GEOLOGICA CARPATHICA, OCTOBER 2005, 56, 5, 455460
www.geologicacarpathica.sk
Shaping of clay fragments during transport:
a theoretical model
JIØÍ FAIMON
Institute of Geological Sciences, Faculty of Sciences, Masaryk University, Koláøská 2, 611 37 Brno, Czech Republic;
faimon@sci.muni.cz
(Manuscript received July 22, 2004; accepted in revised form March 17, 2005)
Abstract: Irregular blocks of cohesive clay (IBC) are transformed into spherical clay balls (SCB) during transport
across a sedimentary basin bed by runoff. This transformation is controlled by rotation of the IBC about a randomly
changing axis. During this process, deformation dominates; abrasion is less significant. Based on a theoretical model,
the equation describing SCB formation,
(
)
ef
d
k
e
1
1
0
π
λ
−
Ψ
=
Ψ
, was derived in terms of the SCB instantaneous
sphericity, Ψ
Ψ
Ψ
Ψ
Ψ, the IBC initial sphericity, Ψ
Ψ
Ψ
Ψ
Ψ
0
, the transport length, λλλλλ, and the effective diameter, d
ef
. The equation was
validated with the new data set obtained from IBC/SCB distribution along the transport path on the bed of an artificial
basin (sandpit quarry). The coefficient k was found to be 2.9×10
4
and 4.8×10
4
for the shaping of two kinds of clays,
respectively. Based on the proposed model, the length of transport can be estimated if sphericity and effective
diameter are known. Despite the verification in the artificial setting, the presented model is valid also for the natural
environment of various clay rich cliffs or scarps.
Key words: Czech Republic, Rudice Formation, modelling, recent process, sedimentology, basin bed, clay ball.
Introduction
The formation of rounded clay pebbles (known as clay
balls or mud balls) was described in various environ-
ments by many authors (e.g. Haas 1927; Bell 1940;
Karcz 1969; Picard & High 1973; Ojakangas & Thomp-
son 1977; Collinson & Thompson 1982; Kale & Awasthi
1993; Tanner 1996; Faimon & Nehyba 1998a,b; Hindson
& Andrade 1999; Sholokhov & Tiunov 2002).
Recently, Faimon & Nehyba (2004) presented the
semi-empirical equation,
,
0
1
)
1
(
1
ef
5
d
10
9
.
8
0
λ
Ψ
Ψ
´
-
=
(1)
describing the SCB formation (Ψ
Ψ
Ψ
Ψ
Ψ is instantaneous sphe-
ricity, Ψ
Ψ
Ψ
Ψ
Ψ
0
is initial sphericity, λλλλλ is transport length, and
d
ef
is effective diameter of the relevant clay ball). Now, a
theoretical model has been proposed, based on which the
former equation was verified by an exact derivation. The
new model was validated with the data collected in a
small artificial sedimentary basin (sandpit quarry, the
Rudice-Seè, Czech Republic, see the map on Fig. 1).
Pit characteristics
The sandpit originated during the industrial mining
of Lower Cretaceous deposits known as the Rudice For-
mation. These deposits, more than 100 m thick, are re-
deposited products of kaolinite-laterite weathering of
non-carbonate rocks (Brno Massif granitoids, Lower
Carboniferous siliciclastics) and carbonate rocks (Juras-
sic limestones with chert and quartz geodes; see Bosák
Fig. 1. Location map of the Moravian Karst and Rudice site.
et al. 1976). The deposits consist of quartz sands (fine to
medium grained with minor sandy gravel) with abundant
456 FAIMON
chert clasts, red coloured ferruginous sandstones and
multicoloured clays (dominance of kaolinite). The sand-
pit was described in detail by Faimon & Nehyba (2004).
Clay characteristics
Red and white clays are typical constituents of the
clay balls. Their chemical and mineralogical composi-
tions are given in Tables 1 and 2, respectively. Both
clays include predominantly kaolinite and quartz, to-
gether with small amounts of other phases.
Processes on quarry slopes
The studied area is situated in the northwestwest part
of the quarry. During heavy rains, small irregular blocks
of clay (IBC) are released from steep slopes by gully ero-
sion and quickly transported to the steep slope base by
runoff and gravitation. The mechanism of the IBC forma-
tion and release was discussed in detail by Faimon & Ne-
hyba (2004).
Runoff rolls the IBC deeper into the basin in an
ephemeral shallow streambed (0.32.0 m wide and up to
0.4 m deep) that gradually transforms into a fan (up to
10 m wide and 0.2 m high). The distance from the steep
slope base to a depositional base is about 135 m with rel-
ative elevation change of 5 m. Based on a rotation about
a randomly changed axis, the IBC are rounded into
spherical clay balls (SCB). These SCB stay spread along
the transport path as the runoff has decayed.
Methods
The shape of IBC/SCB was expressed by the projec-
tion sphericity, Ψ
Ψ
Ψ
Ψ
Ψ (Sneed & Folk 1958) defined by the
equation
3
2
.
)
I
L
(
/
S
=
Ψ
(2)
The symbols L, I, and S stand for the long, intermedi-
ate, and short IBC/SCB dimensions in meters, respective-
ly. The size of IBC/SCB was quantified by an effective
diameter, d
ef
[m],
,
S
I
L
d
3
ef
=
(3)
where L, I, and S have the former meaning. The d
ef
relates
to the cube edge that has the same volume as the block
L×I×S apprehending the IBC/SCB. The d
ef
is roughly
the same for the objects of the same volumes and, thus, it
is independent on the IBC/SCB shape.
Results
The instantaneous IBC/SCB distribution along the
transport path was obtained on one day: August 26,
2001. The frequency histogram of total 322 IBC/SCB is
shown in Fig. 2. The white IBC/SCB are more abundant
(243 pieces), as visible from a comparing the two areas
Table 1: Chemical composition of the clays (analyst P. Kadlec).
Table 2: Mineral composition of the clays. The phases were quali-
tatively detected by X-ray diffraction (Stady P, analyst V. Vávra)
and then quantitatively calculated from the chemical composition.
Fig. 2. Frequency histogram of clay ball distribution along trans-
port path.
below the histogram contours. The large occurrence of
SCB at 6065 and 7075 m indicates that the distribu-
tion copies, to some extent, the basin bed morphology
(local inclinations and streambed spreading).
The frequency histogram of the clay ball sphericity Ψ
Ψ
Ψ
Ψ
Ψ is
presented in Fig. 3. The sphericity is scattered around the
value of 0.75; the mean values are 0.738 and 0.752, the
medians are 0.757 and 0.763 for the red and white
IBC/SCB, respectively. The sphericity exceeding the
red clay [wt. %]
white clay [wt. %]
H
2
O
8.61
13.06
SiO
2
54.94
46.80
TiO
2
2.45
1.12
Al
2
O
3
25.89
37.29
Fe
2
O
3
5.68
0.38
FeO
0.11
0.06
MnO
0.01
0.01
CaO
0.68
0.74
MgO
0.08
0.01
K
2
O
0.62
0.06
Na
2
O
0.16
0.00
CO
2
0.67
0.44
P
2
O
5
0.09
0.04
red clay [mol. %]
white clay [mol. %]
kaolinite
36.0
85.5
quartz
54.3
10.5
calcite
1.8
3.1
K-feldspar
1.9
0.3
albite
0.8
0.0
hematite
5.2
0.6
SHAPING OF CLAY FRAGMENTS DURING TRANSPORT: A THEORETICAL MODEL 457
value of 0.90 indicates nearly ideal spherical shape of
some SCB. The sphericity of the most irregular objects
does not go under 0.45.
The frequency histogram of the effective diameters, d
ef
,
is demonstrated in Fig. 4. The typical d
ef
of IBC/SCB was
1 to 5.5 cm. The mean diameter and median of the red
IBC/SCB were 3.7 and 3.2 cm, the mean diameter and
medians of the white IBC/SCB were 2.8 and 2.6 cm, re-
spectively. In fact, solely the IBC/SCB exceeding 0.5 cm
in diameter were monitored.
The dependence of sphericity Ψ
Ψ
Ψ
Ψ
Ψ on the distance from
steep slope base λλλλλ is shown in Fig. 5. The positive slopes
0.0011 and 0.0013 m
1
for the red and white IBC/SCB,
respectively, indicate the slight increase of Ψ
Ψ
Ψ
Ψ
Ψ with λλλλλ.
The huge data scattering is caused by the dissimilar initial
sphericities 0.450.85 of red IBC/SCB and 0.550.90 of
white IBC/SCB. In addition, some scattering could result
from the shorter transport path of the IBC released from
the ephemeral streambed banks on the basin bed (see Fai-
mon & Nehyba 2004).
Discussion
Data comparison
The found values of sphericity and effective diameter of
IBC/SCB are consistent with those presented by Faimon
& Nehyba (2004). This is not surprising because the local-
ities were the same and conditions similar. Karcz (1969)
reported analogous mud pebble sphericity; 0.60.7 near
wadimouths and 0.551.0 near clayey wadi-banks. How-
Fig. 3. Frequency histogram of clay ball sphericity.
Fig. 4. Frequency histogram of clay ball effective diameter.
Fig. 5. Dependence of sphericity, Ψ
Ψ
Ψ
Ψ
Ψ, on the distance from steep
slope base, λλλλλ.
ever, he referred to somewhat larger dimensions in com-
parison with the IBC/SCB in this study, from 4 to 10 cm
in diameter. Based on findings of Faimon & Nehyba
(2004), the IBC/SCB sizes are controlled by fissures in
clays and by intensity of gully erosion. Ojakangas & Thomp-
458 FAIMON
son (1977) stated the dominance of spheroid mud balls at
first appearance, but a majority of elliptical mud balls at
second appearance, both in an urban environment. The
size of these mud balls was lesser at the first appearance,
from 1.0 to 2.3 cm, and significantly larger at the second
appearance, the largest ball was about 25 cm long.
Discoid and bladelike mud balls, 425 cm long, were
reported in the inter-tidal zone by Kale & Awasthi
(1993). The preponderance of elongated mud balls rang-
ing in size from 2 to 20 cm in diameter was also reported
in the similar environment by Tanner (1996).
Mathematical model
A new model was derived to evaluate the IBC/SCB
shaping at transport. The model is based on the principal
assumption that the clay block shape is controlled by ro-
tation of the IBC about a randomly changing axis (the
axis orientation is presumed to be uniformly distributed
on the interval [0, 2πππππ]). The other assumptions are as fol-
lows: The sphericity ψ
ψ
ψ
ψ
ψ quantifies the instantaneous
shape of clay ball. The condition ψ
ψ
ψ
ψ
ψ=1 characterizes a
steady state shape during transport (i.e. spherical
shape). The deviation from the steady state shape is 1ψ
ψ
ψ
ψ
ψ.
The high deviation (extremely irregular shape) will
change more quickly than the slight deviation. There-
fore, the increment of sphericity with number of rota-
tions, dΨ
Ψ
Ψ
Ψ
Ψ/dn, will be proportional to the deviation 1ψ
ψ
ψ
ψ
ψ,
(
)
,
1
k
dn
d
Ψ
−
=
Ψ
(4)
where k is a dimensionless constant.
The coefficient k itself includes the mechanical prop-
erties of the clay and basin bed. The possible effects of
the IBC/SCB mass, runoff layer thickness, and runoff
intensity were neglected (see discussion in Faimon &
Nehyba 2004). Integrating the equation (4) in the initial
conditions that Ψ
Ψ
Ψ
Ψ
Ψ = Ψ
Ψ
Ψ
Ψ
Ψ
0
if n = 0 (where Ψ
Ψ
Ψ
Ψ
Ψ
0
is the initial
sphericity) gives
.
e
1
1
n
k
0
-
-
-
=
Ψ
Ψ
(5)
Other symbols have their standard meaning. The num-
ber of rotations, n, is a function of transport length and
rotation perimeter. If the distance from the steep slope
base, λλλλλ, is simply assumed to be a transport length, then
n,
p
=
λ
(6)
where p is the rotation perimeter.
The question is how to evaluate the perimeter, p, un-
ambiguously. This is rather difficult because the perime-
ter of non-spherical objects can vary during transport.
For example, the perimeter of as simple and regular ob-
ject as a cube can be very different, depending on wheth-
er the cube has rolled by the way of its edges or its
corners. Furthermore, the perimeter could decrease dur-
ing transport if abrasion has operated. The dependence of
d
ef
on λλλλλ, is presented in Fig. 6. Actually, the negative
slopes, 0.017 and 0.010 m m
1
for the red and white
balls, respectively, indicate a slight decrease of the effec-
tive diameters with the distance from steep slope base.
However, the R-square values, 0.071 and 0.045, docu-
ment very weak correlation. Based on these facts, the de-
crease of d
ef
was neglected and the effective diameter was
assumed constant. This postulates that deformation dom-
inates and abrasion is not significant.
The perimeter was expressed from the effective diame-
ter as
.
d
p
ef
π
=
(7)
Inserting (7) into (6) gives
λ = π d
ef
n. (8)
Rewriting the equation (8) for n and inserting into (5)
yields
(
)
.
e
1
1
ef
d
k
0
π
λ
−
Ψ
−
−
=
Ψ
(9)
As can be seen, the equation (9) is only slightly modi-
fied in comparison with the equation (1). It authorizes
the semi-empirical approach of Faimon & Nehyba
(2004).
On the basis of the data processing, the constant k was
determined. The logarithm of the equation (9) gives the
expression
,)
1
(
ln
d
k
)
1
(
ln
0
ef
Ψ
−
+
π
λ
−
=
Ψ
−
(10)
representing the linear dependence of ln(1 Ψ
Ψ
Ψ
Ψ
Ψ) on λλλλλ/d
ef
with the k/πππππ slope. Plotting data in these coordinates
(Fig. 7) and regressing by the equation (10) give the
slopes 9.2×10
5
and 1.5×10
4
, from which k values,
2.9×10
4
and 4.8×10
4
, were calculated for the red and
white clays, respectively. In Fig. 8, the theoretical func-
tions (equation 9) are plotted for various initial sphericities.
Despite the data scattering, a relatively good agreement
of the calculated curves with the data points is visible.
Comparing the two models
Rewriting the equations (1) and (10) gives
κ
−
=
−
Ψ
−
Ψ
λ
1
1
log
d
0
ef
(11)
and
,
e
log
k
1
1
log
d
0
ef
π
−
=
−
Ψ
−
Ψ
λ
(12)
respectively. The symbol κκκκκ corresponds to the constant
8.9×10
5
from the previous model of Faimon & Nehyba
(2004); e is the Euler number, the base of the natural log-
arithm. Comparing the right sides of the equations (11)
and (12) yields
.
e
log
k
π
=
κ
(13)
Evaluating the former equation with the found k gives
κκκκκ~4.0×10
5
and 6.6×10
5
for the red and white clays, re-
spectively. The two former values are somewhat lower
SHAPING OF CLAY FRAGMENTS DURING TRANSPORT: A THEORETICAL MODEL 459
than the κκκκκ~8.9×10
5
found by Faimon & Nehyba (2004)
for the red clays. This difference could be a consequence
of changeable mechanical properties of clays. For exam-
ple, dry IBC needs a certain time to acquire its deforma-
tion ability. If heavy rains come after a long dry period,
this ability can be reached after the initiation of trans-
port. The calculation using the value of Faimon & Nehyba
(2004), κκκκκ=8.9×10
5
, gives k=6.4×10
4
. Therefore, the
verified interval of the constant k for clay ball shaping is
from 2.9×10
4
to 6.4×10
4
. The most probable value is
k=(4.7±1.75)×10
4
.
Transport length estimation
Rewriting the equation (10) for λλλλλ gives the equation
,
1
1
ln
k
d
0
ef
Ψ
−
Ψ
−
π
=
λ
(14)
which allows estimation of the minimal transport length
so that the given shape was reached. Using the former
value k=4.7×10
4
, the calculation yields the transport
length e.g. of 211, 105 and 53 m for the IBC/SCB of the
effective diameters of 0.02, 0.01 and 0.005 m, respective-
ly, so that the initial sphericity Ψ
Ψ
Ψ
Ψ
Ψ
0
=0.75 increases to
Ψ
Ψ
Ψ
Ψ
Ψ = 0.95.
Conclusions
The presented model is universal; it is valid for the de-
scription of spherical shape formation of various objects
by transport (under the condition that orientation of rota-
tion axis is uniformly distributed on the interval [0, 2π]).
The model was derived for the transformation of small ir-
Fig. 6. Dependence of the effective diameter, d
ef
, on the distance
from steep slope base, λλλλλ.
Fig. 7. Dependence of ln(1 Ψ
Ψ
Ψ
Ψ
Ψ) on λλλλλ/d
ef
.
Fig. 8. Data fitting by the theoretical function.
regular blocks of cohesive clay (IBC) into spherical clay
balls (SCB) during rolling under the effect of runoff. It was
460 FAIMON
based on the assumption that the change of shape with
number of rotations depends on the deviation of ob-
served shape from steady state shape (spherical). Clay
deformation was found to control the shaping; abrasion
seems to be less significant.
The model enables estimation of the length of transport,
during which the IBC changed into the SCB. The parame-
ter k=(4.7±1.75)×10
4
is valid for the sand/clayey envi-
ronment. Although the shaping process was evaluated in
artificial object (sandpit quarry), the model parameters are
believed to be valid for analogous natural environment, such
as clay rich cliffs and scarps in many settings worldwide.
Acknowledgments: The author wish to thank Dr. P. Èej-
chan and Prof. P. Bosák (Institute of Geology, Academy
of Science of the Czech Republic), Prof. D. Vass (Geo-
logical Institute, Slovak Academy of Sciences), and two
anonymous reviewers for their constructive comments.
Dr. V. Vávra and P. Kadlec (Institute of Geological Sci-
ences, Faculty of Sciences, Masaryk University) are
thanked for performing of analyses.
References
Bell H.S. 1940: Armored mud balls their origin, properties, and
role in sedimentation. J. Geol. 48, 131.
Collinson J.D. & Thompson D.B. 1982: Sedimentary structures.
Alen & Unwin, London, 1194.
Bosák P., Glazek J., Gradzinski R. & Wojcik Z. 1976: Genesis and
age of sediments of the Rudice type in fossil-karst depressions.
Èas. Mineral. Geol. 24, 147154.
Faimon J. & Nehyba S. 1998a: Recent formation of spherical mud
balls at Rudice-Seè. Geol. Res. in Morav. Siles. in 1997, ÈGÚ,
Masaryk. Univ. 1, 23 (in Czech).
Faimon J. & Nehyba S. 1998b: The recent formation of spherical
clay balls on sedimentary basin slopes. Book of abstracts: 15
th
Conf. Clay Miner. Petrol., Sept. 310, 1998, Brno. Scripta
Fac. Sci. Nat. Univ. Masaryk. Brun. (Geology) 26, 176.
Faimon J. & Nehyba S. 2004: The formation of spherical clay balls
on the slopes of sandpit quarry, the Rudice-Seè (Czech Repub-
lic). Catena 58, 2340.
Haas W.H. 1927: Formation of clay balls. J. Geol. 35, 150157.
Hindson R.A. & Andrade C. 1999: Sedimentation and hydrody-
namic processes associated with the tsunami generated by the
1755 Lisbon earthquake. Quart. Int. 56, 2738.
Kale S.V. & Awasthi A. 1993: Morphology and formation of ar-
mored mud balls on Revadanda Beach, Western India. J. Sed.
Petrology 63, 809813.
Karcz I. 1969: Mud pebbles in a flash floods environment. J. Sed.
Petrology 39, 333337.
Ojakangas R.W. & Thompson J.A. 1977: Modern armored mud
balls in an urban environment. J. Sed. Petrology 47,
16301633.
Picard M.D. & High Jr. L.R. 1973: Sedimentary structures of
ephemeral streams. In: Development in Sedimenology.
Vol.
17. Elsevier, Amsterdam, 1223.
Sholokhov V.V. & Tiunov K.V. 2002: The first finding of ar-
moured mud balls in Turkmenistan. Lithol. Miner. Resour. 37,
9293.
Sneed E.D. & Folk R.L. 1958: Pebbles in the lower Colorado River,
Texas: a study in particle morphogenesis. J. Geol. 66,
114150.
Tanner L.H. 1996: Armoured mud balls revisited. Atl. Geol. 32,
123125.