2.12 Soil Texture
2.12.3 Particle Size Distribution Functions
After having experimentally obtained a particle size distribution, it is useful to character- ize it by employing a mathematical model allowing specific parameters of the distribution to be obtained. These parameters can then be used to compute other soil physical prop- erties through functions called pedotransfer functions. Apedotransfer functionuses basic pedological information such as textural fractions, particle size distribution parameters, bulk density and others to compute other soil physical properties. There is therefore a transferof information. Commonly, the obtained properties are hydrological properties such as the soil water retention curve or hydraulic conductivity. Since the latter are more difficult and expensive to obtain experimentally, it is convenient to derive them from other basic properties that are readily easier to measure and more commonly available in soil databases.
30 Basic Physical Properties of Soil
Particle size distribution has then been represented by a variety of mathematical functions (Buchanet al., 1993), also called particle size distribution models, to obtain specific parameters to be used as input for the pedotransfer functions. Software is avail- able for derivation of hydrological properties by using pedotransfer functions (Schaap et al., 2001; Acutis and Donatelli, 2003).
Gaussian distribution
Shiozawa and Campbell (1991) and Pieri et al. (2006) found that particle size dis- tribution functions are typically bimodal and that each mode can be represented by a Gaussian function. The Gaussian function used to describe the particle size distribution is
f(x) = 1
σ(2π)1/2
exp
–(x–μ)2 2σ2
(2.26) whereμis the mean ofxandσ is the standard deviation. For a log-normal distribution, the particle diameterxis replaced by its logarithm. Usually, the particle size distribution is represented by a cumulative curve, which, for a Gaussian function, is obtained by integration of eqn (2.26) with respect tox:
F(x) = 1 2
1 + erf
(x–μ) σ√
2 forx> μ (2.27) F(x) = 1
2
1 – erf
(x–μ) σ√
2 forx≤μ (2.28)
where erf[ ] is the error function. For the computation of the error function, a numerical approximation from Abramowitz and Stegun (1970) can be used:
erf(x) = 1 – (0.3480242T– 0.0958T2+ 0.7478556T3) exp(–x2) (2.29) whereT = 1/(1 + 0.47047x).
In a bimodal distribution, the sample is divided into two fractions: the primary frac- tion (sand and silt) and the secondary fraction (clay). The distribution is then a weighted sum of the two fractions:
F(x) =F1(x) + (1 –)F2(x) (2.30) where F1(x) represents the cumulative Gaussian function for the secondary minerals (clay),F2(x) represents the Gaussian function for the primary minerals (sand and silt), and is the weighting factor between the secondary minerals (clay fraction) and the primary minerals (sand and silt).
The mean and the standard deviation of the bimodal distribution are also weighted sums of the mean and standard deviation of the two fractions, called bimodal parameters.
The model can be fitted to experimental particle size distribution data using a nonlinear
Soil Texture 31 least squares fitting procedure (Marquardt, 1963), whereμ1,μ2,σ1, andσ2 are fitting parameters, whileis set equal to the clay fraction at 2µm. The subscripts 1 and 2 for the mean and the standard deviation are the Gaussian parameters for the clay (F1) and the silt and sand fraction (F2) distributions, respectively. A nonlinear least squares fitting algorithm will be presented in Chapter 5. The size distribution function for primary minerals, with sizes between 2 and 2000µm, is approximately log-normal. Secondary minerals, with sizes less than 2µm, form a second distribution (Pieriet al., 2006).
If a complete particle size distribution is not available, means and standard deviations may be obtained from size fractions. The appropriate means and standard deviations are log means and log standard deviations, and they can be computed from
dg= exp milndi
(2.31) and
σg = exp
mi(lndi)2– (lndg)2
(2.32) wheremianddiare respectively the mass fraction geometric mean diameter of separatei.
Equations (2.31) and (2.32) are appropriate for any number of size classes.
If only three textural classes are available, it is possible to derive the geometric means and standard deviations. The geometric means (inµm),dy,dtanddd, are the geometric means for clay, silt and sand (the subscriptsy,tanddrefer to the last letters of the words clay, silt and sand). They are calculated from set values:
dy=√
0.01×2 = 0.14 (2.33)
dt=√
2×50 = 10 (2.34)
and
dd =√
50×2000 = 316.2 (2.35)
The lower and the upper limits for the clay fraction were set at 0.01 and 2, while 2 and 50 represent the lower and upper limits for the silt fraction, and 50 and 2000 represent those for sand, based on the USDA classification (Shiozawa and Campbell, 1991). The geometric mean of the distribution (μm) is given by
dg= exp[my(–1.96) +mt(2.3) +md(5.76)] (2.36) The geometric standard deviation (σm) of the distribution is given by
σg= exp
my(–1.96)2+mt(2.3)2+md(5.76)2– (lndg)2
(2.37)
32 Basic Physical Properties of Soil
Table 2.4Typical sand, silt and clay fractions and geometric mean particle diameter and geometric standard deviation for the 12 textural classes
Texture md mt my dg[µm] σg[µm]
Sand 0.92 0.05 0.03 211.75 4.43
Loamy sand 0.81 0.12 0.07 122.05 8.71
Sandy loam 0.65 0.25 0.1 61.74 12.21
Loam 0.42 0.4 0.18 19.81 16.34
Silt loam 0.2 0.65 0.15 10.52 9.57
Silt 0.06 0.87 0.07 9.11 4.08
Sandy clay loam 0.6 0.13 0.27 25.17 28.40
Clay loam 0.32 0.34 0.34 7.09 23.10
Silty clay loam 0.08 0.58 0.34 3.09 10.96
Sandy clay 0.53 0.07 0.4 11.36 39.70
Silty clay 0.1 0.45 0.45 2.07 13.75
Clay 0.2 0.2 0.6 1.55 22.81
Referring back to eqns (2.31) and (2.36), it should be clear that the logarithm of each size class is just multiplied by the mass fraction for that size class, and the result is summed to get the logarithm of the mean size (Shiozawa and Campbell, 1991). Table 2.4 shows typical silt and clay fractions (USDA) for each of the textural classes, along with their geometric mean diameter and standard deviation.
Fractal distribution
Particle size distribution can be described by other mathematical formulations. For in- stance, Bittelli et al. (1999) proposed to analyse the particle size distribution with a power-law distribution, where the exponent is interpreted as a fractal dimension. A description of this method was presented earlier. The authors utilized a mass-based approach, since this is compatible with the data obtained from experiments. The mass ratio is expressed as
M(r<R)
MT =
R RL,upper
ν
(2.38) whereM(r < R) is the mass of soil particles with a radiusr smaller thanR,MT is the total mass of particles with radius less than RL,upper,RL,upper is the upper size limit for fractal behaviour and ν is a constant exponent. The authors showed that D = 3 –ν, whereDis the fractal dimension of the distribution. Therefore, cumulative particle size
Sedimentation Law 33 distributions in soils can be represented by a power-law distribution, consistent with a fractal fragmentation model, but with scale invariance valid only over a limited domain range. Three domains—clay, silt and sand—were identified in which power-law scaling was applicable (Bittelliet al., 1999). The boundaries between the domains were relatively constant for different soil types, but did not coincide with the traditional boundaries between clay, silt and sand.