**Technical note**
20 Jul 2018

**Technical note** | 20 Jul 2018

# Technical note: Saturated hydraulic conductivity and textural heterogeneity of soils

Carlos García-Gutiérrez Yakov Pachepsky and Miguel Ángel Martín

^{1},

^{2},

^{1}

**Carlos García-Gutiérrez et al.**Carlos García-Gutiérrez Yakov Pachepsky and Miguel Ángel Martín

^{1},

^{2},

^{1}

^{1}Department of Applied Mathematics, Universidad Politécnica de Madrid, Madrid, Spain^{2}USDA-ARS Environmental Microbial and Food Safety Laboratory, Beltsville, MD 20705, USA

^{1}Department of Applied Mathematics, Universidad Politécnica de Madrid, Madrid, Spain^{2}USDA-ARS Environmental Microbial and Food Safety Laboratory, Beltsville, MD 20705, USA

**Correspondence**: Carlos García-Gutiérrez (carlos.garciagutierrez@upm.es)

**Correspondence**: Carlos García-Gutiérrez (carlos.garciagutierrez@upm.es)

Received: 01 Dec 2017 – Discussion started: 05 Dec 2017 – Revised: 14 May 2018 – Accepted: 28 Jun 2018 – Published: 20 Jul 2018

Saturated hydraulic conductivity (*K*_{sat}) is an important
soil parameter that highly depends on soil's particle size distribution
(PSD). The nature of this dependency is explored in this work in two ways,
(1) by using the information entropy as a heterogeneity parameter of the PSD
and (2) using descriptions of PSD in forms of textural triplets, different
than the usual description in terms of the triplet of sand, silt, and clay
contents. The power of this parameter, as a descriptor of ln*K*_{sat}, was tested on a database larger than 19 000 soils.
Bootstrap analysis yielded coefficients of determination of up to 0.977 for
ln*K*_{sat} using a triplet that combines very coarse, coarse,
medium, and fine sand as coarse particles; very fine sand, and silt as
intermediate particles; and clay as fine particles. The power of the
correlation was analysed for different textural classes and different
triplets using a bootstrap approach. Also, it is noteworthy that soils with
finer textures had worse correlations, as their hydraulic properties are not
solely dependent on soil PSD.

This heterogeneity parameter can lead to new descriptions of soil PSD, other than the usual clay, silt, and sand, that can describe better different soil physical properties, that are texture-dependent.

Saturated hydraulic conductivity (*K*_{sat}) is the measure of soil's ability
to conduct water under saturation conditions (Klute and Dirksen, 1986). It is an
essential parameter of soil hydrology. Soil *K*_{sat} affects many aspects of
soil functioning and soil ecological services, like infiltration, runoff,
groundwater recharge, and nutrient transport. Knowing values of soil
*K*_{sat} appears to be essential in designing management actions and
practices, such as irrigation scheduling, drainage, flood protection, and
erosion control.

The dependence of *K*_{sat} on soil texture has been well documented
(Hillel, 1980). Different parametrizations of particle size distributions
(PSDs) were suggested to relate *K*_{sat} and soil texture. It was
proposed that *d*_{10}, *d*_{20}, and *d*_{50} particle diameters
(Chapuis, 2004; Odong, 2007) or slope and intercept of the particle size
distribution curve (Alyamani and Sen, 1993; Arya and Paris, 1980) could be used. Also, various functions were fitted to PSDs, and the fitting
parameters were related to *K*_{sat}. For example, Chapuis et al. (2015)
proposed using two lognormal distributions to fit the detailed particle size
distribution and to use the lognormal distribution parameters to predict the
*K*_{sat}.

A common way to parametrize the PSD for *K*_{sat} estimation purposes is
using the textural triplet that provides the percentage of coarse particles
(sand), intermediate particles (silt), and fine particles (clay). *K*_{sat}
values are estimated using the contents of one or two triplet fractions or
just the textural class (Rawls et al., 1998). Representing PSD by textural
triplets is the common way to estimate a large number of soil parameters
(Pachepsky and Rawls, 2004). The coarse, intermediate, and fine fractions need not
be sand, silt, and clay. Martín et al. (2018) showed that different
definitions of the triplet (e.g. coarse sand, sand, and medium sand as
coarse; fine sand and very fine sand as intermediate; and silt and clay as fine
triplet fractions) provide much better inputs for bulk density estimation
compared with the standard textural triplet. These different parametrizations
of soil texture might put the focus on different soil physical properties,
depending on the different particle sizes represented in the triplet.

The heterogeneity of particle size distributions appears to be an important
factor affecting hydraulic parameters of soils, including the saturated
hydraulic conductivity. Values of *K*_{sat} depend on both
distribution of sizes of soil particles, i.e. soil texture, and the spatial
arrangement of these particles, i.e. soil structure. Soil structure can be to
some extent controlled by soil texture, since packing of particles is
affected by the particle size distributions (Assouline and Rouault, 1997; Gupta and Larson, 1979; Horn et al., 1994; Jorda et al., 2015). Recent studies proposed using the
information entropy as the parameter of the PSD heterogeneity for predicting
soil water retention (Martín et al., 2005) and soil bulk density
(Martín et al., 2018). Previously, information entropy was
used, together with other predictor variables to estimate *K*_{sat},
using multivariate analysis (Boadu, 2000).

The objective of this work was to test the hypothesis that combining two
recent developments – the description of the PSD by different textural
triplets that may represent different soil physical properties dependent on
the particle sizes present in the triplet, and the information entropy, as a
PSD heterogeneity parameter that depends on the triplet used – may linearly
correlate with ln*K*_{sat} and may be seen as a step forward to
study the effect of heterogeneity widely recognized in the majority of works
that studied the particle size–hydraulic-conductivity relationships. By
describing the PSD in terms of different triplets, the input information
would possibly have different physical interpretations. We wanted to link the
heterogeneity of this physical information to the hydraulic behaviour of the
soil. Therefore, we explored the possible relationships between ln*K*_{sat} values and an entropy metric of soil texture heterogeneity
using different size limits of coarse intermediate and fine fractions, using
the large USKSAT database on laboratory-measured *K*_{sat}, which
contains more than 19 000 samples. The triplets with highest correlations will be
understood as the physical sizes that influence the most in the packing of
particles yielding the particular hydraulic behaviour. While pedotransfer functions (PTFs) are a
useful tool to predict difficult-to-measure soil properties, they sometimes
exhibit highly non-linear relationships that are difficult to interpret.
While the objective of this paper was the exploration of the physical
relation of the new tools and the saturated hydraulic conductivity, the
future development of PTFs for prediction purposes is a promising avenue for
expanding this research. We note that research in this work is descriptive. It does not include an explanation of what we have observed. However, any
explanatory research with mechanisms, models, etc. was historically preceded
with the descriptive research.

## 2.1 Database description and textural triplet selection

For this study we used the USKSAT database, about which detailed information
can be found in Pachepsky and Park (2015). This database consists of soils from
different locations of the USA and contains soils from 45 different sources.
We selected only those sources which (a) had data on both *K*_{sat}
and on the seven textural fractions and (b) presented measurements of
*K*_{sat} made in laboratory with the constant head method. From
those, we subset those soils whose sum of mass in the seven textural
fractions, i.e. (1) very coarse sand, (2) coarse sand, (3) medium sand,
(4) fine sand, (5) very fine sand, (6) silt, and (7) clay ranged from 98 to
102 %. The final number of soils considered was 19 121. By USDA textural
classes the total number of soils are 12 068 sands, 1780 loamy sands,
2123 sandy loams, 104 loams, 135 silt loams, 36 silts, 2004 sandy clay loams,
78 clay loams, 41 silt clay loams, 345 sandy clays, 0 silty clays, and
407 clays. All the samples in the database used are undisturbed soil samples.

We used all possible triplets formed from seven textural fractions. Triplets consisted of coarse, intermediate, and fine fractions. The symbols for triplet showed how the fractions were grouped. For example the “coarse” fraction for the triplet “3-2-2” included very coarse sand, coarse sand, and medium sand; the “intermediate” fraction included fine sand and very fine sand; and “fine” included silt and clay. The triplet “5-1-1” was the standard one where “coarse” included all five sand fractions, “intermediate” included silt, and “fine” included clay. The amount of possible triplets with 7 textural fractions was 15.

## 2.2 Heterogeneity metric calculation

The entropy-based parametrization of textures introduced in
Martín et al. (2001) is a central concept in the information entropy (IE)
(Shannon, 1948). Assuming the texture interval divided into *k* textural
size ranges and that the respective textural fraction contents are
${p}_{\mathrm{1}},{p}_{\mathrm{2}},\mathrm{\dots},{p}_{k}$, $\mathrm{1}\le i\le k$ , with ${\sum}_{i=\mathrm{1}}^{k}{p}_{i}=\mathrm{1}$, the
Shannon IE (Shannon, 1948) is defined by

where *p*_{i}log_{2}*p*_{i}=0 if *p*_{i}=0. The IE is a widely accepted measure
of the heterogeneity of distributions (Khinchin, 1957). In the case of three
fractions, the minimum value of IE is zero when only one fraction is
present, and the maximum value is 1.57 when three fractions are present in
equal amounts (see Fig. 1).

For each soil in this study, we grouped the 7 available textural fractions in
the 15 possible triplet combinations and calculated the respective triplet's
IE using formula (1). Figure 2 shows ternary graphs of IE
calculated for all the soils available in this study but using two different
triplets as input. It is clear that, by changing the triplet, the calculated
IE values vary differently along the same textural triangle. IE is a
measure of heterogeneity, but the triplet used is the substrate for this
measure: (IE,*triplet*), i.e. (IE,“5-1-1”).

As we want to compare the linearity (i.e. the proportionality between the
heterogeneity of the particular physical sizes chosen and the hydraulic
behaviour), we used the coefficient of determination, *R*^{2}, as a comparison
statistic. As this statistic is highly sensitive to the number of points in
the regression, we followed the binning method of Martín et al. (2017) to
research the relationship between ln*K*_{sat} and soil heterogeneity.
Specifically, the range of values of IE was divided into 10 bins, and the
average value of ln*K*_{sat} was plotted against the average IE for the
bin, i.e. the bin midpoint. This way, the number of points in each
relationship was always the same. We want to state that this way, the
particular value of *R*^{2} is irrelevant, but it is only to be used as a
comparison tool among these regressions.

Linear regressions “bin midpoint vs. average bin ln*K*_{sat}” were
computed. Besides the coefficient of determination value for comparison
purposes, the goodness-of-fit of these regressions was tested using the
root-mean-square error, RMSE:

where ${\widehat{y}}_{t}$ are the predicted and *y*_{t} are the measured values of ln*K*_{sat}, and *n* is the number of soils.

In order to make some inference on these parameters we employed the bootstrap
method, which has been used in a very similar context by
Schaap and Leij (2000). The bootstrap method is a tool for assessing
statistical accuracy. It assumes that one can obtain multiple samples from a
single data set, by randomly drawing data with replacement from the original
sample. Thus, one can perform the same statistical analysis multiple times in
different data sets, obtaining slightly different regression models, thus
resulting in an uncertainty in each of the parameters of the model. All of
the samples used have the same size as the original sample they were drawn
from, so they are generated by random sampling with replacement. We used 1000 bootstrap data sets, resulting in 1000 linear regression models. In
particular we obtained not just one *R*^{2} and one RMSE value for each IE
vs. triplet regression, but 1000 of them, which were summarized into a
mean and a standard deviation values. More information on this method can be
found in Efron and Tibshirani (1993) and Hastie et al. (2003).

We took 1000 samples with size equal to the total amount of soils, with
repetition, and calculated, for each sample, the coefficient of
determination (*R*^{2}) and the RMSE. Finally, the mean
and standard deviation from these two values for the 1000 samples were
calculated.

These regressions were obtained for each of 15 triplets and for those of USDA textural classes that were represented in the selected database by more than 50 samples, i.e. all of them except silty clay loams and silts.

## 2.3 IE variation in the textural triangle

Ternary graphs were used to visually correlate the IE values calculated
with the ln*K*_{sat} values of the soils in the study. Also, a less visual,
but more quantifiable approach, to find out how much of ln*K*_{sat} could
be explained through IE variation was to find out what ranges of IE are
available for soils in different textural classes and compare them to the
range of ln*K*_{sat} values of soils inside those same textural classes.
Also, in order to compare the new tool (IE triplet), we compared these
ranges to the ranges computed for (IE,“5-1-1”), i.e. to the values of the
IE computed with the usual description of soil texture. We wanted to find
out if, by changing the triplet, we would obtain a wider range of variation
in IE for a given range of ln*K*_{sat}. This way we compared if the new
descriptions of texture, in the form of different triplets, might be suitable for
explaining soil physical properties, in particular ln*K*_{sat}.

For each textural class, we did a sensitivity analysis by calculating the
ratio of the range of ln*K*_{sat} values inside the textural class versus
the range of ln*K*_{sat} values of all the soils in the study. The same was
done for IE for each triplet.

## 3.1 The data set overview

Figure 3 presents the 19 121 soils used in this study in the USDA textural triangle and in the modified “3-2-2” triangle. The density of points reflects the dominance of coarse textural soils in the database. When the triplet is changed, the distribution of points across the triangle changes. By setting the textural fractions to be the “3-2-2” triplet, the distribution of soils in the new textural triangle spreads. While there is still a high concentration of soils with more than 85 % of the coarse fraction, where coarse 3 includes very coarse sand, coarse sand a medium sand, now those soils spread fully from 10 to 100 % of the intermediate-2 fraction, where intermediate-2 contains fine and very fine sand. On the USDA textural triangle, most of the soils are clustered in the subtriangle limited by the lines “more than 70 % sand” and “less than 20 % silt”. This new textural triangle allows for a finer look into the sand fraction, revealing the distribution of soils within the USDA sandy textural classes. This finer look might prove itself useful to study physical properties of these soils that are mainly related to the type and amount of sand in them.

Table 1 shows the *K*_{sat} statistics for the
soils in the study. A total of 19 420 soils were used in this study, from
which 299 (1.53 %) had to be rejected due to missing values. The textural
class sand comprises the 63.1 % of all the soils, followed by sandy loam
(11.1 %) and sandy clay loam (10.48 %). Five textural classes were
poorly represented, with percentages less than 1 % of the total soils.
The *K*_{sat} values varied between 0.0005 and 841 cm h^{−1},
22.57 being the mean value.

## 3.2 Regression in binned data: IE as a predictor of *K*_{sat} and ln*K*_{sat}

Linear regressions for ln*K*_{sat} were done to find out the predictive
power of the proposed parameter, (IE,triplet), with the 15 possible
different triplets that could be archived by grouping the available textural
data. Table 2 shows the computed *R*^{2}
and RMSE values for the linear regressions using 10 interval bins.

The best triplet in terms of highest mean *R*^{2} value was “4-2-1”, with a
mean of 0.977 and a standard deviation of 0.002, but the lowest mean RMSE
(ln (cm h^{−1})) value (mean = 0.207, SD = 0.030) was attained
with the “1-2-4” triplet. Figure 4 shows a
ternary representation of the ln*K*_{sat} values of the soils of the
study on the textural triangle compared to a ternary representation of the IE
values of the same soils computed using the “4-2-1” triplet. There is a
high visual similarity between these two images, with high ln*K*_{sat} value zones, near the lower corners (sandy and silty soils)
that correspond to low (IE,“4-2-1”) values. The ln*K*_{sat} values
tend to decrease towards the centre of the triangle. However, the (IE,“4-2-1”) values tend to
increase around this point.

The standard triplet (“5-1-1”) yielded, for the ln*K*_{sat} regression, the
*R*^{2} value with this triplet had a mean of 0.960 and a standard deviation of
0.005; the RMSE mean value was 0.339 with a standard deviation of 0.021. The
average of the *R*^{2} mean values of the regressions with all possible
triplets for ln*K*_{sat} was 0.727.

## 3.3 Predictive power of IE among the USDA textural classes

In this section we show how IE works differently among textural classes: using different triplets we can find that the textural classes are predicted differently; what works for some is for others counterproductive.

Table 3 shows the best triplet, chosen
in terms of the highest mean *R*^{2} value of all the possible regressions, for
each textural class that had *N*>50. In the table are shown the mean and
standard deviation for *R*^{2}, of the 1000 bootstrap samples for ln*K*_{sat} linear regressions. The best *R*^{2} values were obtained for
the regression of the sand textural class against (IE,“5-1-1”), i.e. the IE
computed with the standard “5-1-1” clay–silt–sand USDA triplet. The mean
value was equal to *R*^{2}=0.987 for all the regressions. A possible
explanation for this triplet being the best among all the other possible
triplets, is that sandy soils are the ones that contain percentages of the
sand fraction higher than 70 %, so their distribution is highly
heterogeneous. Minor fractions are now silt and clay, and the information
about these two fractions could be very important for the hydraulic
properties of the soil; thus, the (IE,“5-1-1”) triplet yielded the best
regression result. One might think that, with such a high concentration of
sand particles, it is now silt and clay fractions that made the difference in
the packing properties, and thus in the saturated hydraulic conductivity
values. The high value of *R*^{2} indicates
that the relation is very strong in this case.

Almost all sandy textural classes had the highest regression coefficients.
Table 3 suggested grouping the
textural classes into two super-classes: `SC1`

, comprising the textures
sandy, sandy clay loam, sandy loam, and loamy sand, and `SC2`

, with
sandy clay, clay, clay loam, loam, and silty loam. Soils in `SC1`

are
mostly sandy soils, with the exception of the sandy clay textural class which
is within the `SC2`

soils which are mostly clayey and loamy soils.
The lowest mean *R*^{2} value for the log*K*_{sat} regressions in the
`SC1`

superclass was 0.742 and the highest one for the `SC2`

class was 0.604. Total number of soils in `SC1`

was 17 975
(94.06 % of total soils in the database). `SC2`

contained 1069
soils (5.59 % of total). Tables 4 and
5 show the *R*^{2} and RMSE values for all
regressions for the soils in `SC1`

and `SC2`

.

For the `SC1`

we observed that the best regression (*R*^{2}=0.986,
RMSE = 0.184) against ln*K*_{sat} was reached with (IE,“4-1-2”). This
triplet creates a division among the sand fractions, grouping together very
coarse, coarse, medium, and fine, and leaving the very fine sand alone.
Finally, the fine fractions contains only the silt and clays. Comparing this
to the sandy textural class results, where the best triplet was “5-1-1”, we
observed that now more information from the sandy fraction was required to
infer hydraulic properties. The area that the `SC1`

soils cover in the
textural triangle and the hydraulic property variation of these soils can be
related with a heterogeneity metric associated to triplets that distinguish
well among the predominant fraction in that area of the triangle, i.e. sand.

For the `SC2`

, the best triplet was “1-1-5”, with *R*^{2}=0.623. Regression
results were worse than for `SC1`

, but this might be just provoked by
the nature of `SC2`

itself: these are soils with less sand and thus
higher content in clays and aggregating particles. The packing – and
consequently the *K*_{sat} – of these soils is not just mainly affected by the
PSD, but also by aggregation, which cannot be accounted for in the IE
value, regardless of the triplet used.

Furthermore, the best triplet, “1-1-5”, also pointed in this direction: the
fine fraction contains medium sand, fine sand, very fine sand, silt, and sand
particles, while the intermediate fraction contains only the coarse sand,
leaving the coarse fraction with the very coarse sand, thus giving more
importance to the possibly aggregating particles than a triplet, like
“1-4-2”,
which had *R*^{2} values equal to and 0.033.

## 3.4 Triplets and scaling break

In the regressions made with all the soils, the behaviour
of (IE,“3-1-3”) was noteworthy. The average value of all triplets was 0.727, but
(IE,“3-1-3”) gave an exceptionally low *R*^{2} value of 0.087, the next
lowest being (IE,“2-2-3”), with a mean *R*^{2} value of 0.235.

The “3-2-3” triplet groups fine sand with silt and clay, and coarse and very
coarse sand with medium sand. Kravchenko and Zhang (1998), Wu et al. (1993), and Tyler and Wheatcraft (1992)
reported the break in scaling where the power law scaling of soil texture
occurred in the size range of fine sand. The particle size distribution scales
in a different way in two different regions of the size intervals, and
the change of scaling is produced around the fine sands. The triplet “3-1-3”
separates these two regions, maybe bringing forth this scaling break effect.
Figure 5. shows how the relationship between ln*K*_{sat} and (IE,“3-1-3”) could be non-linear, maybe due to the absence of
global self-similarity showed in the scaling break.

However, it is also noteworthy that regressions against
(IE,“3-1-3”) were actually quite good (*R*^{2}=0.939) in the `SC1`

,
while in the `SC2`

they were moderate (*R*^{2}=0.045).

When all the soils are considered together, then (IE,“3-1-3”) might fail, due to the scaling break, but when we restrict the study to a certain part of the textural triangle, that effect might diminish to a point where this triplet is even useful to predict some texturally derived properties, or maybe the scaling break effect is also restricted to some textural classes and should be further investigated.

As results show, IE is not a powerful ln*K*_{sat} predictor by itself, but
can be when it is combined with an input triplet. By changing the triplet, we may focus on
certain physical aspects of the soils, but it is also important to keep in
mind that this might not work statistically for random groupings of soils
that belong to different textures.

## 3.5 IE variation as a spatial function in the textural triangle

Table 6 shows, for each textural class, the ratio of
the percentage of (IE,“5-1-1”) against the percentage of the ln*K*_{sat}
range. The same ratio was also calculated using IE for the triplet that
gave the best *R*^{2} value in the linear regression against ln*K*_{sat}.
These values can be thought of as how much range of (IE,*triplet*) can
be used to explain a certain variation of ln*K*_{sat} inside each textural
class, i.e. as how much parametrizing power is available to the IE. In all
the textural classes the parametrizing power of the alternative triplet was
higher than the one using the usual clay–silt–sand triplet. For the sand
textural class, the triplet which gave the best *R*^{2} regression was
“5-1-1”, and thus the results are the same; the average value of the parametrizing power
for the usual triplet was 0.50, while when we change the triplet we obtain
0.79. This shows how, by considering different triplets, combined with IE,
a better description or parametrization of ln*K*_{sat} can be reached.

## 3.6 Final comments

Textural heterogeneity is a crucial factor affecting soil *K*_{sat}, but it
acts alongside many other ecological factors, such as animal activity, root exudates,
soil aggregation, etc. In this work we showed that a proper representation of
textural heterogeneity, by IE, allows one to (1) demonstrate its effect on
ln*K*_{sat} by binning samples based on the textural heterogeneity and (2) to statistically parametrize this effect for some textures.

This work has limitations, in particular the limited available texture data
of only seven fractions in the database. The boundaries between coarse,
intermediate, and fine fractions can be moved with data from continuous
measurements of texture in the fine sand–silt–clay range of particle sized.
This may bring the improvements in mean bin ln*K*_{sat} estimates for
non-sandy soils that could not be achieved in this work.

Although globally the IE computed from different triplets shows a potential
to reflect the effect of soil texture on the ln*K*_{sat} values, the
different relationship between the IE and the ln*K*_{sat} depending on
the triplet used might have different possible explanations. While the
IE ∕ ln*K*_{sat} relationship is found to be satisfactory in some textural
classes, results seem to indicate that the IE parameter cannot reflect with
the same efficiency the ln*K*_{sat} values in other classes predominating
fine particles, in which other processes such as aggregation or weathering cannot
be elucidated by the single textural data input.

Overall, the heterogeneity parameter, IE, combined with the different
triplet information, appears to be a strong candidate as an input for the
development of new PTFs to predict ln*K*_{sat} and
probably other soil physical parameters that are strongly dependent on soil
particle size distribution.

The PSD coarse, intermediate, and fine fractions in soil textural triplets can be redefined from standard “sand–silt–clay” to other fraction size ranges. The textural heterogeneity parameters obtained for some of the new triplets correlate with soil saturated hydraulic conductivity averaged by ranges of the heterogeneity parameters. This approach allows one to quantify the effect of the textural heterogeneity of saturated hydraulic conductivity of soils. Given that size boundaries of sand, silt, and clay fractions have not originally been established for the purposes of prediction of soil hydraulic conductivity, it may be beneficial to look for other size-based subdivisions of particle size distributions that, when used along with other soil properties such as bulk density and organic matter content, may provide better predictions of the saturated hydraulic conductivity.

The data we have used come from the reference Pachepsky and Park (2015). Requests for the data can be directed to the authors of that paper.

The authors declare that they have no conflict of interest.

This research work was funded by Spain's Plan Nacional de Investigación
Científica, Desarrollo e Innovación Tecnológica (I+D+I), under ref.
AGL2015-69697-P.

Edited by: Roberto Greco

Reviewed by: Robson André Armindo and one anonymous referee

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*K*

_{sat}) is an important soil parameter that highly depends on soil's particle size distribution (PSD). The nature of this dependency is explored in this work in two ways, (1) by using the information entropy as a heterogeneity parameter of the PSD and (2) by using descriptions of PSD in forms of textural triplets, different than the usual description in terms of the triplet of sand, silt, and clay contents.

*K*

_{sat}) is an important soil parameter that highly depends on...