves: the relationship between the amount of water present in the soil pores and the matric potential (termed soil water characteristic or soil water retention curve), and the relationship between the hydraulic conductivity and either matric potential or water content (the soil hydraulic conductivity curve). Numerical solvers of Richards' equation for water flow in unsaturated soils require these curves as descriptors of the soil in which the movement of water should be calculated. Many parametric expressions for the retention curve and fewer for the hydraulic conductivity have been developed for that purpose (see the Supplement, Leij et al., 1997; Cornelis et al., 2005; Durner and Flühler, 2005; Khlosi et al., 2008; Assouline and Or, 2013).

A brief overview of retention curve parameterizations is given in the following while the references to the parameterizations in question are given in the Supplement and Sect. 2, where their equations are presented. The earliest developed parameterizations focused primarily on the wet end of the curve since this is the most relevant section for agricultural production. Numerical models were struggling with the discontinuity of the first derivative at the air-entry value. Observations with methods relying on hydrostatic equilibrium (Klute, 1986, pp. 644–647) typically gave a more smooth shape around the matric potential where the soil started to desaturate as an artefact of the sample height, as was later demonstrated by Liu and Dane (1995). This led to the introduction of parameterizations that yielded a continuously differentiable curve.

The interest in the dry end of the retention curve was triggered by an increased interest in water scarcity issues (e.g., Scanlon et al., 2006; UN-Water, FAO, 2007; UNDP, 2006). For groundwater recharge under deep vadose zones, the dry end of the soil water retention curve affects both slow liquid water movement in film and corner flow (Tuller and Or, 2001; Lebeau and Konrad, 2010) and vapor phase transport (Barnes and Turner, 1998; de Vries and Simmers, 2002). The earlier parameterizations had an asymptote at a small (or at zero) water content. This often gave poor fits in the dry end, and several parameterizations emerged in which the dry branch was represented by a logarithmic function that reached zero water content at some point.

A nonparametric approach was advocated by Iden and Durner (2008). They estimated nodal values of volumetric water content from evaporation experiments and derived a smooth retention curve by cubic Hermite interpolation. They extrapolated the retention function to the dry range and computed a coupled conductivity function based on the Mualem model.

Liu and Dane (1995) were the first to point out that the smoothness of observed curves around the air-entry value could be an artefact related to experimental conditions. Furthermore, it became apparent that a particular parameterization that gave a differentiable curve led to unrealistically large increases of the soil hydraulic conductivity near saturation (Durner, 1994; Vogel et al., 2001). This was eventually linked to the nonzero slope at saturation (Ippisch et al., 2006), implying the existence of unphysically large pores with air-entry values up to zero. This led to the reintroduction of a discrete air-entry value.

Most of the parameterizations are empirical, curve-fitting equations (Kosugi et al., 2002). One exception is the very dry range, where measurement techniques are often not so reliable (e.g., Campbell and Shiozawa, 1992) and were not always employed. The proportionality of the water content in this range to the logarithm of the absolute value of the matric potential that has frequently been invoked conforms to the adsorption theory of Bradley (1936), which considers adsorbed molecules to build up in a film consisting of layers, with the net force of electrical attraction diminishing with every layer (Rossi and Nimmo, 1994).

The empirical power-law relationship between water content and matric potential introduced by Brooks and Corey (1964) was later given a theoretical foundation by Tyler and Wheatcraft (1990), who showed that the exponent was related to the fractal dimension of the Sierpenski carpet used to model the hierarchy of pore sizes occurring in the soil. The sigmoid shape of the Kosugi's (1996, 1999) retention curve was derived rigorously from an assumed lognormal distribution of effective pore sizes, making this the only parameterization discussed in this paper developed from a theoretical analysis.

Some soils have different types of pore spaces: one type appears between individual grains. Its architecture is determined by soil texture, and by the geometry of the packing of the individual grains. The second type appears on a larger scale: the soil may consist of aggregates (e.g., Coppola, 2000, and references therein), and the pore space between these aggregates is very different from those between the grains. Biopores formed by roots that have since decayed, soil fauna, etc. can also create a separate type of pore space. In shrinking soils, a network of cracks may form. The volume and architecture of these pore spaces are essentially independent of the soil texture (Durner, 1994), even though a certain texture may be required for these pores to form. In soils with such distinct pore spaces, the derivative of the soil water retention curve may have more than a single peak, and for this reason multimodal retention curves have been proposed, e.g., by Durner (1994) and Coppola (2000). Most of the parametric expressions for the soil water retention curve are unimodal though. Durner (1994) circumvented this by constructing a multimodal retention curve by summing up several sigmoidal curves of van Genuchten (1980) but with different parameter values. He presented excellent fits of bimodal retention functions at the price of adding three or four parameters, depending on the chosen parameterization. Priesack and Durner (2006) derived the corresponding expression of the hydraulic conductivity function. Romano et al. (2011) developed a bimodal model based on Kosugi's (1994) curve and derived the associated hydraulic conductivity function. Coppola (2000) used a single-parameter expression for the intra-aggregate pore system superimposed on a five-parameter expression for the inter-aggregate pores, thereby reducing the number of fitting parameters and the degree of correlation among these. The primary focus of this paper is on unimodal functions, but we briefly discuss three multimodal models as well.

The wealth of parameterizations for the soil water retention curve calls for a robust fitting method applicable to various parameterizations and capable of handling data with different data errors. These errors arise from the various measurement techniques used to acquire data over the full water content range. Parameter fitting codes are available (e.g., Schindler et al., 2015), but they do not fit the parameterizations focusing on the dry end. The first objective of this paper is to introduce a parameter fitting procedure that involves an objective function that accounts for varying errors, embedded in a shell that allows a wide spectrum of retention function parameterizations to be fitted.

The analysis by Ippisch et al. (2006) of the effect of the shape of the soil water retention curve on the hydraulic conductivity near saturation considered van Genuchten's (1980) parameterization in combination with Mualem's (1976) conductivity model only. Iden et al. (2015) approached the same problem but only examined the conductivity curve. They too focused on the van Genuchten–Mualem configuration only. The analysis of Ippisch et al. (2006) could well have ramifications for other parameterizations. A second objective of this paper is therefore the development of a more general analysis based on Ippisch et al. (2006) and its application to other parameterizations of the retention and conductivity curves.

Several hydraulic conductivity parameterizations that relied only on observations of soil water retention data have been developed (see the reviews by Mualem, 1992 and Assouline and Or, 2013). Many of these consider the soil layer or sample for which the conductivity is sought as a slab of which the pore architecture is represented by a bundle of cylindrical tubes with a given probability density function (PDF) of their radii. This slab connects to another slab with a different pore radius PDF. By making different assumptions regarding the nature of the tubes and their connectivity, different expressions for the unsaturated hydraulic conductivity can be found (Mualem and Dagan, 1978). Raats (1992) distinguished five steps in this process: (1) specify the effective areas occupied by connected pairs of pores of different radii that reflect the nature of the correlation between the connected pore sizes; (2) account for tortuosity in one of various ways; (3) define the effective pore radius as a function of both radii of the connected pairs of pores; (4) convert the pore radius to a matric potential at which the pore fills or empties; and (5) use the soil water retention curve to convert from a dependence upon the matric potential to a dependence upon the water content. Only step 5 constitutes a direct effect of the choice of the retention curve parameterization on the conductivity curve. Choices made in steps 1–3 result in different conductivity curves associated with any particular retention curve parameterization.

These conductivity parameterizations give the hydraulic conductivity as a function of matric potential or water content relative to the value at saturation. They therefore require a value for the saturated hydraulic conductivity, either independently measured or estimated from soil properties. Assouline and Or (2013) review numerous expressions for the saturated hydraulic conductivity. Interestingly, approaches have emerged to estimate the saturated hydraulic conductivity from the retention curve parameters (Nasta et al., 2013; Pollacco et al., 2013, 2017).

The functions based on the pore bundle approach discussed by Mualem and Dagan (1978), Mualem (1992), and Raats (1992) that have found widespread application in numerical models can be captured by Kosugi's (1999) generalized model. In this paper, we limit ourselves to three parameterizations as special cases of Kosugi's general model, and discuss them in more detail in Sect. 2. In doing so, we add to the existing body of comparative studies of parametric retention curves by explicitly including the associated hydraulic conductivity curves according to these conductivity models. Papers introducing new parameterizations of the soil water retention curve as well as reviews of such parameterizations typically show the quality of the fit to soil water retention data (e.g., van Genuchten, 1980; Rossi and Nimmo, 1994; Cornelis et al., 2005; Khlosi et al., 2008). The role of these parameterizations is to be used in solutions of Richards' equation, usually in the form of a numerical model. Their performance can therefore be assessed through the water content and water fluxes in the soil calculated by a numerical Richards solver. This is not often done, one exception being the field-scale study by Coppola et al. (2009) comparing unimodal and bimodal retention curves and the associated conductivity curves in a stochastic framework on the field scale, for a 10-day, wet period. A third objective therefore is to carry out a numerical modeling exercise to examine the differences in soil water fluxes calculated on the basis of various parameterizations by the same model for the same scenario. By doing so, the inclusion of the conductivity curves in the comparison is taken to its logical conclusion by carrying out simulations for all possible combinations of retention and conductivity models.

Should the differences in the fluxes be small, the choice of the parameterizations can be based on convenience. If they are significant, even if the fits to the data are fairly similar, this points to a need for a more thorough selection process to determine the most suitable parameterization.

Numerous functions have been proposed to describe the soil water retention curve, several of them reviewed below. Fewer functions exist to describe the soil hydraulic conductivity curve. When these rely on the retention parameters, one can use the retention curve to predict the conductivity curve. However, when both retention and conductivity data exist, a single set of parameters does not always fit both curves well, even if both sets of data are used in the fitting process. It may therefore be prudent to attempt to find a retention–conductivity pair of curves that shares a number of parameters that could be fitted on retention data only and has additional parameters that only occur in the expression for the hydraulic conductivity.

Various theoretical models exist to determine the unsaturated hydraulic
conductivity

Driven by the occasionally unrealistic shape of Mualem's (1976) hydraulic
conductivity curve near saturation, Ippisch et al. (2006) rigorously analyzed
the version of Eq. (3) specific to Mualem's (1976) model. They concluded that
the integrand must approach zero near saturation in order to prevent
unrealistically large virtual pores dominating the hydraulic conductivity of
very wet soils, a point raised earlier by Durner (1994). We generalize their
criterion for prohibiting excessively larger pores from dominating the
conductivity near saturation for arbitrary parameter values (after converting
d

Iden et al. (2015) argued that limiting the maximum pore size of the pore-bundle models that gave rise to models of the type of Eq. (1) eliminated the large pores that caused the excessively rapid rise of the hydraulic conductivity near saturation. By only modifying the conductivity function without changing the water retention function, a discrepancy emerges between the retention curve (which reflects the presence of unphysically large pores) and the conductivity curve (which does not). Retention curves with a distinct air-entry value maintain the desired consistency, at the price of having noncontinuous derivatives. Computational tests by Ippsisch et al. (2006) suggest that state-of-the-art numerical solvers of Richards' equation are capable of handling this.

The Supplement reviews 18 parameterizations of the soil water retention curve. Their derivatives are presented and used to verify the physical plausibility of the hydraulic conductivity near saturation according to Eq. (4). In this section only those equations that satisfy the criterion in Eq. (4) are presented, together with the associated hydraulic conductivity functions. For comparison, the most widely used parameterization is also included here. To facilitate cross-referencing between the Supplement and the main text, the equations lifted from the Supplement into the main text have the same number in the main text as in the Supplement.

The water retention function of Brooks and Corey (1964) is

Van Genuchten's (1980) formulation is continuously differentiable:

Ippisch et al. (2006) proposed to introduce an air-entry value and scale the
unsaturated portion of VGN by its value at the water-entry value:

This equation is labeled VGA below. With the common restriction of

Rossi and Nimmo (1994) preferred a logarithmic function over the
Brooks–Corey power law at the dry end to
better represent the adsorption processes that dominate water retention in
dry soils, as opposed to capillary processes in wetter soils. They also
implemented a parabolic shape at the wet end as proposed by Hutson and
Cass (1987). Rossi and Nimmo (1994) presented two retention models, but only
one (the junction model) permitted an analytical expression of the
unsaturated hydraulic conductivity. Here, we modified the junction model by
removing the parabolic expression for the wet end of the retention curve in
favor of the discontinuous derivative at the air-entry value:

Rossi and Nimmo (1994) required the power law and logarithmic branches as
well as their first derivatives to be equal at the junction point (

This gives the fitting parameters

Fayer and Simmons (1995) used the approach of Campbell and Shiozawa (1992) to
have separate terms for adsorbed and capillary-bound water. If the capillary
binding is represented by a Brooks–Corey-type function, the retention model
becomes

This expression is denoted FSB below. Note that this model is valid if

In the original equations as presented by Fayer and Simmons (1995), the
adsorbed water content reached zero at

In the Supplement we argue that most of the retention curves examined result
in conductivity curves with physically unacceptable behavior near saturation,
even though several of these expressions were derived with the explicit
purpose of providing closed-form expressions for the hydraulic conductivity.
Only the Brooks–Corey function (1964) (BCO, Eq. S1a), the junction model of
Rossi and Nimmo (1994) without the parabolic correction (RNA, Eq. S9a), and
the model of Fayer and Simmons (1995) based on the Brooks–Corey (1964)
retention function (FSB, Eq. S12a) lead to an acceptable conductivity model
with full flexibility (three free parameters:

The multimodal model of Durner (1994) is a weighted sum of van Genuchten's (1980) retention functions (Eq. S4a) with zero residual water content. The bimodal retention model of Coppola (2000) adds a rapidly decaying asymptotic function representing the aggregate pore space to Eq. (S4a), also with zero residual water content. Because they are derived from Eq. (S4a), neither multimodal retention model meets the criterion of Eq. (4). The asymptotic nature of the dry ends of either multimodal retention model limits their usefulness under very dry conditions.

The bimodal model of Romano et al. (2011) consists of two of Kosugi's (1994)
retention functions. Romano et al.'s expression for the derivative shows that
at least for

Data were obtained from Schelle et al. (2013), who measured soil water
retention curves for a range of soil textures (clay, silt, silt loam, and
sand). They took undisturbed and disturbed samples of a silt loam, a silt,
and a sand near Braunschweig (northern Germany) and of a clay near Munich
(southern Germany). The retention data were measured on soil samples using
different laboratory methods and cover the moisture range from saturation to
near oven dryness at pF of approximately 7. For silt, silt loam, and sand they
used data obtained by suction plates, pressure plates, and the dew-point
method. For clay they used data from the evaporation method
HYPROP^{®} (UMS, 2015) (until pF 3), pressure
plate and dew-point methods. Here, we trimmed the disproportionally large
data set in the HYPROP^{®} range by stratifying
the data into intervals of 0.5 on the pF scale and then randomly picking one
data point for each interval. This ensured an adequate sensitivity of the fit
in the dry range for all textures. For some of the soil samples, hydraulic
conductivity data were available, including the values at saturation
(unpublished). Hydraulic conductivity data were obtained by the evaporation
method according to Peters and Durner (2008).

Undisturbed samples of 4.0 cm height and 100 cm^{®} setup worked with an undisturbed
sample of 5.0 cm height and 250 cm

The fitting routine uses the variance of the data error to determine the
weighting factor each data point. We estimated these on the basis of
estimated measurement errors of water level readings, pressure gauges, sample
masses, etc. Typically, the estimated standard deviation in the matric
potential was 0.05 cm for

When the three conductivity parameters are set to the values dictated by Burdine (1953), Mualem (1976), or Alexander and Skaggs (1986), hydraulic conductivity curves can be derived from soil water retention data only, supplemented by an estimate for the saturated hydraulic conductivity. For the soils with available conductivity data we compared the hydraulic conductivity curves to the direct measurements.

We selected 21 soils from the UNSODA database (Nemes et al., 2001; National Agricultural Library, 2018). The database has relatively many records for sandy soils, and hardly any in heavy clays. The selected soils have no organic matter contents that would lead to considering them as organic soils, have texture data records that allow their texture class to be determined, are fairly uniformly distributed over the textures covered by the database, have data points on the main drying curve, and have measurements over a sufficiently wide range of matric potentials to allow retention curves to be fitted to them.

We classified the texture of the selected soils according to the USDA classification as well as the hydrologically oriented classification developed by Twarakavi et al. (2010). The latter distinguishes 12 texture classes, grouped in three sets (A, B, C) of four each (1 through 4). Soils with (nearly) 100 % sand, silt, or clay are classified as A1, B1, and C1, respectively. Numbers larger than 1 identify texture classes that must have at least two of the components sand, silt, and clay. B3 and C4 are the only categories that must have all three components. The differences with the USDA classification are considerable for clayey and silty soils, and we refer to Twarakavi et al. (2010) for full details. Figure 1 shows the distribution of the selected soils over the soil texture triangle.

The textures of the soils used to test the fitting capability of selected soil water retention curve parameterizations. The numbers next to the data points are the identifiers used in the UNSODA database to distinguish individual soils.

We fitted the original Brooks–Corey (BCO, Eq. S1a) and van Genuchten (VGN, Eq. S4a) parameterizations, and the derivates thereof that do not lead to
unrealistic hydraulic conductivities near saturation: FSB (Eq. S12a) and RNA
(Eq. S9a), both of which emerged from BCO, and VGA (Eq. S7a), which emerged
from VGN. Thus, BCO, FSB, and RNA all have a power law shape in the mid-range
of the matric potential (and for BCO over the full range below the air-entry
value). The slope therefore monotonically increases with decreasing water
content. VGN and VGA have a sigmoid shape and therefore are able to fit
curves that have an inflection point. As Groenevelt and Grant (2004) pointed
out,

The fitting parameters for five parameterizations, their physically permitted ranges, and their fitted values for four textures. The three-character parameterization label is explained in the main text.

All three conductivity models are compatible with BCO, FSB and RNA. Burdine's (1953) and Mualem's (1976) conductivity models can be used with VGA. VGN does not meet the criterion of Eq. (4) but is very often used in conjunction with Mualem's conductivity model (1976). It was therefore included for comparison.

A set of parameters describing the soil water retention curve must be
optimized to provide the best fit to an arbitrary number of data points. To
do so, an objective function was minimized, construed by the sum of weighted
squares of the differences between observed and fitted values. The fitted
values depend on the parameter values in the parameter vector

The definition of the objective function

For relatively wet soils (0 >

If and only if the standard deviation of the measurement error of the
individual observations is known, a maximum-likelihood estimate of the soil
hydraulic parameters can be obtained (Hollenbeck and Jensen, 1998). To
ensure this, the weighting factors in vector

Data points for a retention curve over the whole moisture range cannot be
obtained by a single method. Furthermore, measurement errors occur in both

In the code, the gradient is approximated by

The calibration algorithm employed here is the shuffled complex evolution
algorithm introduced by Duan et al. (1992) with parameter adjustments
of Behrangi et al. (2008). The strategy of this algorithm is to form out of

The SCE algorithm used here is configured with two complexes each consisting
of (

As stated in the Introduction, previous tests of parametric expressions of
soil water retention functions mostly focused on the quality of the fit to
direct observations of points on the water retention curve. Here, we will
also examine how the various parameterizations affect the solution of
Richards' equation by simulating water fluxes and soil water profiles for a
scenario involving infiltration and evaporation. We set up a hypothetical
999-day scenario representative of a desert climate with prolonged drying,
infiltration into dry soil, and redistribution after rainfall, permitting a
comprehensive test of the parameterizations. We used the HYDRUS 1-D model
version 4.xx (Šimůnek et al., 2013,

We considered an unvegetated uniform soil profile of 1 m depth, initially in
hydrostatic equilibrium with

The record of daily rainfall sums from Riyadh city that was used in the numerical scenario study. Three rainfall clusters are visible. The largest daily rainfall amount (5.4 cm) fell on day 656. The observation period starts on 4 June 1993, and ends on 27 February 1996.

The simulation period involved large hydraulic gradients when water infiltrated a very dry soil, limited infiltration of small showers followed by complete removal of all water, and deeper infiltration after clusters of rainfall that delivered large amounts of water followed by prolonged periods in which flow of liquid water and water vapor occurred simultaneously. These processes combined permitted a comprehensive comparison of the various parameterizations. We were interested in the magnitude of the fluxes of liquid water and water vapor and the partitioning of infiltration into evaporation, storage change, and deep infiltration under various conditions, and the effect on these fluxes and storage effects of the choice of parameterization. We did not intend or desire to carry out a water balance study. Under semiarid conditions this would have required a much longer meteorological record, which was not available.

The various parameterizations are not implemented in HYDRUS. We therefore used the MATER.IN input file to supply the soil hydraulic property curves in tabular form to the model. The retention models BCO, FSB, and RNA permitted all three conductivity models (Burdine, B; Mualem, M; and Alexander and Skaggs, AS) to be used. VGA only gives useful expressions for Burdine and Mualem. VGN only allows Mualem's conductivity model. Thus, there are 12 combinations of retention and conductivity curves that we tested on four different textures, leading to 48 different simulations (and MATER.IN files) in total.

Table 1 presents the fitted parameters for all combinations of texture and
parameterization for the soils used in the simulations. The parameter with
the best-defined physical meaning is

Observed and fitted retention curves for the different soil textures.

In 3 of the 48 parameter estimation runs, the fits pushed one of the
parameters to one of its bounds (even after expanding these to their
physical limits), irrespective of their initial guess: FSB for clay (we
fixed

The root mean square error (RMSE) of the fits (Table 2) illustrate why VGN has been very popular for over 3 decades. It gives the best fit in three cases (sand, silt, and silt loam) and the second-best fit in the fourth (clay). BCO performs poorest in three cases (sand, silt, and silt loam) and second-poorest in one (clay). The other three have varying positions, with no clearly strong or weak performers. FSB has the best performance in the finest soil (clay).The overall difference in the RMSE values between textures reflects the different scatter in the underlying data clouds.

Root mean square errors (RMSEs) for the different parameterizations.

The soil water retention curves defined by the different parameterizations
are plotted in Fig. 3. The models that were not developed with dry
conditions in mind (BCO, VGA, and VGN) have relatively high water contents
in the dry end of clay and silt loam. The logarithmic dry end of FSB and RNA
eliminates this asymptotic behavior. The cutoff to zero of the FSB
parameterization is quite strong in fine-textured soils. The fixed value of

In the intermediate range, all fits are close to one another. RNA underperforms in sand and silt compared to the others. In the wet range, the absence of an air-entry value in VGN results in a poor fit for sand. Here, the contrast between VGN and VGA is very clear. Overall, the inclusion of the water-entry value as a parameter seems beneficial to the fits. FSB has the most satisfactory overall performance.

For sand, silt, and silt loam, independent observations of

The observed and fitted hydraulic conductivity curve according to
Burdine (1953), Mualem (1976), and Alexander and Skaggs (1986) using the
fitted parameters of the Fayer and Simmons soil water retention curve (1995)
for

For all simulations, the vapor flux within the profile was of little consequence compared to the liquid water flow. For that reason it will not be discussed in detail here. Vapor flow may play a larger role under more natural conditions with day–night temperature cycles and in the presence of plant roots.

We start the analysis by examining the flux at the bottom of the soil profile. Figure 5a–e shows all combinations of parameterizations of the retention and conductivity curves.

The cumulative bottom fluxes leaving a silt soil column for the
different combinations of soil water retention curve and hydraulic
conductivity parameterizations. Panels

Cumulative evaporation from a silt soil column for the different
combinations of soil water retention and hydraulic conductivity
parameterizations. Panels

Cumulative infiltration in a silt profile for the VGA
parameterization (see Table 1) with conductivity functions according to
Mualem (1976) and Alexander and Skaggs (1986)

As in Fig. 5, but for a sandy soil column. Unlike Fig. 4, the results
of Burdine's (1953) conductivity curve are shown

Cumulative evaporation from a sandy profile for the different
combinations of retention curve parameterizations (see Table 1) and hydraulic
conductivity functions: Burdine (1953)

Pressure head hBot and flux density vBot at the bottom of the sand
column for the FSB parameterization (see Table 1) and the conductivity
functions of Mualem (1976)

As in Fig. 10, but for the RNA parameterization (see Table 1).

As in Fig. 10, but for the BCO parameterization (see Table 1).

Cumulative bottom fluxes from a silt loam profile for all
combinations of parameterizations (see Table 1) and
Mualem's (1976)

Cumulative evaporation from a silt loam profile for all
parameterizations (see Table 1) with Mualem's (1976) conductivity
function

Cumulative infiltration from a silt loam profile for four
parameterizations (see Table 1) with the Alexander and Skaggs (1986)
conductivity function

As in Fig. 13, for clay.

As in Fig. 14, for clay.

As in Fig. 15, for clay.

The early rainfall cluster event at around

For the individual parameterizations, Mualem and Burdine gave reasonably similar results in which the second and third rainfall cluster generated a little more downward flow for B than for M. In all cases, Alexander and Skaggs gave a more rapid response of a very different magnitude. Clearly visible is a sustained, constant flux leaving the column during prolonged dry periods for the AS conductivity curves. This is physically implausible.

Figure 5f shows the substantial effect of the parameterization of the water
retention curve on bottom fluxes when the M-type

Figure 5g shows the similar comparison of all parameterizations for the
AS-type

The evaporative flux was nearly identical for B and M conductivity functions
(Fig. 6a–c). Since their bottom fluxes differed, this necessarily implies
that the storage in the soil profile must also be different for B and M. The
AS parameterization gave a much more spiky response of evaporative flux to
rainfall than B or M, with zero evaporation most of the time (Fig. 6a–d). In
terms of cumulative evaporation, AS responded more strongly to the second
rainfall cluster around

Given the nonphysical behavior of the bottom flux of AS for VGA in particular (Fig. 5d), we also examined the infiltration. We first compare infiltration for VGA with M- and AS-type conductivity (Fig. 7a) and clearly see the zero infiltration for VGA during periods without rain contrasted to the impossible nonzero infiltration rates for AS during dry spells. For the other water retention parameterizations in combination with AS, the effect is less pronounced (Fig. 7b). Still, the AS conductivity should be used with care and the results and mass balance checked.

Table 3 summarizes the bottom and evaporative fluxes. For evaporation, the differences are inconsequential except for the markedly low values for RNA. For the bottom flux, the difference between B and M is small enough to be within the margin of error for typical applications. The effect of the parameterization of the retention curve is an order of magnitude between the smallest bottom flux (for VGA) and the largest (for RNA).

Cumulative bottom and evaporative fluxes (positive upwards) for
silt from day 281 (the start of the first rainfall) onwards for Burdine and
Mualem conductivity functions with the different parameterizations. The
hydraulic conductivity at

The relationship between the bottom (Fig. 8) and evaporative fluxes (Fig. 9) as generated by the various parameterizations for the sandy soil were comparable to those for silt, and the analysis applied to the silt carries over to sand. The bottom fluxes in sand responded faster and with less tailing than in silt, and the third rainfall cluster near the end of the simulation period produced a clear signal (Fig. 8).

The FSB (Fig. 8b) and RNA (Fig. 8c) parameterizations were both in their logarithmic dry range when bottom fluxes occurred, and both gave comparable values. BCO is not well adapted for dry conditions, and this is reflected by a bottom flux that is 4 times lower than the others (Fig 8g).

The bottom fluxes for BCO and FSB with AS-type

For both B and M conductivity functions, the evaporation (Fig. 9a and b) and
the bottom flux (Fig. 8a, f, and g) for BCO differed from the other
parameterizations. These differences seem to have been dominated by the
complementary responses of evaporation and bottom fluxes to the rainfall
events around

The AS-type

Coarse-textured soils have the sharpest drop in the hydraulic conductivity as the soil desaturates. We therefore used the result for the sandy column to study the relationship between the matric potential at the bottom of the column and the bottom flux in order to evaluate water fluxes in dry soils. The free drainage lower boundary condition ensures there is always a downward flux that is equal to the hydraulic conductivity at the bottom at any time. Particularly for coarse soils this can still lead to negligible bottom fluxes for considerable periods of time. We first consider FSB and RNA, these being the parameterizations specifically developed to perform well in dry soils.

The difference in matric potentials between FSB and RNA is immediately clear
from Fig. 10a, b and 11a, b. The effect of the conductivity function is
manifest by including Figs. 10c and 11c in the comparison. The effect of the
first rainfall cluster is visible in the matric potential in all cases (Figs. 10
and 11), but not enough to generate a significant flux. A flux through the
lower boundary first occurs when the matric potential there exceeds (i.e.,
becomes less negative than)

The second rainfall cluster at 600 <

The AS-type

The AS conductivity function led the soil to dry out so completely that the atmospheric matric potential during dry spells was reached at 1 m depth in a few months (Figs. 10c and 11c). This seems unrealistic, and seems to be related to the significant overestimation of the unsaturated hydraulic conductivity by AS evidenced in Fig. 4.

For comparison, the bottom matric potentials and fluxes are given for BCO as well (Fig. 12). They are very different and, given the poor suitability of BCO for dry soils and the poor fitting performance, probably incorrect. The differences between the parameterizations illustrate the need to carefully consider the suitability of the parameterization for the intended purpose.

The bottom fluxes from the clay and the silt loam soil for all combinations
of parameterizations for the soil water retention and hydraulic conductivity
curves were similar to those for the silt soil (Figs. 13 and 16), with two
notable exceptions: for RNA, there was a much more damped response to the
rainfall around

The behavior of the evaporative fluxes from the silt loam and the clay soil for all combinations of parameterizations for the soil water retention and hydraulic conductivity curves was essentially similar to that for the silty soil (Figs. 14 and 17). The main difference was the less gradual response of the evaporation for VGA, particularly for clay, which was, in fact, rather similar to the notoriously spiked response of the AS-type conductivity function. The relative amounts of evaporation of the various parameterizations varied from one texture to another.

For AS in combination with the VGA retention curve, there was significant
infiltration during periods of zero rainfall (Figs. 15 and 18). This
numerical artefact led to erroneous simulations of the bottom flux. This is
the most significant occurrence of mass balance errors that plague the
simulations with AS-type

The fits for the clayey soils selected from the UNSODA database (Fig. S1,
first panel) show that with data ranging to pF

Rossi and Nimmo (1994) fixed the matric potential in their parameterization
at which the water content became zero. The fits for the soils used in the
simulations showed that fixing

Soil 1180 (Fig. S1, first panel) had a large discrepancy between the porosity and the unsaturated water contents. The effect on the shape of FSB points to the effect of the weighting factors: the accuracy of the porosity was assumed to be higher than that of the water content measurements. Because the weighting factors of the data points are inversely proportional to the measurement error as quantified by its estimated standard deviation, the outlier was given more weight in this case. If weighting factors are manipulated to improve the quality of the fit, the fitted parameter values can no longer be qualified as maximum likelihood estimates.

For silty soils (Fig. S1, second panel), the fits were generally good, with some evidence that the fitted residual water contents were somewhat high for some soils (3260, 3261). The extrapolations to zero water content by FSB and RNA appeared plausible even though they differed significantly in some cases (3251, 4450), highlighting the desirability of data points in the dry range.

For sandy soils with some clay and/or silt (A3 and A4, Fig. S1, third
panel), residual water contents for BCO, VGN, and VGA were often large
(1120, 1143, 2110, 1133). When the data range was limited (below pF

For sandy soils (A1 and A2, Fig. S1, fourth panel), the fits were good if the data covered the full water content range. In all cases, VGA and VGN fitted the residual water content close to driest data point, which is very unrealistic if the dry range was not covered (1142).

The RMSE values in Tables S5–S8 in the Supplement reflect the observations
based on the curves above. If the curves have a clear inflection point, which
is the case for the sands and some of the silty soils, the van
Genuchten-based curves (VGN and VGA) outperform the Brooks–Corey-based curves
(BCO, FSB, RNA) (Tables S6–S8). With two exceptions in clays and silty soils,
VGA and VGN have very similar RMSE values. As discussed above, the upper
limit of

For the fits of the four soils used for the simulation and the 21 soils, sets of three optimizations were independently run for all five parameterizations, with initial guesses that covered the full range over which the parameters were allowed to vary. In about a quarter of the cases we found no more than a single acceptable fit, and we ran these again with other sets of initial guesses (again widely different from one another) and/or expanded parameter ranges. For only two of the 125 fitted parameter sets did this procedure not lead to convincing convergence.

In none of the cases did the three independent runs yield parameter estimates that differed by more than 10 % while the sum of squares of the fits differed by less than 10 %, even though in all cases the initial guesses were very different, thereby ensuring that the starting points of the different searches were located in completely different regions of the parameter space. We take this as evidence of the absence of parameter correlations, since one would expect correlated parameters to vary over a considerable range, with the RMSE of different combinations of parameter values remaining nearly constant. We found that the fitted values obtained from the different runs were very similar, with an occasional outlier in a local minimum with a considerably larger RMSE.

In order to determine the correlation matrix of the fitted parameters correctly, a Markov Chain–Monte Carlo approach would be required for each of the 125 combinations of soils and parameterizations. Given the lack of evidence that significant correlations exist, we considered this beyond the focus of and the computational resources available for this work.

Some of the data sets displayed multimodality. None of the parameterizations we tested can account for that, which is why we did not examine this further in this paper. If one wishes to reproduce this by summing several curves of the same parameterization but with different parameter values (advocated by Durner, 1994), one needs a sigmoidal curve. If physically realistic conductivity curves near saturation are deemed desirable, VGA is the only viable parameterization for this purpose among those evaluated in this paper.

We found that 14 out of 18 parameterizations of the soil water retention curve were shown to cause nonphysical hydraulic conductivities when combined with the most popular (and effective) class of soil hydraulic conductivity models. For one of these cases (VGN), Ippisch et al. (2006) demonstrated convincingly that their alternative (VGA) significantly improved the quality and numerical efficiency of soil water flow model simulations, and our simulations confirmed the profound effect of this modest modification on the model results. We hope that the general criterion we developed for verifying the physical plausibility of the near-saturated conductivity will be used in the selection of suitable soil hydraulic property parameterizations for practical applications of numerical modeling of water flow in soils, and likewise will be of help in improving existing parameterizations (as we have done in a few cases here) and developing new ones.

Replacing the residual water content in a retention curve parameterization
by a logarithmic dry branch generally improved the fits in the dry range for
many soils. If data in the dry range were lacking, the logarithmic extension
provided a physically realistic extrapolation into the dry range, but the
spread between the different fits showed the level of uncertainty in this
extrapolation caused by the limited range of the data. The cutoff to zero
water content of FSB could be excessive for fine-textured soils, but this is
only a problem if the soil actually so far that it reaches

The ability of both Burdine's (1953) and Mualem's (1976) models of the soil hydraulic conductivity function to predict independent observations of the soil hydraulic conductivity curve on the basis of soil water retention parameters fitted on water content data only is reasonably good, at least for the limited data available to test this. The conductivity model of Alexander and Skaggs (1986) overestimated the conductivity of the soils for which independent data were available. This resulted in a rapid and unrealistically strong response to changes in atmospheric forcings even at 1 m depth, as shown in our simulation study.

The simulations with different parameterizations showed that under the given boundary conditions the choice of the parameterization had a modest effect on evaporation but strongly affected the partitioning between soil water storage and deep percolation. The uncritical use of a default soil hydraulic parameterization or selection of a parameterization solely based on the quality of the fit to soil water retention data points entails the risk of an incomplete appreciation of the potential errors of the water fluxes occurring in the modeled soil. This points to the importance of carefully considering the soil hydraulic parameterization to be used for long-term water balance studies. Such studies typically aim to determine or predict the variation of seasonal water availability to plants or long-term groundwater recharge to assess the sustainability of extractions from an underlying aquifer. If at all possible, observations during dynamic flow (water contents, matric potentials, fluxes) should be included in the parameterization selection process. In this context it would be interesting to see if parameter-estimation processes based on inverse modeling of a nonsteady unsaturated flow experiment would lead to a different choice of parameterization than fitting parameters to data points obtained at hydrostatic equilibrium. This requires the inclusion of all the parametric expressions of interest in the numerical solvers of Richards' equation capable of running in parameter estimation mode.

The parameter optimization code is available upon request from Gerrit Huibert de Rooij. At a later time we intend to make the code available through a website.

RM gathered the soil hydraulic functions from the literature. RM and GHdR carried out the parameter optimization runs with the SCE-based code. RM and GHdR designed the test problem (column size, initial and boundary conditions) for the test simulations with HYDRUS-1D. RM set up, ran, and analyzed these model simulations. GHdR wrote the shell of the optimization code, selected the UNSODA soils, and carried out the mathematical analysis of the soil hydraulic functions. HM carried out the experiments that generated the data for the soils used in the simulations. JM wrote the SCE parameter optimization code. RM and GHdR wrote the paper. All authors were involved in checking and improving the paper.

Gerrit Huibert de Rooij is a member of the HESS Editorial Board. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: Alberto Guadagnini Reviewed by: Antonio Coppola and four anonymous referees