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Geoderma 161 (2011) 177–181

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Geoderma
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / g e o d e r m a

Determining the inﬂuence of stones on hydraulic conductivity of saturated soils using
numerical method
Viliam Novák a,⁎, Karol Kňava a, Jirka Šimůnek babInstitute of Hydrology, Slovak Academy of Sciences, Račianska 75, 83102 Bratislava, Slovakia
Dept. of Environmental Sciences, Univ. of California Riverside, Riverside, CA 92521, USAa r t i c l e

i n f oArticle history:
Received 24 May 2010
Received in revised form 3 December 2010
Accepted 21 December 2010
Available online 21 January 2011
Keywords:
Saturated hydraulic conductivity
Stony soil
Soil coarse fraction
HYDRUSa b s t r a c t
Mountainous soils usually contain a large number of rock fragments (particle diameter N 2 mm), whichinﬂuence soil hydraulic and retention properties. Data characterizing the properties of these soils usuallydescribe only the ﬁne earth (particle diameter b 2 mm). To quantitatively describe soil water movement instony soils, their most important characteristic, i.e., the effective saturated hydraulic conductivity (Kse), must
be known. The objective of this study was to use a numerical method for estimating the saturated soil
hydraulic conductivity that depends on relative stone content (stoniness), and sizes and shapes of stony parts.
The method is based on a numerical version of the classical experiment of Darcy. The steady-state water ﬂow
under a unit hydraulic gradient through hypothetical soils containing stones was simulated using the twodimensional simulation model HYDRUS-2D. Four soil textural types were considered. Special attention was
paid to the moraine soil from the FIRE site in High Tatras. A relationship between the relative saturated
hydraulic conductivity, Kr, and the relative stone content, Rv, was derived. Kr(Rv) function is decreasing slower
for larger stones. Numerical results were used to propose an empirical equation to estimate Kr of soils
containing rock fragments of a spherical shape of various diameters.
© 2011 Elsevier B.V. All rights reserved.

1. Introduction
A soil, often described as a weathered upper layer of the Earth's
surface (Hillel, 1971), is the source of water and nutrients, and an
ideal growth environment for plants (Hraško & Bedrna, 1998; Jury &Horton, 2003). Soil particles, which are a product of weathering, have
many different sizes and shapes. It is generally accepted to divide soil
particles into two fractions: the ﬁne earth with a diameter of particles
less than 2 mm, and the rock fragments with particles larger than
2 mm.
Water moves through pores, the size and shape of which dependon sizes, shapes and arrangements of solid particles. Soil water
movement is usually described on a macroscopic scale, since for thequantitative description of water movement in pores (a microscopic
approach) one would need to describe the geometry of porous
pathways. The most frequently used macroscopic equation describing
soil water movement in homogeneous (or layered) soils is theRichards equation (Richards, 1931). This equation needs the “macroscopic” soil hydraulic characteristics as an input.
The soil hydraulic properties (i.e., the hydraulic conductivity
function and the retention curve) are in the macroscopic approach⁎ Corresponding author. Tel.: +421 2 44259383; fax: +421 2 44259404.
E-mail address: novak@uh.savba.sk (V. Novák).
0016-7061/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.geoderma.2010.12.016

characterized for the so-called representative elementary volume(REV) of the soil (Bear, 1972; Nordahl & Ringrose, 2008). REV is such a
volume of the porous media (soil), that hydraulic characteristics
would change when a smaller soil volume was used for theirdetermination, and would remain unchanged when a larger soilvolume was used. REV has to, therefore, contain enough pores and soil
particles to represent the soil macroscopically, so that the Darcy's
approach can be used.
Baker and Bouma (1976), Lichner (1994), and Kutilek and Nielsen(1994), recommended to deﬁne REV as a soil sample with its crosssection containing no less than 20 elementary units of the soil structure(particles, peds, or macropores). A cross-sectional area corresponding to
REV is usually between 10 cm2 and 1 m2 (Kutílek & Nielsen, 1995).
Current methods for determining basic soil texture and soil hydrauliccharacteristics usually use soil samples with a volume of 100 cm3(Kopecky cylinders). Kopecky cylinders fulﬁll the above requirement
only for the ﬁne soil fraction. To estimate hydraulic characteristics ofsoils containing rock fragments, it is necessary to either use larger soil
samples, to measure hydraulic characteristics in the ﬁeld, or to develop
new methods for their determination. For soils containing stones a soil
sample of 1 m3 volume would be needed.
Since arable soils, after decades or centuries of stone removal, areusually composed mainly of ﬁne earth, standard procedures can beapplied to evaluate their hydraulic characteristics. On the other hand,
mountainous (or forest) soils usually also contain coarse rockfragments (Mehuys, et al., 1975). At the site under consideration

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178

V. Novák et al. / Geoderma 161 (2011) 177–181(FIRE site in High Tatras), at a depth of 50 cm below the soil surface,
up to 49% of the soil volume consists of rock fragments larger than
1 cm in diameter (Novák et al., 2008). Such high relative content ofstones signiﬁcantly inﬂuences both soil water retention and hydraulicconductivity and their determination thus needs a special attention.
The ﬁrst objective of this study is to propose a method for
determining the effective saturated hydraulic conductivity Kse of soils
containing signiﬁcant amount of stone fragments, and to compare
obtained results with those for the ﬁne earth of the same soil. The
second objective is to compare Eqs. (1)–(5) to estimate effective
saturated hydraulic conductivity of stony soils (Kse) with the proposed
method.
2. Theory and methods
2.1. Saturated hydraulic conductivity of stony soils
Multiple equations have been derived to calculate the saturated
hydraulic conductivity of soils containing rock fragments Kse, LT−1.
Equations for Kse calculation developed by Corring and Churchill(1961) and Vasiliev and Tanaeva (1971) are based on the analyticalsolution of heat ﬂux in dispersed systems.
The hydraulic conductivity of the ﬁne earth saturated with water
Ks, LT−1 is constant and the hydraulic conductivity of rock isapproximately zero (it can be neglected when compared with Ks of
the ﬁne earth). As a result, equations derived by Corring and Churchill(1961) can be written for stony soils as follows:
Kse = Ks
2ð1−Rv Þ
2 + Rv

Kse = Ks

1−Rv
1 + Rvð1Þð2Þ

where Rv is the relative volume of the dispersed phase, i.e., the ratio ofthe volumetric stone content (Vr) and the soil volume (V). This ratio issometimes called stoniness. Eq. (1) was derived for spherical stones
and Eq. (2) for cylindrical stones. The orientation of the cylindricalstones with ﬂow direction is not mentioned. Peck and Watson (1979)(acc. to Bagarello & Iovino, 2007) derived an equation identical with
Eq. (1).
Bouwer and Rice (1984) intuitively derived an equation for
calculation of Kse in the form of the relative saturated hydraulicconductivity Kr:
Kr =

Kse
e
= se
Ks
es

ð3Þ

where ese and es are void ratios of the soil with stones and of the ﬁnesoil fraction alone, respectively. Brakensiek et al. (1986) showed, thatEq. (3) can be approximated to:
Kr =

Kse
= 1−RwKs

ð4Þ

where Rw (= mr / m) is the relative stone content (stoniness),
expressed in mass units; mr is the mass of stones in soil, and m is
the mass of soil with stones in a given volume of soil.
3. Method
As it was stated above, the second objective of this work will beachieved by numerically simulating steady-state water ﬂow under aunit hydraulic gradient through hypothetical soils containing stones
using the two-dimensional simulation model HYDRUS-2D (Šimůnek
et al., 2006, 2008). As will be shown below unsaturated zones develop
below the stones depending on their sizes and thus variably-saturated

Fig. 1. Distribution of water pressure heads during steady-state water ﬂow through thesoil containing stones (calculated using HYDRUS-2D). A cross-sectional area of the soil
sample is 1 m2, a diameter of dispersed stones is 10 cm, and Rv = 0.071. Steady-statewater pressure heads ranged from −2 to 6 cm.

ﬂow model is needed to solve this problem. HYDRUS-2D is a computermodel that solves numerically highly nonlinear Richards equation fora selected set of soil hydraulic functions. Note that the Richards
equation can be linearized using the Kirchhoff transformation with an
exponential conductivity function and then analytical solutions can bederived for relatively simple geometries and homogeneous (e.g.,
Kacimov, 2000; Philip, 1968; Raats, 1971) or heterogeneous (Bakker &Nieber, 2004; Sviercoski et al., 2009; Warrick and Knight, 2002) soils.
For example, Warrick and Knight (2002) presented an analyticelement solution for a single embedded cylindrical inhomogeneity
having a different saturated hydraulic conductivity than the surrounding uniform soil. However, for soils characterized using soil
hydraulic functions of a general shape (e.g., van Genuchten, 1980), theeffect of embedded inhomogeneities on vadose zone ﬂow andtransport can be quantiﬁed only using numerical solutions of the
Richards equation (e.g., Ju and Kung, 1993, 1997).
Water ﬂow in a vertical cross-section (1 * 1 m) of a hypotheticalsoil sample of a 1-m3 volume (1 * 1 * 1 m) with impermeable side faces
was simulated using HYDRUS-2D (Šimůnek et al., 2006, 2008). Stones
were approximated in this cross-sectional area as circles, representing
thus impermeable cylinders in the full three-dimensional sample.
As 1 cm of water head was applied at both upper and lower
surfaces, the pressure water head difference between the upper andlower boundaries was exactly 1 m. For a unit pressure head gradient
the rate of water ﬂow equals to the saturated hydraulic conductivity
Kse.
The stone content (stoniness) of this virtual soil sample differed
between individual experiments. First, water ﬂow was simulated
through a homogeneous soil consisting only of the ﬁne earth fraction

Table 1
A number (n) and a diameter (D) of spherical stones, and the relative stone content Rv
in hypothetical stony soil samples used in simulations; d1 is a diameter of a single(integral) stone sphere corresponding to a particular Rv.
n

D [cm]

Rv[cm3 cm-3]ndi [cm]Rv[cm3 cm-3]

9
20
30
40

10

0.071
0.157
0.235
0.314

1

24
50.5
61.8
80

0.045
0.2
0.3
0.5

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V. Novák et al. / Geoderma 161 (2011) 177–181

179

Table 2
Textural characteristics of the soil ﬁne earth at the FIRE site (using data of Bärwolf,
2006).
Depth cm

Particles diameter
range [mm]Percentage of soil
fraction (averages)

Percentage of soil
fraction (range)

Soilfraction

0-110

N 0.050.05–0.002
b 0.002

60
28
12

55–65
26–29
7–15

Sand
Silt
Clay

of the stony soil. Here, the resulting ﬁnal water ﬂux must be equal to
the saturated hydraulic conductivity of the ﬁne earth (Ks). Next,
numerical experiments were performed for relative volumes of stones
equal to 0.071, 0.157, 0.235, and 0.31. Spherical stones, represented byimpermeable circles with a diameter of 10 cm were chosen to
represent stones in the soil proﬁle. This diameter was close to theaverage diameter found at the FIRE ﬁeld site. To cover a wide range ofreal stones, numerical experiments were conducted also for stones
with a diameter of 5, 10, and 20 cm, and for different stoniness.
Stones as impermeable circles were regularly distributed in thevirtual soil sample to obtain required relative volumes. To evaluate
the inﬂuence of the different stone size on Kse at the same relative
stone content, the stone contents (0.09, 0.2, 0.3, and 0.4) of soil
samples with stones of diameter 10 cm were reproduced with one big
stone of the spherical (circular in 2D) shape in the second set ofexperiments. Two different values of Kse for the same relative stone
content Rv were thus obtained.
4. Results and discussion
An example of spatial distributions of stones and pressure heads in
one simulation is shown in Fig. 1. Note positive pressure heads on the
top of stones and negative pressure heads below the stones.
Numerical solution of this ﬂow problem is required because of thesenegative pressure head zones below stones. If the entire soil proﬁlewas saturated, including areas below stones, the highly nonlinear
Richards equation would simplify into a linear Laplace equation,
which could be solved using various analytical or semianalyticalmethods.
Characteristics of rock fragments (stones) and virtual soil samples
used in numerical experiments are given in Table 1. Water ﬂow
simulations were performed for four types of the ﬁne earth fraction, toestimate their inﬂuence on the effective hydraulic conductivity ofstony soils. The particle distribution of the ﬁne earth fraction from themoraine soil sample from the FIRE site in High Tatras is given in
Table 2. This soil represented in this study the actual soil. Additional
simulations were carried out for three textures, i.e., sandy loam, loam,
and clay, that were taken from the HYDRUS database. Soil hydraulicparameters of the four soils considered in this study are given in
Table 3.
Results of water ﬂow simulations of steady-state water ﬂow forsoil samples with different stone contents are presented in Fig. 2. Fig. 2

Fig. 2. Effective saturated hydraulic conductivities Kse as a function of the relative stone
content Rv calculated using numerical experiments. Loamy sand from the FIRE site inHigh Tatras (1), sandy loam (2), loam (3), and clay (4). Full lines represent soils withuniformly dispersed stones of a 10-cm diameter, dashed lines represent soils with asingle large stone and the same Rv. Soil characteristics are given in Tables 1 and 3.

shows that there is an inverse relationship between the effective
hydraulic conductivity, Kse, and the relative stone content of the soil,
Rv, for all four studied soils. Fig. 2 also shows that the effective
hydraulic conductivity of a soil from the FIRE site with a given Rv islower when the soil contains a single “large” stone than when it
contains multiple 10-cm diameter stones. Similar results wereobtained also for a clay soil. For the other two textures, i.e., loam
and sandy loam, the effective saturated soil hydraulic conductivities
Kse were similar for soils with a single stone or multiple stones withthe same stoniness. The differences in Kr for different stone diameterare very small, which simpliﬁes the calculation of Kr in this range of Rv.
Results of numerical experiments with soils containing stones ofvarious sizes are presented in Fig. 3. Note that the decrease of Kse withincreasing Rv is different for soils with stones of different sizes. The
effective hydraulic conductivity Kse for the soil containing the“smallest” stones with a diameter of 5 cm decreases faster, comparedto the soil containing stones with “bigger“ diameters. It can beconcluded that the bigger the stones in the soil, the slower thedecrease of its effective saturated hydraulic conductivity withincreasing relative stoniness. Differences in the Kse(Rv) relationshipfor different sizes of rock fragments can be explained by different
hydraulic losses due to bypassing of the stones. The smaller thestones, the higher the resistance to ﬂow at a given stoniness, because
for the same Rv the number of small stones increases.
Fig. 4 shows a comparison of the Kr(Rv) relationships obtainedusing Eqs. (1), (2), and (4) with those obtained using numericalexperiments for loamy sand from the FIRE site in High Tatras with

Table 3
Soil hydraulic characteristics of the soil ﬁne earth of four soils with different texture. θrand θs are the residual and the saturated soil water contents, respectively, α and n are
the van Genuchten (1980) shape parameters, and Ks is the saturated hydraulicconductivity of the ﬁne earth.
Soil

θr

θs

α [cm−1]nKs [cm min−1]

Sandy loam
Loam
Clay
Loamy sand (FIRE)

0.065
0.078
0.068
0.040

0.410
0.430
0.380
0.622

0.075
0.036
0.008
0.106

1.890
1.560
1.090
1.230

0.074
0.017
0.003
0.465

Fig. 3. Relative saturated soil hydraulic conductivities Kr as a function of the relative
stone content Rv (calculated using HYDRUS-2D). Loamy sand from the FIRE site in High
Tatras with three different stone diameters (5, 10, and 20 cm) was used.

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V. Novák et al. / Geoderma 161 (2011) 177–181

the identiﬁcation of stone properties (sizes, shapes and theirdistribution), it can be expressed using a linear equation as follows:
Kr = 1−a Rv :Fig. 4. Relative saturated soil hydraulic conductivities Kr as a function of the relative
stone content Rv (calculated using HYDRUS-2D). The loamy sand from the FIRE site withuniformly dispersed stones of a 10-cm diameter (curve 4), a single big stone and thesame Rv (curve 5). Other Kr(Rv) functions were calculated using Eq. (1) (curve 1),
Eq. (2) (curve 2), Eq. (4) (curve 3), and a linear relationship (line 6). Soil hydraulicparameters used in calculations are given in Table 3.

either multiple regularly distributed stones with a diameter of 10 cm
or a single big stone. The Kr(Rv) relationships obtained with numericalexperiments are similar to those evaluated using analytical solutionsof the heat transport equation and applied to porous media. While thesolutions would be identical if the soil proﬁles were fully saturated, forwhich the nonlinear Richards equation would convert into a linear
Laplace equation, some deviations between the two solutions could be
due to unsaturated zones that developed in the ﬂow “shadows” of thestones. As the numerical solution of the Richards equation implemented in HYDRUS-2D can account for the effects of these zones, it
should provide more realistic results. Analytical solutions of the linear
Fourier heat transport equation with a constant thermal conductivity
are thus only approximations. Notice that Corring and Churchill's(1961) Eqs. (1) and (2) obtained for heat transport through a
conductive medium with dispersed bodies of spherical and cylindricalshapes, respectively, produce lower values of Kr for cylindricalimpermeable objects.
Sauer and Logsdon (2002) identiﬁed, using inﬁltration tests, asmall increase of Kse with an increase of the stone content in two soil
types at the pressure head of −12 cm. In several other cases Kse did
not respond to changes in Rv. Sauer and Logsdon (2002) estimated theKse(Rv) relationship using inﬁltration tests and some of their results
can likely be explained by spatial variability of both variables. A
possible explanation for an Kse increase with increasing Rv at positive
pressure heads are the so-called lacunar pores, i.e., stable pores
existing at the interface between stones and ﬁne soil particles. Such
relatively stable macropores preserve their size and shape, and theirvolume is proportional to the stone content (Fiés et al., 2002). Thesetypes of pore are speciﬁc for some types of stony soils. In our case (aloamy sand with stones at the FIRE site), the soil was classiﬁed as themoraine soil and no macropores at the interfaces between stones andthe soil matrix were observed. Therefore, estimated Kr values decrease
with an increase in Rv.
The Kr(Rv) relationship is in general nonlinear. However, as a ﬁrst
approximation for calculation of Kr, when there are uncertainties inTable 4
Parameters a in Eq. (7) for soils with different texture and different stone sizes.
Soil typea (stone diameter = 10 cm)

a (a single stone)

Loamy sand (FIRE)
Sandy loam
Loam
Clay

1.2
1.1
1.1
1.32

1.66
1.32
1.32
1.93

ð5Þ

Values of the coefﬁcient a were estimated to be between 1.1 and 1.32
for soils containing stones with a diameter of 10 cm. The lower value
was estimated for sandy clay, the higher value for clay (Table 4).
Fine textured soils containing single stones can be characterizedusing a higher coefﬁcient a, generally within the range from 1.32 to
1.93. Coefﬁcients a for all four analyzed soil types and multiple andsingle stones are given in Table 4. Eq. (5) can be used in the range of
0 b ≤ 0.4. It can be seen in Fig. 2, Eq. (5) can really cover Kr (Rv) linear
relationship in the Rv range, under consideration.
Numerical results indicated several conclusions: a) Kr(Rv) relationships are nonlinear, b) Kr decreases with the relative stoniness, and c)
Kr(Rv) decreases faster for stones with a smaller diameter.
5. Conclusions
Hydraulic conductivities of saturated stony soils (Kse) decrease
nonlinearly as a function of the relative stone content Rv. Their values
are in general lower than those expressed by the linear function (5)(line 6, Fig. 4), because of the additional hydraulic resistance of therock fragments to ﬂow. Kse decreases with an increase in stoniness(Rv), and is additionally also sensitive to the stone size. Kse (Rv),decreases slower with increasing stone diameter. A relationshipbetween the relative saturated hydraulic conductivity, Kr, and therelative stone content Rv, can be approximately expressed by thelinear Eq. (5). The coefﬁcient a varies from 1.1 to 1.32 for sphericalstones of a 10-cm diameter, depending on the soil matrix. Eq. (5) canbe used to estimate the relative saturated hydraulic conductivity ofstony soils for soils with different texture as an alternative to Eqs. (1),(4), which do not consider soil texture and stone diameter.
The method presented here can be used to calculate Kse of stonysoils with various stoniness, dimensions and shapes of stones, using
coefﬁcients a (Table 4).
Acknowledgements
The authors are grateful to the Slovak Grant Agency APVV (Grant
No. 51-030205) and VEGA (Project No. 2/0021/10) for the partial
support of this work.
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