Characterization of Particle-Size Distribution in Soils with a Fragmentation Model
Marco Bittelli,* Gaylon S. Campbell and Markus Flury
ABSTRACT
to be best suited. The Shiozawa and Campbell model
divides the particle distribution into two parts domi-
Particle-size distributions (PSDs) of soils are often used to estimate
nated by primary (sand and silt) and secondary (clay)
other soil properties, such as soil moisture characteristics and hydraulic
conductivities. Prediction of hydraulic properties from soil texture minerals, respectively. However, as pointed out by Bu-
requires an accurate characterization of PSDs. The objective of this
chan et al. (1993), the assumption of a lognormal distri-
study was to test the validity of a mass-based fragmentation model
bution in the clay fraction cannot be justified because
to describe PSDs in soils. Wet sieving, pipette, and light-diffraction
Shiozawa and Campbell (1991) had no data available
techniques were used to obtain PSDs of 19 soils in the range of 0.05
in that range.
to 2000 m. Light diffraction allows determination of smaller particle
One of the latest developments in the study of PSDs
sizes than the classical sedimentation methods, and provides a high
in soils has focused on the use of fractal mathematics
resolution of the PSD. The measured data were analyzed with a mass-
to characterize particle sizes in soil (Turcotte, 1986;
based model originating from fragmentation processes, which yields
Tyler and Wheatcraft, 1992; Wu et al., 1993). However,
a power-law relation between mass and size of soil particles. It was
questions remain about the validity and applicability of
found that a single power-law exponent could not characterize the
PSD across the whole range of the measurements. Three main power- fractal concepts to PSDs. There has been some discus-
law domains were identified. The boundaries between the three do- sion about the proper use and definition of the term
mains were located at particle diameters of 0.51 0.15 and 85.3
 fractal in the literature (Young et al., 1997; Pachepsky
25.3 m. The exponent of the power law describing the domain be-
et al., 1997; Baveye and Boast, 1998). Different concepts
tween 0.51 and 85.3 m was correlated with the clay and sand contents
of fractals are used, and these concepts lead to different
of the soil sample, indicating some relationship between power-law
interpretations of fractal dimensions obtained. There-
exponent and textural class. Two simple equations are derived to
fore it is essential to clearly specify the type of fractal
calculate the parameters of the fragmentation model of the domain
model used.
between 0.51 and 85.3 m from mass fractions of clay and silt.
Particle- and aggregate-size distributions are often
rendered as cumulative functions, either as number of
particles larger than a certain diameter, or as mass
article-size distribution in soil is one of the more
smaller than a certain diameter. These cumulative distri-
Pfundamental soil physical properties. It is widely
bution functions have been analyzed with power-law
used for the estimation of soil hydraulic properties such
relations and the exponents interpreted as fractal di-
as the water-retention curve and saturated as well as
mensions. Tyler and Wheatcraft (1989, 1992) analyzed
unsaturated conductivities (Arya and Paris, 1981;
particle-size data ranging from 0.5- to 5000- m radii,
Campbell and Shiozawa, 1992). Generally, a conven-
and observed that the fractal power law was not valid
tional particle-size analysis involves the measurement
across the entire extent of particle sizes. It is expected
of the mass fractions of clay, silt, and sand. These frac-
that there are lower and upper limits to the validity
tions may be used to find the textural class using a
of fractal relations (Turcotte, 1986). Wu et al. (1993)
textural diagram, commonly in form of a textural trian-
measured PSDs down to 0.02- m radius by using light-
gle (e.g., Gee and Bauder, 1986). However, soil samples
scattering techniques, and found a power-law relation
that fall into a certain textural class may have consider-
between number of particles and particle radius valid
ably different PSDs. For example, the textural class of
across a range of particle radii with a lower cutoff be-
 clay in the USDA classification scheme (Gee and
tween 0.05 and 0.1 m and an upper cutoff between 10
Bauder, 1986) contains soil samples that vary in clay
and 5000 m. Assuming that the exponent of a power-
content between 40 and 100%. The size definitions of
law relation is a fractal dimension, Wu et al. (1993)
the three main particle fractions of clay, silt, and sand,
found a dimension of D 2.8 0.1 for the four soils
used as diagnostic characteristics in most classification
studied and suggested that this might be a universal
schemes, are rather arbitrary, and they do not provide
value of an underlying structure. Kozak et al. (1996)
complete information on the soil PSD.
analyzed PSDs of 2600 soil samples and found that for
A more accurate description of texture is obtained
50% of the samples power-law scaling of particle num-
by defining a PSD function. Commonly, PSDs are re-
bers vs. size was not applicable across the whole range
ported as cumulative distributions, and different func-
of particle sizes between 2 and 1000 m. The authors
tions have been proposed to fit experimental data. Bu-
indicate that power-law scaling might be applicable for
chan et al. (1993) fitted several of these models to
a narrower range of particle sizes, although this was not
experimental data and found the bimodal lognormal
analyzed in their study.
distribution proposed by Shiozawa and Campbell (1991)
Most applications of fractal concepts to particle- and
aggregate-size distributions are based on the fragmenta-
M. Bittelli, G.S. Campbell, and M. Flury, Department of Crop and
tion model of Matsushita (1985) and Turcotte (1986).
Soil Sciences, Washington State University, Pullman, WA 99164. Re-
ceived 26 Aug. 1998. *Corresponding author ([e-mail]).
Abbreviations: PSD, particle-size distribution; RMSE, root mean
Published in Soil Sci. Soc. Am. J. 63:782 788 (1999). square error.
782
BITTELLI ET AL.: FRAGMENTATION MODEL TO CHARACTERIZE PARTICLE-SIZE DISTRIBUTION 783
Table 1. Soil classification, geological parent material, percentage of sand, silt and clay by weight, and organic C content for the 19 soils
used. Particle-size data were obtained by sieving and light-diffraction methods. Textural classes are according to the USDA classification.
Soils Soil classification Geological parent material Sand Silt Clay OCż
%
Affoltern Typic Hapludalf moraine 47.4 48.5 4.1 2.2
Aeugst Typic Hydraquent fluvial deposits 38.7 55.6 5.7 1.9
Buelach Typic Hapludalf gravel deposits 57.2 40.4 2.4 2.2
Les Barges Mollic/Aquic Udifluvent fluvial deposits 74.2 25.5 0.3 1.0
Mettmenstetten Lithic Ustorthent moraine 55.6 40 4.4 0.4
Murimoos Lithic Medihemist fluvial deposits 69.9 29.6 0.5 1.0
Obermumpf Lithic Rendoll limestone 25.6 69.2 5.2 0.5
Obfelden Typic Hydraquent fluvial deposits 36.3 59.6 4.1 0.6
Palouse! Ultic Haploxeroll loess 13.2 68.6 18.2 naÅ›
Reckenholz Vertic/Typic Eutrochrept moraine 23.8 70.7 5.5 1.3
Red Bluff! Ultic Palexeralfs fluvial deposits 17.9 36.5 45.6 na
Rheinau Arenic Eutrochrept fluvial gravels 68.1 29.2 2.7 0.8
Royal! Ultic Haploxeroll glaciofluvial sediments 30.7 63.1 6.2 na
Salkum! Xeric Palehumults glacial drift 11.9 59.7 28.4 na
Walla Walla! Typic Haploxeroll loess 8.3 78.4 13.3 na
Wetzikon 1 Lithic Ruptic-Alfic Eutrochrept moraine 48.9 46.7 4.4 0.9
Wetzikon 2 Rendollic Eutrochrept moraine 59.7 37.2 3.1 0.9
Wuelflingen Vertic/Typic Eutrochrept anthropogenic deposits 32.7 60.5 6.8 0.7
Zeiningen Ultic Hapludalf floess 40.5 55.4 4.1 0.4
U.S. soil taxonomy.
! Soils from USA.
ż OC, organic C percentage by weight, determined with Walkley-Black method (Nelson and Sommers, 1982).
Å› na, not available.
dried at 105 C, gently crushed, and passed through a 2-mm
In this model, the fragmentation of an initially intact
sieve. Each sample was tested for the presence of carbonates
particle into smaller particles leads to a power-law rela-
using cold 1 M HCl, and if carbonates were present, the sample
tion between (i) number or (ii) mass of particles as a
was treated with 0.5 M sodium acetate at 75 C for at least 1 h.
function of particle size. These two types of fragmenta-
After acetate treatment, samples were washed with deionized
tion relations are known as number-based and mass-
water. The five soil samples from the USA were further pre-
based approaches (Turcotte, 1992). The power-law ex-
treated by destroying organic matter using H2O2 (30%, w/w)
ponent of the number-based approach can be inter-
at 65 C. The 14 Swiss soil samples were not pretreated for
preted as fractal dimension (Matsushita, 1985; Turcotte,
organic matter. The absence of pretreatment for organic mat-
1986). It is worth noting that the fragmentation model
ter could in some cases have affected the dispersion of particles
does not lead to a geometrical fractal with the fractal for the Swiss soils, leading to incomplete segregation, and
therefore to an underestimation of small particle fractions.
dimensions confined between Euclidian dimensions.
Organic matter contents of the Swiss soils, determined with the
The sorting of particles by size in the fragmentation
Walkley-Black method (Nelson and Sommers, 1982), ranged
model results in fractal dimensions ranging theoretically
from 0.4 to 2.2% by weight (Table 1).
between the limits of 0 and 3 (Turcotte, 1986). Borkovec
After pretreatment, all samples were dried at 105 C for 24
et al. (1993) experimentally determined fractal dimen-
h. Prior to particle-size analysis, all soil samples were dispersed
sions of fragmentation and surface areas of soil particles
in 1 g L 1 hexametaphosphate solution and shaken for 24 h
and found the two dimensions to be 2.8 0.1 and 2.4
to destroy aggregates. For the pipette analysis, samples were
0.1, respectively.
wet sieved with the hexametaphosphate solution at 1000-,
The objective of this study was to test the mass-based
500-, 250-, 125-, and 53- m mesh sizes. The material smaller
fragmentation approach proposed by Turcotte (1986) than 53 m was then analyzed by the pipette method (Gee
and Bauder, 1986). To obtain four size classes between 2 and
for characterizing PSDs, and to determine the range of
50 m, sedimentation techniques based on Stoke s law were
particle diameters where power-law scaling is applica-
used to obtain the following diameters: 2, 5, 10, and 20
ble. To test the general validity and the extent of power-
m. For the light-scattering technique, the soil samples were
law scaling it is of fundamental importance to have
wet sieved down to a size of 250 m for the Swiss soils and
data that span several orders of magnitude. Traditional
500 m for the U.S. soils. The particles passing the smallest
sedimentation and hydrometer techniques for the mea-
sieve mesh were collected in a bucket, dried at 105 C, and
surement of PSDs yield only limited data in the clay
subsequently analyzed by light diffraction. A 3-g aliquot of
fraction smaller than 2 m. Light-scattering methods
the dried material was introduced into an ultrasonic bath unit
overcome this problem and provide data between 0.05
of a small-angle light-scattering apparatus (Malvern Master
to 1000 m. Sizer MS20, Malvern, England) equipped with a low-power
(2 mW) Helium-Neon laser with a wavelength of 633 nm as
the light source.1 Suspension concentrations were adjusted to
MATERIALS AND METHODS
an obscuration of the primary beam of H"0.1 to 0.2%. The
Particle-Size Analysis
obscuration values were set to optimize between best signal/
noise ratio and negligible multiple scattering effects. If the
Nineteen soils were used in this study, five of them were
sample concentration is too low, the obscuration and the inten-
from the USA and 14 from Switzerland. The soils were chosen
such that they represent a wide variety of parent materials,
1
weathering conditions, and textures. Characteristic properties
Reference to company name does not reflect endorsement of
of these soils are summarized in Table 1. All soil samples were particular products by Washington State University.
784 SOIL SCI. SOC. AM. J., VOL. 63, JULY AUGUST 1999
sity of the scattered light are low, leading to noisy data. If the
sample concentration is too high, then the light scattered from
a particle may be scattered again by a second particle, causing
errors in the final particle-size analysis. Prior to measurement,
samples were dispersed by sonication in an ultrasonic bath
for 25 min. A focal length of 300 mm was used with an ordinary
Fourier Optics configuration, and a focal length of 45 mm was
used for the inverse Fourier Optics configuration. The inverse
configuration allows the accurate measurement of scattering
at high angles in order to correctly measure the very fine
particles (sizes down to 0.01 m). Particle-size distribution
was obtained by fitting full Mie scattering functions for spheres
(Kerker, 1969).
Data Analysis
Soils are formed by weathering of geological parent mate-
rial. The weathering results in a fragmentation of the initial
solid rock or sediment. It has been recognized that the prod-
Fig. 1. Cumulative particle-size distributions for four soils obtained
ucts of fragmentation in nature can often be described with
by two different experimental methods.
fractal concepts. For different types of objects, a power-law
relation between the number and size of objects has been
shown below by our experimental data and discussed in the
proposed (Mandelbrot, 1982; Matsushita, 1985; Turcotte,
literature (Turcotte, 1992), the power-law relation given in
1986)
Eq. [2] has also a lower limit of validity. The radius R of
N(r R) CR D [1]
particles satisfying Eq. [2] is confined between RL,lower
R RL,upper.
where N(r R) is the number of objects per unit volume
The mass-based fragmentation approach was used to ana-
having a radius r larger than R, C is a constant of proportional-
lyze experimentally determined PSD data. The lower and up-
ity, and D is the fractal dimension. For soil particles, Turcotte
per limits RL,lower and RL,upper as well as the power-law exponent
(1986) and Tyler and Wheatcraft (1992) pointed out that it is
D 3 v were determined by the following procedure. A
generally more convenient to express the number-based power
linear regression was used to fit Eq. [2] on a log-log plot to
law (Eq. [1]) as a mass-based form. The mass-based approach
the experimental data. The entire range of experimental data
is compatible with data obtained from experimentation, where
was used first and the residuals were calculated. Subsequently,
usually mass fractions rather than number fractions are mea-
the upper- and lower-range data points were eliminated and
sured. The mass-based form of Eq. [1] is expressed as (Tur-
new residuals and root mean square errors (RMSE) were
cotte, 1986; Tyler and Wheatcraft, 1992)
calculated. In an iterative procedure, the RMSE error was
v
M(r R) R minimized by eliminating data points at the upper and
[2]
lower boundaries.
MT RL,upper
where M(r R) is the mass of soil particles with a radius
RESULTS AND DISCUSSION
smaller than R, MT is the total mass of particles with radius
less than RL,upper, RL,upper is the upper size limit for fractal behav-
Comparison between Pipette and
ior, and v is a constant exponent. This power law can be
Light-Diffraction Methods
related to the fractal number relation by taking incremental
values as shown by Matsushita (1985) and Turcotte (1992).
Most of the textural data reported in the literature
Taking the derivatives of Eq. [1] and [2] with respect to the
have been measured by sedimentation techniques, such
radius R yields, respectively,
as hydrometer or pipette. It is therefore illustrative to
briefly compare experimental results obtained by pi-
dN R D 1dR [3]
pette and light-scattering methods. The results obtained
and
by the two techniques were in excellent agreement in
dM Rv 1 dR [4] our study. Figure 1 shows a qualitative comparison be-
tween pipette and light-scattering methods for four soils.
Assuming a constant density of soil particles, the volume of
Experimental differences in the cumulative fraction at
a particle with radius r is proportional to its mass m, hence
a given particle size obtained by the two methods were
r3 m; therefore, for incremental particle numbers and masses
in the order of 0.3 to 11.7%. Similar results were ob-
we have
tained by Wu et al. (1993), who found that sedimenta-
R3dN dM [5]
tion and light-scattering techniques were in good agree-
Substituting Eq. [3] and [4] into [5] gives (Turcotte, 1992) ment for the majority of the soil samples used in their
experiment.
R D 1 R 3Rv 1 [6]
from which it follows that
Characterization of Particle-Size Distribution
D 3 v [7]
In Fig. 2, cumulative mass fractions are plotted as a
Equation [7] relates the exponent v of the mass-based ap- function of particle diameter on double logarithmic
proach to the exponent D of the number-based approach. As scale for four soils. The plots clearly show that a single
BITTELLI ET AL.: FRAGMENTATION MODEL TO CHARACTERIZE PARTICLE-SIZE DISTRIBUTION 785
ary was 0.51 m and of the silt sand domain boundary
was at 85.3 m, with a coefficient of variation of 15 and
25%, respectively. The consistent occurrence of three
power-law domains in all 19 soils and the close agree-
ment of the domain scales indicate similarity between
the different soils, particularly when considering the
wide textural variability of the samples ranging from
0.3 to 46% clay. In a similar study on four soils, Wu et
al. (1993) also found three domains where a power law
was applicable, but the limits between the domains were
located at 0.05 to 0.1 and 10 to 5000 m. The consistency
of the limits for the three domains needs therefore to
be investigated across a greater number of soils.
We denote the three fractal dimensions determined
in our study as Dclay, Dsilt, and Dsand. The fitted values of
the fractal dimensions obey in all cases the relation: Dclay
Dsilt Dsand. In the clay domain, the fractal dimension
Fig. 2. Log-log plots of particle-size distributions for four soil samples.
ranged from 0.118 to 1.21, in the silt domain from 1.728
Symbols denote experimental data, solid lines denote model fits.
and 2.792, and in the sand domain from 2.839 to 2.998
(Table 2). These data are consistent with the limits of the
power law cannot describe the data across the entire
fragmentation approach given as 0 D 3 (Turcotte,
range of measured particle sizes. There is evidence that
1986). The generally high value of the coefficient of
different power laws apply for three domains in all of
determination R2 shows that the fragmentation models
the 19 soils. The solid lines in Fig. 2 are the curves of
are good descriptions of the PSDs in the three domains.
Eq. [2] fitted to the different domains of particle sizes
Some soils showed poor power-law agreement in the
on the log-log plots.
sand domain (e.g., Obfelden and Reckenholz). Experi-
Optimized parameters of the fragmentation model
mental data in the sand as well as in the clay domain
together with the median particle diameter for all the
are limited by the experimental procedures, namely the
19 soils are shown in Table 2. The median diameter has
maximum particle size as allowed by the 2-mm sieve
been calculated from the measured PSDs by linearly
mesh and the minimum particle size determined by the
interpolating the 50% quantile (e.g., Sokal and Rohlfs,
light-scattering technique.
1995). The identified power-law domains separate the The model used in the derivation of Eq. [2] is based
particle sizes in three classes, which we denote as clay, on the fragmentation of an initially intact particle into
silt, and sand domains. The diameter boundaries be- smaller particles (Matsushita, 1985; Turcotte, 1986). An
tween the clay and silt domains ranged from 0.33 to intact cubical particle of size h is fragmented into eight
0.99 m, and between silt and sand domains from 45.3 identical cubes of size h/2. Each of these smaller cubes
to 126.7 m. The average of the clay silt domain bound- is further divided in cubes with size h/4, and so forth.
Table 2. Fragmentation fractal dimensions, median particle diameter, and cutoff boundaries, estimated from particle-size distribution
data obtained by the light-diffraction method for the 19 soils.
Silt domain
Clay domain Silt domain Sand domain
Median Lower Upper
Soils Dclay R2 Dsilt R2 Dsand R2 diameter d50 boundary boundary
m m m
Affoltern 0.808 0.96 2.239 0.99 2.930 0.95 46.19 0.42 94.35
Aeugst 0.606 0.97 2.294 0.99 2.979 0.94 36.51 0.38 93.93
Buelach 0.596 0.96 2.122 0.99 2.898 0.96 60.01 0.99 98.41
Les Barges 0.808 0.99 1.768 0.99 2.839 0.98 74.96 0.53 124.58
Mettmenstetten 0.792 0.96 2.297 0.98 2.858 0.98 109.79 0.58 74.99
Murimoos 0.255 0.99 1.728 0.99 2.948 0.91 74.44 0.56 112.92
Obermumpf 0.701 0.96 1.801 0.99 2.974 0.98 30.96 0.40 56.93
Obfelden 1.210 0.97 2.152 0.98 2.969 0.85 24.86 0.40 69.98
Palouse 0.118 0.96 2.504 0.99 2.996 0.91 14.34 0.44 54.21
Reckenholz 0.789 0.96 2.238 0.99 2.998 0.81 29.22 0.40 71.58
Red Bluff 0.174 0.96 2.792 0.99 2.921 0.99 2.88 0.51 77.92
Rheinau 0.799 0.96 2.251 0.99 2.815 0.97 77.58 0.42 126.73
Royal 0.987 0.95 2.269 0.99 2.981 0.94 35.56 0.56 90.46
Salkum 0.214 0.97 2.618 0.98 2.953 0.99 8.18 0.63 45.31
Walla Walla 0.896 0.94 2.384 0.98 2.973 0.91 16.57 0.61 50.86
Wetzikon 1 0.796 0.99 2.249 0.99 2.931 0.98 50.04 0.38 98.57
Wetzikon 2 0.808 0.95 2.201 0.98 2.901 0.95 67.71 0.33 122.84
Wuelflingen 0.896 0.96 2.279 0.99 2.994 0.99 36.44 0.55 71.28
Zeiningen 0.795 0.96 2.182 0.99 2.991 0.96 30.85 0.34 101.71
Average 0.51 85.3
Standard deviation 0.15 25.3
Coefficient of variation, % 15 25
786 SOIL SCI. SOC. AM. J., VOL. 63, JULY AUGUST 1999
Fig. 3. Fragmentation fractal dimensions D and probabilities p of
fragmentation. The solid line represents Eq. [8], symbols are calcu-
lated with Eq. [8] from experimentally determined fractal di-
mensions.
The fragmentation of a cube has a certain probability
p, which is assumed to be constant for all orders of
fragmentation. A cube can maximally disintegrate into
eight smaller cubes (p 1) and minimally into one
smaller cube (p 1/8). As shown by Turcotte (1986),
the fragmentation probability p is related to the fractal
Fig. 4. Fragmentation fractal dimension of the silt domain Dsilt vs.
dimension D by
clay and sand percentage. Data from Tyler and Wheatcraft (1992)
were obtained from the entire range of the particle-size distribution
log(8p)
used in their study.
D [8]
log 2
The fractal dimensions reported by Tyler and Wheatcraft
where the range of possible fractal dimensions is 0
(1992), plotted in this figure, were obtained by applying
D 3. Figure 3 shows a plot of Eq. [8] along with values
Eq. [2] to the entire range of the PSD data, which ranged
calculated from the analysis of our experimental data.
from 1 to 50 m, 0.5 m to 5 mm, and 16 mto 1 mm
A scale-independent fragmentation process would have
for different data sets. Fractal dimensions given by Tyler
a constant fragmentation probability. Evidently, frag-
and Wheatcraft are therefore not directly comparable
mentation probabilities varied across almost the entire
with our Dsilt, but nevertheless, Fig. 4 shows a trend
range of 1/8 p 1. It is interesting that the fractal
between the D value and the clay and sand contents.
dimensions for the three domains are typically Dclay
The fractal dimension increases with clay content, and
Dsilt Dsand. It appears that for the 19 soils studied, the
decreases with sand content. These results suggest that
probability of fragmentation is scale dependent, and
the power-law relation of Eq. [2] can be used to charac-
in particular it decreases with decreasing size of the
terize PSD in soils, and may be an alternative to the
particles. There is experimental evidence that fragmen-
conventionally used approaches, such as the lognor-
tation of soil and sediment aggregates is scale dependent
mal distribution.
(Perfect et al., 1993; Rasiah et al., 1993). Larger aggre-
gates tend to fracture more easily than smaller aggre-
Calculation of Parameters of the Fragmentation
gates (Perfect, 1997). Considering soil particles as prod-
Model from Mass Fractions of Clay and Silt
ucts of a fragmentation process, our results are in
qualitative agreement with observations from aggregate It is evident that no single power law can characterize
failure studies. the PSD of a soil across the entire scale usually measured
Particle-size distribution measurements are strongly in a particle-size analysis. For the majority of the sam-
influenced by the experimental methods of dispersion ples, 46 to 86% (with an average of 71%) of the total
of the soil particles. The dispersion itself can be regarded mass is carried by particles with diameters between 0.51
as a fragmentation process. Organic matter increases and 85.3 m, the silt domain of the distribution. On a
aggregate stability and hence leads to less fragmentation log-log scale, the PSD of the silt domain is a straight
(Rasiah et al., 1993). Therefore the omission of organic line and is therefore characterized by two parameters,
matter removal in the Swiss soils probably leads to less the intercept and the slope of the power-law distribu-
dispersion of smaller particles. This explains the smaller tion. If we know any two points on this line, we can
clay fractions determined in the Swiss soils compared calculate the model parameters of the silt domain. As
with the U.S. soils (Table 1). Based on the fragmentation Table 2 shows, the USDA boundaries between clay and
model, we would also expect smaller fractal dimensions silt (2 m), and between silt and sand (50 m) are within
for the clay fraction of the Swiss soils compared with the silt domains for all 19 soils. Therefore we can use
the U.S. soils; however, there is no evidence that this these standard particle-class fractions to calculate the
is the case (Table 2). two parameters of a power-law particle-size distribution
Following Tyler and Wheatcraft (1992), Dsilt vs. clay in the silt domain. As the second parameter besides
and sand fraction was plotted in Fig. 4 to demonstrate the fractal dimension D we choose the median particle
the relation between fractal dimension and soil texture. diameter d50 of the PSD. The median particle diameter
BITTELLI ET AL.: FRAGMENTATION MODEL TO CHARACTERIZE PARTICLE-SIZE DISTRIBUTION 787
CONCLUSIONS
There is evidence that cumulative PSDs in soils follow
a power-law distribution, consistent with a fractal frag-
mentation model. The mass-based approach suggested
by Matsushita (1985) and Turcotte (1986) showed good
agreement between the fractal model and our experi-
mental data. Three main domains a clay, silt, and sand
domain were identified where power-law scaling was
applicable. The limits between the domains were rela-
tively constant for different soil types, but do not coin-
cide with the traditional boundaries between clay, silt,
and sand. Fragmentation fractal dimensions of the three
domains increased in the order: clay silt sand do-
main. A method is imposed to estimate the parameters
of the fragmentation model of the PSD in the silt domain
from standard textural data of clay, silt, and sand
fractions.
ACKNOWLEDGMENTS
We thank Alan Busacca and Sandra Lilligren for assistance
Fig. 5. Experimental and calculated values of median diameter d50
during the laboratory analyses. The manuscript benefitted
and of fragmentation fractal dimension of the silt domain Dsilt for
from fruitful discussions with Claudio O. Stockle, Sally D.
all 19 soils used in this study. Calculated values are from Eq. [9]
Logsdon, and Philippe Baveye.
and [10]. RMSE is the root mean square error.
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1469 1470.
(2 Dsilt) log 2 log(mclay)
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d50 exp [9]
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3 Dsilt
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Buchan, G.D., K.S. Grewal, and A.B. Robson. 1993. Improved models
and
of particle-size distribution: An illustration of model comparison
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techniques. Soil Sci. Soc. Am. J. 57:901 908.
Dsilt 3
Campbell, G.S., and S. Shiozawa. 1992. Prediction of hydraulic proper-
log 50 log 2
ties of soils using particle-size distribution and bulk density data.
p. 317 328. In M.Th. van Genuchten, F.J. Leij, and L.J. Lund.
log(1 msilt/mclay)
3 [10] (ed.) Indirect methods for estimating the hydraulic properties of
log 25
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Gee, G.W., and J.W. Bauder. 1986. Particle-size analysis. p. 383 411.
where mclay and msilt are the mass fractions of clay and silt,
In A. Klute (ed.) Methods of soil analysis. Part 1. 2nd ed. Agron.
log is the natural logarithm, and the median diameter d50 Manag. 9. ASA and SSSA, Madison, WI.
is given in micrometers. Note that the median d50 in Eq. Kerker, M. 1969. The scattering of light and other electromagnetic
radiation. Academic Press, New York.
[9] is a non-log-transformed parameter. Mass fractions
Kozak, E., Y.A. Pachepsky, S. Sokolowski, Z. Sokolowska, and W.
of clay, silt, and sand are readily available for many
Stepniewski. 1996. A modified number-based method for estimat-
soils, and from these data the median diameter and
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the fractal dimension of the silt domain can then be
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computed with Eq. [9] and [10]. Note that in the deriva- Mandelbrot, B.B. 1982. The fractal geometry of nature. Freeman and
Co., San Francisco.
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Measuring Saturated Hydraulic Conductivity using a Generalized Solution
for Single-Ring Infiltrometers
L. Wu,* L. Pan, J. Mitchell, and B. Sanden
ABSTRACT
meter is a combination of both vertical and horizontal
flow (Tricker, 1978). A method to calculate the Ks from
Saturated hydraulic conductivity is a measure of the ability of a
data obtained from a pressure or ring infiltrometer for
soil to transmit water and is one of the most important soil parameters.
New single-ring infiltrometer methods that use a generalized solution both early-time and steady-state infiltration was devel-
to measure the field saturated hydraulic conductivity (Ks) were devel- oped by Reynolds and Elrick (1990), Elrick and Rey-
oped and tested in this study. The Ks values can be calculated either
nolds (1992), and Elrick et al. (1995). Their steady-
from the whole cumulative infiltration curve (Method 1) or from
state method uses a shape factor that was numerically
the steady-state part of the cumulative infiltration curve by using a
calculated based on Gardner s (1958) relationship be-
correction factor (Method 2). Numerical evaluation showed that the
tween hydraulic conductivity and matric pressure head.
Ks values calculated from the simulated infiltration curves of represen-
Groenevelt et al. (1996) further extended this concept
tative soil textural types were in the range of 87 to 130% of the real
by developing a method to define the critical time that
Ks values. Field infiltration tests were conducted on an Arlington fine
separates early-time and steady-state infiltration.
sandy loam (coarse-loamy, mixed, thermic, Haplic Durixeralfs). The
By applying scaling theory, Wu and Pan (1997) devel-
geometric means of the Ks values calculated from the field-measured
infiltration curves by Method 1 and Method 2 were not significantly oped a generalized solution for single-ring infiltromet-
different. The geometric mean of the Ks calculated from the detached
ers. Wu et al. (1997) showed further that the infiltration
core samples, however, was about twice that of the Ks calculated from
rate of a single-ring infiltrometer was approximately f
the infiltration curves, which was consistent with earlier findings.
times greater than the one-dimensional (1-D) infiltra-
Unlike the earlier approaches, Method 1 calculates Ks values from
tion rate for the same soil, where f is a correction factor
the whole infiltration curve without assuming a fixed relationship
that depends on soil initial and boundary conditions and
( Ks / ) between saturated hydraulic conductivity and matric
m
ring geometry. For a relatively small ponded head, the
flux potential .
m
1-D final infiltration rate of a field soil is approximately
equal to the field saturated hydraulic conductivity (Ks),
which is valuable information for computer modeling,
aturated hydraulic conductivity is an important
as well as for irrigation management. The objectives of
Ssoil parameter that measures the ability of a soil to
this research were (i) to develop alternative methods
transmit water. Measurement of field saturated hydrau-
to calculate Ks by best fit of a generalized solution to
lic conductivity (Ks) is often done by borehole permea-
the infiltration curves that are measured by single-ring
meters (Amoozegar and Warrick, 1986; Elrick and
infiltrometers, and (ii) to compare and evaluate Ks val-
Reynolds, 1992). In many cases, however, measurement
ues calculated from infiltration curves of single-ring in-
of the soil surface Ks is essential, especially in infiltra-
filtrometers with those measured by the single head
tion-related applications, such as irrigation manage-
(SH) method (Elrick and Reynolds, 1992) and detached
ment.
soil core samples (Klute and Dirksen, 1986).
Ring infiltrometers are often used for measuring the
water intake rate at the soil surface. Water flow from
THEORY
a single-ring infiltrometer into soil is a three-dimen-
A generalized infiltration equation developed by Wu and
sional (3-D) problem (Reynolds and Elrick, 1990). The
Pan (1997) has essentially the same form as the truncated
total flow rate into the soil from a single-ring infiltro-
Philip (1957) model of vertical infiltration. We propose here
to measure infiltration curves in the field and then utilize the
L. Wu, Dep. of Environmental Sciences, Univ. of California, River- generalized equation to fit to the data in order to obtain
side, CA 92521; L. Pan, Earth Sci. Div., Lawrence Berkeley National the relevant parameters for estimating Ks. The generalized
Lab., Univ. of California, Berkeley, CA 95720; J. Mitchell, Kearney
equation (Wu and Pan, 1997) is
Agri. Center, Univ. of California, Parlier, CA 93648; and B. Sanden,
i/ic a b(t/Tc) 0.5 [1]
Univ. of California Coop. Ext., Kern County, Bakersfield, CA 93307.
Received 8 June 1998. *Corresponding author ([e-mail]).
Published in Soil Sci. Soc. Am. J. 63:788 792 (1999). Abbreviations: SH, single head.