Published in Soil Sci. Soc. Am. J. 67:1703-1706 (2003).
© 2003 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
DIVISION S-1—NOTES
TIME-MASS SCALING IN SOIL TEXTURE ANALYSIS
Roberto R. Filgueiraa,b,
Yakov A. Pachepsky*,c and
Lidia L. Fourniera
a Area Física Aplicada, Facultad de Ciencias Agrarias y Forestales, National University of La Plata, CC 31, 1900 La Plata, Argentina
b CONICET, Rivadavia 1917, 1033 Buenos Aires, Argentina
c Animal Waste Pathogen Lab., USDA-ARS-BA-ANRI-AWPL, Bldg. 173, Rm. 203, BARC-EAST, Powder Mill Road, Beltsville, MD 20705
* Corresponding author (ypachepsky{at}anri.barc.usda.gov).
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ABSTRACT
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Popular methods of textural analysis employ the relationship among time, the travel distances, and the radii of particles subject to sedimentation in a viscous liquid. The purpose of this note is to present and test an explicit relationship between time and soil suspension density in the course of the hydrometer analysis procedure applied to particles with the fractal mass-size distribution. The relationship between the logarithms of mass of particles remaining in the solution and of the observation time is linear with the slope equal to -(3 - D)/2 where D is the fragmentation fractal dimension. This relationship was tested with samples of Typic Argiudoll and 24 soils from Imperial Valley, CA, and gave good approximation of hydrometer data on sedimentation of silt fraction 2 to 50 m. Monitoring soil mass in the solution presents a way of testing the applicability of fractal fragmentation model to particle-size distributions (PSDs) and estimating the fragmentation fractal dimension.
Abbreviations: PSD, particle-size distribution
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INTRODUCTION
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SOIL TEXTURE DATA are often used to infer soil functioning and use. Particle-size distributions have been studied extensively. The model of fractal fragmentation has attracted attention as a possible method to describe observed PSDs with a minimum set of parameters (Tyler and Wheatcraft, 1992). This model has presented often satisfactory, albeit not universal, way to simulate PSDs (Wu et al., 1993; Kozak et al., 1996; Bitelli et al., 1999).
The common source of textural data is the sedimentation of particles in water and the most popular techniques are the hydrometer method and the pipette method. Both methods are based on the Stokes law and employ the relationship among time, travel distance, and radius of a particle subject to sedimentation in a viscous liquid (Gee and Bauder, 1986). Assuming a fractal mass-size distribution for particles, one should expect the aforementioned relationship to reflect parameters of fractal scaling. The purpose of this note is to present and test an explicit relationship between time, soil suspension density, and the fragmentation fractal dimension in the course of the hydrometer analysis procedure.
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Theory
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The velocity v of a particle of radius r subject to sedimentation in a viscous liquid is
 | [1] |
where B = 2g
/(9
), is the difference between densities of particles and the liquid, is the dynamic viscosity of liquid, and g is the acceleration due to gravity. If the length of the sedimentation cylinder is L, then, at time t, radii of particles remaining in solution are smaller than r* where
 | [2] |
The mass of particles remaining in the liquid above the depth L is
 | [3] |
where h(r,t) = vt =Br2t, m(r) is the distribution function of particle mass in the sample, rmin is the radius of a minimum particle under consideration.
Assuming the particle mass obeys the fractal fragmentation distribution (Tyler and Wheatcraft 1992; Bitelli et al., 1999), one has for m(r)
 | [4] |
where
 | [5] |
rL,upper is the upper size limit for the fractal behavior, MT is the total mass of particles with r less than rL,upper, and D is the fragmentation fractal dimension. According to the fractal fragmentation model of Turcotte (1986), values of D must be <3 since the inequality D > 3 would require that the probability of grain fragmentation be 
1. The lower boundary for D is zero. At D = 0 the cumulative number of grains obtained after fragmentation does not change as the grain size decreases.
Substitution of Eq. [4] into Eq. [3] results in
 | [6] |
This equation can be rearranged as
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While r* >> rmin, the second term in the braces is much <1, and the Eq. [7] reduces to
 | [8] |
Using definitions of r* from Eq. [2] and C from Eq. [4], one obtains:
 | [9] |
These equations express the scaling relationship between the mass of particles in suspension and the time from the beginning of sedimentation that should exist if the PSD follows the fractal fragmentation model. Equation [9] can be rearranged in log-log scale as
 | [10] |
thus providing the way of finding fractal dimension from the slope of regressions log t vs. log(Mrem/MT).
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Materials and Methods
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Both published and our own experimental data were used to test the applicability of the proposed scaling. Soil samples were taken at the Experimental Farm of the School of Agronomy and Forestry Engineering, National University of La Plata, Argentina, located at 34°54' S latitude and 57°57' W longitude. The soil is a Typic Argiudoll, with texture varying from silt loam in the 0- to 20-cm layer to silty clay loam in the 20- to 40-cm layer. Samples were taken from four plots. At each plot, soil cores were taken with a 100-mm i.d. cylindrical sampler from 0- to 10-, 10- to 20-, 20- to 30-, and 30- to 40-cm depths intervals. The samples were crushed and spread thinly (in layers 2- to 3-cm thick) on trays to air-dry at room temperature. Samples were then passed through a 2-mm sieve to exclude fractions >2 mm.
The particle-size analysis was done with a standard Bouyoucos hydrometer (American Society for Testing and Materials, 1985; Gee and Bauder, 1986). To disperse aggregates, 50 g of soil were poured into a 600-mL beaker with 250 mL of distilled water and 100 mL of Na-hexametaphosphate (HMP) solution. Samples were soaked overnight and then transferred to a dispersing cup and mixed for 5 min with an electric mixer. The suspension was poured into a sedimentation cylinder, and distilled water was added to bring the volume to 1 L. The cylinder was end-over-end shaken for 1 min. The hydrometer was then lowered into the suspension and readings were taken after 40 s, and 2, 8, 15, 30, 60, 120, 240, and 1440 min. The hydrometer was removed, rinsed, and wiped after every measurement. A similar cylinder with the HMP solution without soil was maintained in the same room, to calibrate the hydrometer-scale reading with the blank solution. The temperature was recorded at each time to allow for viscosity and density corrections.
Published data on frequent hydrometer reading were taken from the work of Kaddah (1974). In this study, samples of 24 soils from Imperial Valley, CA, were air-dried and 2-mm sieved. Fifty-gram subsamples were treated with H2O2 to remove organic matter. The readily soluble salts and gypsum were removed by shaking the soil with 250 mL of H2O for 30 min, centrifuging the suspension, and decanting the supernatant solution. The suspension was filtered, and soil was dispersed before application the hydrometer method. Hydrometer readings were taken after 0.5, 1, 2, 3, 4, 5, 10, and 30 min; and after 1, 2, 3, 4, 8, 24, and 36 h.
Fractal dimensions were estimated from slopes of the regression Eq. [10] fitted to data on log(t) vs. log(Mrem/MT). The software package SigmaPlot v. 8 (SPSS, Inc.) was used for plotting and regression analysis.
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Results and Discussion
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The Eq. [10] fitted the data well, with all but one determination coefficients of regressions, R2, >0.96. Figure 1
shows an example of the fit that Eq. [10] has provided for one of the replications. The fractal fragmentation model was applicable to the PSD range observed in this work. There was no statistically significant difference between fractal dimensions at 0 to 10 and 10 to 20 cm and between dimensions at 10 to 20 and 20 to 30 cm; statistically significant differences were found between average values of fractal dimensions at 0 to 10, 20 to 30, and 30 to 40 cm (Table 1).
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Fig. 1. Measured (symbols) and simulated with the fractal fragmentation model (lines) sedimentation of the silt fraction for the experimental data set from Argentina.
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Ranges of diameters for particles that have settled during the observation period showed that sedimentation of silt fraction (2–50 µm) was dominant during the observation period (Table 1), and that the fractal dimensions reflected the fragmentation scaling within the silt fraction of texture. Values of fractal dimension derived from our data increased with clay content. A similar effect of clay content on fractal dimensions of silt fraction was also noted by Tyler and Wheatcraft (1992) and Bitelli et al. (1999). Fractal dimensions ranged from 2.40 to 2.70 for our experimental data. Tyler and Wheatcraft (1992) found fragmentation fractal dimensions of silt fraction in the same range, whereas Bitelli et al. (1999) observed much wider range of values from 1.77 to 2.80. We note that both soil sample pretreatment and method of particle-size analysis (light diffraction) in the Bitelli et al. (1999) work were different from the one employed in this work.
Results of application Eq. [10] to data of Kaddah (1974) are shown in Fig. 2
. The proposed scaling has been applicable to this data set. Twenty of 24 values of determination coefficient R2 of the regression Eq. [10] applied to this data were >0.96. The remaining four values of R2 were 0.93, 0.95, 0.89, and 0.95 (Table 2).
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Fig. 2. Measured (symbols) and simulated with the fractal fragmentation model (lines) sedimentation of the silt fraction for the experimental data set from California (Kaddah, 1974).
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Fractal dimensions can be estimated directly from PSDs using Eq. [4] (Bitelli et al., 1999). We performed such estimation for our samples and found differences between time-based and PSD-based values <4% (data not shown).
The standard time scale for taking hydrometer measurements (2, 8, 15, 30, 60, 120, 240, and 1440 min, see Gee and Bauder, 1986) appears to be appropriate for using Eq. [10] in regression, because the measurement times are approximately equidistant in log scale. The initial and the final points could be augmented with additional points at 4 and 480 min. The presence of fractal scaling within the silt fraction indicates that the linear interpolation of PSD within these fraction boundary radii can be better made in ‘log mass–log diameter’ coordinates than in ‘mass-log diameter' coordinates. That may be one explanation for the unsatisfactory performance of the linear interpolation of texture in ‘mass-log diameter’ coordinates observed by Nemes et al. (1999). The applicability of the power-law distribution Eq. [4] to the PSD in the silt range of sizes does not imply that this type of distribution is applicable to the whole range of particle sizes encountered in soil texture analysis. Kozak et al. (1996) showed that power law is applicable only to PSD of 20% of 2000 soils that they analyzed. Bitelli et al. (1999) demonstrated that fractal scaling is applicable separately to particles in sand-, silt-, and clay-size ranges. Posadas et al. (2001) suggested using a multifractal model for the whole range of particles in PSD. Hwang et al. (2002) showed that the Fredlund model was the best PSD model for a wide range of textures. We note that the applicability of the fractal power law distribution to specific fraction size ranges allows to address properly the issue of selection of the representative fraction particle-size which appears to be a function of the fractal dimension (Kozak et al., 1996).
Overall, the time-mass scaling in soil textural analysis followed from a combination of the fractal fragmentation model and the Stokes law. Monitoring soil mass remaining in the solution presents a way to test the applicability of fractal fragmentation model to the PSD within a textural fraction as it was done for silt fraction in this work.
Received for publication September 17, 2003.
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REFERENCES
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- American Society for Testing and Materials. 1985. Standard test method for particle-size analysis of soils. D 422–63 (1972). 1985 Annual Book of ASTM Standards 04.08:117–127. American Society for Testing and Materials, Philadelphia, PA.
- Bitelli, M., G.S. Campbell, and M. Flury. 1999. Characterization of particle-size distribution in soil with a fragmentation model. Soil Sci. Soc. Am. J. 63:782–788.[Abstract/Free Full Text]
- Gee, G.W., and J.W. Bauder. 1986. Particle-size analysis. p. 383–411. In A. Klute et al. (ed.) Methods of soil analysis. Part 1. 2nd ed. Agron. Mongr. 9. ASA and SSSA, Madison, WI.
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- Kaddah, M.T. 1974. The hydrometer method for detailed particle-size analyzis: 1. Graphical interpretation of hydrometer readings and test of method. Soil Sci. 118:102–108.
- Kozak, E., Ya.A. Pachepsky, S. Sokolowski, Z. Sokolowska, and W. Steniewski. 1996. A modified number-based method for estimating fragmentation fractal dimensions of soils. Soil Sci. Soc. Am. J. 60:1291–1297.[Abstract/Free Full Text]
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- Turcotte, D.L. 1986. Fractals and fragmentation. J. Geophys. Res. 91(B2):1921–1926.
- Tyler, S.W., and S.W. Wheatcraft. 1992. Fractal scaling of soil particle-size distributions: Analysis and limitations. Soil Sci. Soc. Am. J. 56:362–369.[Abstract/Free Full Text]
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TOP
ABSTRACT
INTRODUCTION
