Particle size distribution models, their characteristics and fitting capability

Accepted Manuscript Particle size distribution models, their characteristics and fitting capability Hossein Bayat, Mostafa Rastgo, Moharram Mansouri Zadeh, Harry Vereecken PII: DOI: Reference:

S0022-1694(15)00682-4 http://dx.doi.org/10.1016/j.jhydrol.2015.08.067 HYDROL 20690

To appear in:

Journal of Hydrology

Received Date: Revised Date: Accepted Date:

26 April 2015 21 August 2015 30 August 2015

Please cite this article as: Bayat, H., Rastgo, M., Zadeh, M.M., Vereecken, H., Particle size distribution models, their characteristics and fitting capability, Journal of Hydrology (2015), doi: http://dx.doi.org/10.1016/j.jhydrol. 2015.08.067

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Particle size distribution models, their characteristics and fitting capability Abstract Many attempts have been made to characterize particle size distribution (PSD) curves using different mathematical models, which are primarily used as a basis for estimating soil hydraulic properties. The principle step in using soil PSD to predict soil hydraulic properties is determining an accurate and continuous curve for PSD. So far, the characteristics of the PSD models, their fitting accuracy, and the effects of their parameters on the shape and position of PSD curves have not been investigated. In this study all developed PSD models, their characteristics, behavior of their parameters, and their fitting capability to the UNSODA database soil samples were investigated. Results showed that beerkan estimation of soil transfer (BEST), two and three parameter Weibull, Rosin and Rammler (1 and 2), unimodal and bimodal Fredlund, and van Genuchten models were flexible over the entire range of soil PSD. Correspondingly, the BEST, two and three parameter Weibull, Rosin and Rammler (1 and 2), hyperbolic and offset renormalized log-normal models possessed a high fitting capability over the entire range of PSD. The few parameters of the BEST, Rosin and Rammler (1 and 2), and two parameter Weibull models provides ease of use in soil physics and mechanics research. Thus, they are seemingly fit with acceptable accuracy in predicting the PSD curve. Although the fractal models have physical and mathematical basis, they do not have the adequate flexibility to contribute a description of the PSD curve. Different aspects of the PSD models should be considered in selecting a model to describe a soil PSD. Keywords: Fitting; Mathematical PSD models; Models parameters; Particle size distribution; 1

Introduction

The particle-size distribution (PSD) is a fundamental soil physical property. The PSD of a certain soil is attributed to several complex hydrological, geological, (geo) physical, chemical and biological processes (Filgueira et al., 2006; Hillel, 1980; Shi et al., 2012). Particle size

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distribution is one of the static properties (Sumner, 1999) and affects soil chemical properties such as adsorption properties of chemicals (Hillel, 1980), cation exchange capacity (Ersahin et al., 2006), organic carbon content (Broersma and Lavkulich, 1980) and buffering capacity (Rizea et al., 2009). Soil physical properties including available water capacity, thermal conductivity (Hillel, 1980), water and solute movement in the soil (Ghafoor et al., 2013; Kays and Patterson, 1982), permeability, groutability (Vipulanandan and Ozgurel, 2009), erodibility (Beke et al., 1989), water retention (Arya and Paris, 1981), residual water content (Rawls et al., 1982), specific surface area (Ersahin et al., 2006), soil aggregate formation (Semmel et al., 1990), bulk density (Aşkin and Özdemir, 2003), porosity (Nimmo, 2004), aggregate stability (Molina et al., 2001), soil color (Kone et al., 2009), soil aeration (Russel, 1957) and saturated as well as unsaturated hydraulic conductivities (Vereecken et al., 2010) are affected by PSD. The PSD has been correlated to soil mechanical properties such as Atterberg limits (Panayiotopoulos et al., 2004), soil strength (Vipulanandan and Ozgurel, 2009), soil penetration resistance (Buchanan et al., 2010), compactability (Brady and Weil, 2010), tensile strength and friability (Imhoff et al., 2002), compression index (Imhoff et al., 2004), soil classification system (Craig, 2004), and void ratio (Buchanan et al., 2010). Also, PSD has an influence on soil biological properties such as biomass (Chiu et al., 2006), total nitrogen and total sulfur (Ashagrie et al., 2005), nitrogen mineralization (Burgos et al., 2006), and decomposition of organic matter (Brady and Weil, 2010). Availability of nutrients for the plant root (Ma et al., 2011), growth of plants (Huang et al., 2013), carbon mineralization (Gómez-Muñoz et al., 2011), fixation and de-fixation of ammonium (Nieder et al., 2011) and ability to store plant nutrients (Brady and Weil, 2010) are the soil fertility properties that have been affected by PSD. Particle size distribution in the soil profile is strongly related to erosion, deposition, and physical and chemical processes. In other words, soil particle-size distribution is one of the

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most important soil physical attributes due to its great influence on water movement, productivity, and soil erosion. In areas with high soil erosion rates induced by water such as rainfall and runoff, fine particle-size fractions (accompanied by nutrients) are selectively removed or deposited during soil erosion processes (Xu et al., 2013). A better characterization of soil texture can be gained by describing the PSD by means of mathematical models. Many alternative models have been suggested to characterize PSD. The PSD measurement is a simple, yet informative test routinely performed in soil mechanics to classify soils. Mathematically showing the PSD offers several benefits. First, the soil may be classified making use of the best-fit parameters. Second, a mathematical equation can be applied as the basis for analysis related to estimating the soil water characteristic curve. Third, a mathematical equation can offer a method of showing the entire curve between measured data points. Forth, representing the soil as a mathematical function also provides increased flexibility in searching for similar soils in databases (Fredlund et al., 2000). Fifth, many important soil parameters gained from PSD such as the effective particle size (D 10), the coefficient of uniformity, and the fines content that have been correlated to the soil properties like strength and permeability (Vipulanandan and Ozgurel, 2009). Sixth, mathematical representation of PSD can be used in filter design as well as grading of soils for backfill applications. On the other hand, the mathematical equations need to be simple to understand and apply to the existing data. There are only few investigations which have attempted to compare different PSD models. Buchan et al. (1993) analyzed 71 PSD data sets collected from two regions of New Zealand and compared five soil PSD models. . Hwang et al. (2002) by studying on 1387 soil samples compared seven models for soil PSD and Hwang (2004) used the same dataset to study the effect of soil texture on the performance of nine different PSD models. Also, Bagarello et al.

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(2009b) investigated the performances of the Fredlund’s model and beerkan estimation of soil transfer (BEST) model. Nevertheless the above investigations have been done and many different models have been suggested to describe PSD data, their characteristics, accuracy and the effect of their parameters on the shape and position of PSD curve have not been investigated, so far. Also, the fitting capability of all the proposed soil PSD models has not been simultaneously investigated in an study. Another motivation of the present review study was that all the PSD models were proposed on a specific problem-solving context, and may not be investigated in different aspects. Therefore the objectives of the present study were (1) performing a comprehensive review of the suggested PSD models and describing their characteristics and accuracy, (2) to present the mathematical attributes of all PSD models and investigating the effect of PSD models parameters on the shape and position of PSD curve, (3) shortly giving a review of PSD applications, the parameters that can be calculated from PSD and its measurement methods and (4) to evaluate the ability of the PSD models in fitting to the PSD data of the soil samples of the UNSODA database (UNsaturated SOil hydraulic Database (Leij et al., 1996)) with a wide range of soil textures.

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An overview of the properties and importance of soil particle size distribution

Particle size distribution is one of the static properties and affects soil properties such as tensile strength (TS) and Atterberg limits and can be used to predict them (Table 1). Particle size distribution was used to classify soils for engineering and agricultural purposes. In other words, the PSD of a soil is important in understanding its physical and chemical properties. An overview of the properties, applications and importance of soil PSD are shown in Fig. 1. Fig. 1.

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Multiple regression analysis showed that TS was positively related with clay + silt content, and organic matter. Also, the poorly crystalline Fe oxides were the constituent of clay + silt fractions that most contributed to TS. Also, multiple regression analysis showed that plastic limit (PL) and liquid limit (LL) were poorly related with sand, whereas shrinkage limit was positively related with clay. Some of these regression equations are given in Table 1. Table 1. Recent studies have made use of the PSD as a basis for the estimation of other soil properties such as soil hydraulic properties (Arya and Paris, 1981; Gupta and Larson, 1979; Haverkamp and Parlange, 1986). Several models that were developed to estimate saturated hydraulic conductivity (Ks) are presented in Table 2 (Ferrer Julia et al., 2004). Mathematical relationships between the particle-size and hydraulic properties tend to be fairly good for sandy soils, but not as accurate for soils with larger fractions of clay (Cornelis et al., 2001). Furthermore, textural analysis supported the decisions related to management of crop production, soil conservation, and diagnostics of soil health (Filgueira et al., 2006). Table 2. 3

Mathematical models of PSD

Various models have been proposed for describing soil PSDs. A good mathematical representation is the model that reduces the discrepancy between the observed and predicted data. Nevertheless, it should be mentioned that, in general, when the number of fitted parameters increases, the fitting performance of the model gets better. However, our unpublished data showed that there is no correlation between the number of parameters and fitting performance of the PSD models. Even though the models in the literature offer good predictions of the PSD data, the widespread usage of these equations has been limited due to the complexity of the equations (Vipulanandan and Ozgurel, 2009). An overview of the mathematical models of soil PSD is shown in Fig. 2, 3 and 4.

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Fig. 2. Fig. 3. Fig. 4.

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Examining the effect of the fitted parameters on the PSD curve

The ranges of the constants of each model were determined according to the ranges reported in the related original papers, if available. If not, the ranges of the constants were obtained by trial and error or using the information reported in the other works from the literature. Note that the effect of the fitted parameters on the PSD curve was shown only for one model of each group of the PSD models and most of them were put into a supplementary material. 4.1

Power law models

4.1.1 Beerkan estimation of soil transfer (BEST) model (Lassabatere et al., 2006) In this model the effect of parameters on the PSD curve has been shown in Fig. 5. Fig. 5 (A) shows a plot of the BEST model with n and m parameters constant and d g varying. The dg parameter affected the position of the PSD curve. The coarser the soil PSD, the larger the d g parameter. The dg parameter is an index of the largest particle sizes. Fig. 5 (B) shows a plot of the BEST model with dg and m parameters constant and n varying. The n parameter is a PSD index and affected the shape of the PSD curve. The more uniform the soil PSD, the larger the n values and the steeper the PSD curve. Fig. 5 (C) shows a plot of the BEST model with n and d g parameters constant and m varying. It seems that the m parameter is related to the asymmetry of the soil PSD curve. By increasing the m parameter the PSD curve becomes more uniform and shifts towards the coarser particles region. Fig. 5.

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4.2

Exponential-power models

4.2.1 Rosin and Rammler (1933) (1) model Although the model of Rosin and Rammler (1933) was originally derived for powdered coal, but it can be applied to many other materials (Allen, 1996). Weibull (1951) has shown wide applicability of statistical distribution function based on work published by Rosin and Rammler (1933) in many fields of human activity other than particle size distribution. The effect of the k1 and k2 parameters of Rosin and Rammler (1933) (1) model (Fig. 6) on the PSD curve were the same as those of dg and m parameters of the BEST (Fig. 5) model. Nevertheless, by increasing dg parameter of the BEST model and decreasing k1 parameter of the Rosin and Rammler (1933) model, the PSD curve tends to coarser particles region. Fig. 6.

4.3

Logarithm models

4.3.1 Logarithm model (Zhuang et al., 2001) In this model the effect of parameters on the PSD curve has been shown in Fig. 7. Fig. 7 (A) shows a plot of the logarithm model with b constant and a varying. The a parameter affected the shape of the PSD curve. The larger the value of a the steeper the PSD curve. The point of maximum slope along the PSD is an index of the dominant particle size in the soil. Fig. 7 (B) shows a plot of the logarithm model with a constant and b varying. The b parameter does not affect the shape of the curve, but provides a shift in the curve towards the finer or coarser particles regions of the plot. The PSD curve was shifted towards the finer particles region, as b parameter was increased. Fig. 7.

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4.4

Hyperbolic models

4.4.1 Hyperbolic model (Vipulanandan and Ozgurel, 2009) In this model the effect of parameters on the PSD curve has been shown in Fig. 8. Fig. 8 (A) shows a plot of the hyperbolic model with c parameter constant and A varying. Each curve tends to coarser and uniform particles and becomes steeper as A is increased. Fig. 8 (B) shows a plot of the hyperbolic model with A parameter constant and c varying. Each curve tends to coarser and uniform particles and becomes steeper as c is decreased. In fact both of the parameters affect both of the shape and position of the PSD curve but, with the contrary effect. Fig. 8. The hyperbolic (tan h(x)) model parameters were correlated to the particles median diameter d50, particle size range (d10-d90), and fine particles content (Vipulanandan and Ozgurel, 2009). 4.5

Statistical distribution models

4.5.1 Weibull two parameter model The Weibull function (Weibull, 1951) was originally used to characterize ‘the size effect on failures in solids’ or the sequential fragmentation of materials. This is called the Rosin and Rammler (1933) distribution when it was used to give a description of the PSD. In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Waloddi Weibull, who described it in more detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin and Rammler (1933) to give a description of a particle size distribution. The Weibull distribution is used for different purposes such as survival analysis, reliability engineering and failure analysis, industrial engineering, extreme value theory, and weather forecasting.

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The 2-parameter Weibull distribution is used to give a description of PSD, and predicts fewer fine particles than the log-normal distribution. The Weibull model is generally most accurate for narrow PSD. In the geoscience, the Weibull distribution has been used to give a description of the size distributions of fractured rock (Boadu and Long, 1994; Froehlich and Benson, 1996) and fractured ice (Tuhkuri, 1994), aeolian sediments as well as volcanic ash (Wohletz et al., 1989). The effect of the b and c parameters of the 2-parameter Weibull model on the PSD curve were the same as those of L and m parameters of the general expression of the Rosin and Rammler (1933) model (Fig. 5 of the supplementary material), respectively. Zobeck et al. (1999) reported that the Weibull model estimated the PSD very well for the particles <50 μm diameter. Nevertheless, estimates of particles with diameters greater than 50 μm deviated somewhat from their predicted values, possibly in the light of differences in transport processes for larger particles. Further support for the utility of the Weibull model was demonstrated by how well it fitted for many dust distributions described in the literature.

4.6

Logarithm-exponential models

4.6.1 Jaky (1944) one-parameter model In this model the effect of parameters on the PSD curve has been shown in Fig. 9. Fig. 9 shows a plot of the Jaky (1944) model with p parameter varying. As shown, by decreasing p parameter, the position and the shape of the curve change and the curve tends to coarser and more uniform particles. Fig. 9.

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4.7

Fractal models

4.7.1 Single-segment fractal models Fractal geometry was suggested with the objective of describing very irregular forms that are too complex to be described by Euclidean geometry (Mandelbrot, 1983a). Fractals are selfsimilar objects and applicable to scale-invariant systems (Carpinteri and Pugno, 2003; Turcotte, 1997). In spite of the Euclidean geometry, in which a point, a line, a plane and a volume have zero, one, two and, three dimensions, respectively, that all are integer values, the dimension (D) of a fractal is not necessarily an integer as in Euclidean geometry and it varies from 0 to 3 for true fractal sets. In fact, it is a fraction with a range between the values of Euclidean geometry for a line and a volume. A value of fractal dimension higher than 3 cannot be set out physically (Bartoli et al., 1991). Fractal geometry has been paid a lot of attentions in the latest developments in the study of PSD. PSD are frequently described as cumulative function, either as the number of particles larger than a certain size, or as the mass of particles smaller than a certain size (Turcotte, 1986; Tyler and Wheatcraft, 1992; Wu et al., 1993), or the mass of particles smaller than a certain size versus time (Filgueira et al., 2003). The physical basis of the theory was the attractive feature of this approach. The proper use and definition of the term “fractal” were also discussed by some authors (Baveye and Boast, 1998; Pachepsky et al., 1997; Young et al., 1997). Since, different fractal models have been applied to PSD data; therefore it was resulted in different interpretations of gained fractal dimensions. Thus, it is essential to clearly determine the type of fractal model that was used. However, there are lower and upper limits to the validity of fractal models for natural objects (Turcotte, 1986). Mandelbrot (1983a) noticed that several natural phenomena could be well described by a single power law function. The exponent of this power law function, called fractal dimension, is a non-negative real number that can assume fractional or non-fractional values.

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Nevertheless, for PSD, the fractal exponent, D, is usually greater than 3 (Tyler and Wheatcraft, 1989), which does not have a clear physical meaning. Kozak et al. (1996) indicated that power-law scaling might be applicable for a narrower range of particle sizes. Bittelli et al. (1999) reported that no single power-law exponent could characterize the PSD across the whole range of the measurements. They identified three main power-law domains. The domain between 0.51 and 85.3μm was representative for the silt domain, and the fragmentation fractal dimensions were found ranged from 1.728 to 2.792. The fractal dimension values obtained for the clay domain ranged from 0.118 to 1.210, and the fractal dimension values obtained for the sand domain ranged from 2.815 to 2.998. Determination of a single parameter that can replace soil particle size composition in describing soil texture, would be very usefull in characterising soil texture. To this end, soil fractal dimension can provide some useful insghits into not only characterization of the soil particle size composition but also the reflection of the uniform degree of soil texture (Yang et al., 1993).

4.7.2 Two-segment fractal models 4.7.2.1 Segmented model (Millan et al., 2003) Fig. 10 (A and D) shows the plots of the segmented model with k 1 and k2 parameters varying, respectively, and other parameters constant. The k1 and k2 parameters affected only the position of the curve. Each curve is shifted towards the coarser particles region as k 1 or k2 is increased. Fig. 10 (B and E) shows the plots of the segmented model with c1 and c2 parameters varying, respectively, and other parameters constant. The c1 and c2 parameters are related to the PSD index. The more uniform the soil PSD, the larger the value of c 1 and the smaller the value of c2 parameter.

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Fig. 10 (C and F) show the plots of the segmented model with D1 and D2 parameters varying, respectively, and other parameters constant. The D1 and D2 parameters affected the shape of the soil PSD curve. The segmented fractal model is a flexible model and can describe the PSD of a wide range of soils. Millan et al. (2003) reported that, they could not fit the segmented fractal model to the standard PSD. In their case, the computational algorithm did not converge, either for k =0, 1

k ≠0, or k1≠0 k =0, or for k ≠0, k2≠0. They attributed the lack of convergence to either the 2

2

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large number of estimation parameters (k , k , c , c , D , and D2), or to the small number of 1

2

1

2

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data pairs, or both. Fig. 10.

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Inflection point of the PSD models

Inflection points are spots where the growth of the curve begins to slow (going from concave up to concave down), or increase (going from concave down to concave up) (Solomentsev, 2001). In other words, inflection point helps to better highlight the changes. When the equation lacks inflection point, in this case, the curve of this function has only up concave or down concave. To obtain the inflection point, we should obtain the second derivative of the equation and set it equal to zero. Only some of the PSD models had the inflection point, as presented in the Table 3. No correlation has been found between the inflection point and the performance of PSD models; there were models with inflection points that showed good performance while some others showed a poor performance (Table 6).

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Table 3. 6

Fitting soil PSD models on the PSD data of the soil samples in the UNSODA database (UNsaturated SOil hydraulic Database (Leij et al., 1996)) .

6.1

Data

The unsaturated soil hydraulic database (UNSODA) was developed to provide a source of unsaturated hydraulic data and some other soil properties for users and researchers. This database contains measured soil water retention, hydraulic conductivity and water diffusivity data as well as pedological information of 790 soil horizons (codes) from around the world including 11, 20, 360, 355, 12 and 32 soil samples from Africa, Asia, Europe, North America, Pacific Region and No Data, respectively. The information on 790 soil horizons (codes) were contributed either by individual scientists or obtained by Leij et al. (1996) from the literature. Data are stored in 36 tables, three of which contain data on particle size distribution, aggregate size distribution, and mineralogy. The particle size table contains 713 soil samples with particle size data, with a total of 4174 size/fraction data pairs (Nemes et al., 2001). There were several reasons for choosing UNSODA dataset for this study: the UNSODA dataset covered all textural classes (Fig. 11 and Table 5) and also contains a wide range of particle sizes. Table 4 summarizes the statistics of the clay (0-0.002mm), silt (0.002– 0.05mm) and sand (0.05–2.0mm) contents (n=713). Also, this dataset covered soils from different regions in the world such as Africa, Asia, Europe and North America. On the other hand, there are different measured points (readings) for the soils of this dataset ranging from 3 to 16 which may affect the performance of the PSD models. Therefore, it could be a suitable dataset for evaluating the performance of the PSD models for a review research.

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Table 4. Table 5. Fig. 11. 6.2

Evaluation criteria for models comparison

The coefficient of determination (R2) and root mean square error (RMSE) were used as relative measure of the goodness-of-fit of the PSD models to the observed data of soil PSD. The corrected Akaike’s information criterion (AICc) (Burnham and Anderson, 2002) that imposes penalties for additional fitting parameters was used to compare the quality of model fits. 6.3

Fitting procedure

To find the values of the fitting parameters, all models were fitted to the experimental PSD data using the least square curve fitting toolbox of the MATLAB R2012a software (Release, 2012). To compare the fitting accuracy of PSD models the mean values of AICc, R2 and RMSE were compared with Duncan test by the SAS 9.1 software. 6.4

Cluster analysis

Cluster analysis or clustering is the grouping of a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense or another) to each other than to those in the other groups (clusters). It is a main task of exploratory data mining, and a common technique for statistical data analysis, and is used in many fields, including machine learning, pattern recognition, image analysis, information retrieval, and bioinformatics. Cluster analysis can be performed by various algorithms that differ significantly in their notion of what constitutes a cluster and how to efficiently distinguish them. Popular notions of clusters include groups with small distances among the cluster members, dense areas of the data space, intervals, or particular statistical distributions. The appropriate clustering

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algorithm and parameter settings (including values such as the distance function to use, a density threshold or the number of expected clusters) depend on the individual data set and the intended use of the results. Cluster analysis as such is not an automatic task, but an iterative process of knowledge discovery or interactive multi-objective optimization that involves trial and error. It will often be necessary to modify data preprocessing and model parameters until the result meets the desired properties (Bailey, 1994). Determining the number of clusters in a data set is a frequent problem in data clustering, and is a distinct issue from the process of actually solving the clustering problem. For a certain class of clustering algorithms (in particular, k-means, k-medoids and expectation– maximization algorithm), there is a parameter commonly referred to as k that specifies the number of clusters to detect. Other algorithms do not require the specification of this parameter; hierarchical clustering, which used in this study, eliminates the problem altogether (Bailey, 1994). In this study, selecting the number of clusters was determined automatically by the SPSS 17 software. To classify the PSD models based on their fitting accuracy (using RMSE criterion as a variable), cluster analysis with Ward's method and Euclidean distance between the groups was performed by SPSS 17 software (Statistics, IBM SPSS. "17.0." 2008). Ward's minimum variance method is a criterion applied in hierarchical cluster analysis and is a special case of the objective function approach, where the criterion for choosing the pair of clusters to merge at each step is based on the optimal value of an objective function such as sum of squares error (Hartigan, 1975; Sarle, 1990). Ward's method starts with n clusters of size 1 and continues until all the samples are contained into one cluster. 6.5

Comparing the accuracy of the PSD models

Table 6 shows the comparison of the accuracy of PSD models in fitting to the PSD data and Fig. 12 shows the result of cluster analysis. The results showed that the hyperbolic model, 15

performed better (significantly, P < 0.05) than the other models in terms of RMSE and AICc criteria. In the cluster analysis, hyperbolic, ORL and three parameter Weibull models were classified in the same class and they had not a significant (P < 0.05) difference in terms of R2 criterion. Vipulanandan and Ozgurel (2009) by studying ten sandy soils that have been used for chemical grouting studies reported that the Fredlund unimodal model performed better than the hyperbolic model, according to the average coefficient of determination (R 2). Then, their result was different from the results of this study in terms of the RMSE, R 2 and AICc criteria. The soil samples used by Vipulanandan and Ozgurel (2009) were only sandy soils, while the soil samples of this study distributed in all texture classes (Table 5). Therefore, less clay content and a few numbers of the soil samples of the Vipulanandan and Ozgurel (2009) research may be a reason for this disagreement.

In the cluster analysis, two parameter Weibull, Rosin and Rammler (1933) (2), Fredlund bimodal, Fredlund unimodal, Rosin and Rammler (1933) (1), BEST and log-normal models were classified in the same class and they had not a significant (P < 0.05) difference in terms of RMSE (with the exception of log-normal model) and R2 (with the exception of Rosin and Rammler (1933) (1) model) criteria. Fredlund unimodal model performed better (significantly, P < 0.05) than the Fredlund bimodal model in terms of AICc criterion. This result may be due to the penalties imposed by the AICc (Burnham and Anderson, 2002) for additional fitting parameters of the Fredlund bimodal comparing with the Fredlund unimodal model. Because, there is no significant difference between these two models in terms of RMSE and R2 criteria. Hwang (2004) used the Fredlund unimodal, Kravchenko and Zhang (1998), logarithm, power law (2), logarithm–exponential, Weibull (3) and van Genuchten models to describe the PSDs

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of clay, clay loam, loam, loamy sand, sandy clay loam, sandy loam and silty clay soil classes and reported that the performance of Fredlund unimodal model was slightly better than other models for the majority of the studied soils, according to the coefficient of determination criterion. Fredlund et al. (2000) reported that the performance of the Fredlund model would be increased as the clay content increased. The soil samples used by Hwang (2004) were heavy to moderate textures, while the soil samples of this study distributed in all texture classes (Table 5). Therefore, more clay content of the soil samples of the Hwang (2004) research may be a reason for the better performance of the Fredlund model in his study. Hwang et al. (2002) evaluated the accuracy of seven PSD models (i.e., SL, ORL, ONL, Shiozawa and Campbell, Jaky, Gompertz and the Fredlund unimodal models) in predicting PSD for a wide range of soil textures and reported that the Fredlund model with four parameters showed the best performance for the majority of the soils. Also, their results showed that texture could affect the performance of the PSD models and the Fredlund model showed the best fit for all soil textural classes. The BEST model performed better (significantly, P < 0.05) than the Fredlund unimodal model, according to the AICc due to the low number of the parameters of the BEST model. However, there is not found a significant difference (P < 0.05) between the fitting accuracy of the Fredlund unimodal and BEST models according to the RMSE and R2 criteria. This result was different from that of the Bagarello et al. (2009b), who fitted the Fredlund unimodal and BEST models on the experimental data of PSD and reported that the Fredlund unimodal model performed better than the BEST model, according to the mean relative error criterion. Fredlund unimodal model performed better (significantly, P < 0.05) than the van Genuchten model according to the RMSE, AICc and R2 criteria. This result may be due to the more flexibility of the Fredlund unimodal model in comparison with the van Genuchten model. Also, Bagarello et al. (2009a) by comparing two soil PSD models of the van Genuchten and

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Fredlund unimodal reported that the Fredlund unimodal model performed better than the van Genuchten model for the majority of the studied soils. Among the fractal models, Millan model performed better (significantly, P < 0.05) than the other fractal models in terms of RMSE and AICc criteria (except for the Perrier and Bird (2002, 2003) model according to AICc criterion). After the Millan model, segmanted fractal model had a higher fitting accuracy (significantly, P < 0.05) according to the R2 (except for the Millan model) criterion, in comparison with the other fractal models. The large number of the parameters of the segmanted fractal model could make some difficulties in fitting this model to PSD data that would resulted in lower fitting accuracy in comparison with that of Millan model (Millan et al., 2003). In the cluster analysis, power law (1), Mandelbrot (1983a), Bird et al. (2000), van Genuchten, GGS, logarithm-exponential, Perrier et al. (1999) and power law (2) models were classified in the same class and they had not a significant (P < 0.05) difference in terms of RMSE criterion. A poor performance of the van Genuchten model in most texture classes was reported by Hwang (2004). Hwang (2004) by comparing the Fredlund et al. (2000), Fractal model, logarithm, power law (1), logarithm–exponential, Weibull, and van Genuchten models observed that the performance of the PSD models improved as the clay content of soils increased, with the exception of the van Genuchten model. Also, Hwang (2004) reported that the logarithm-exponential, power law (1) and logarithm models showed a poorer performance in most texture classes, according to the coefficient of determination criterion. ac as- arc a et al. (2004) fitted the Rosin and Rammler (1933) (2) and the GGS models on the experimental data of PSD and reported that the Rosin and Rammler (1933) (2) model performed better than the GGS model that is in agreement with our results. The better

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performance of the Rosin and Rammler (1933) (2) model may be due to its physical basis (Brown and Wohletz, 1995) that could describe the PSD accurately. The fitting accuracy of the power law (1 and 2) models had not a significant (P < 0.05) difference, in terms of RMSE and AICc criteria, but, power law (1) model performed better (significantly, P < 0.05) than power law (2) in terms of R2 criterion. This result may be due to the complex form of the power-law (2) model in comparison with the simple form of the power-law (1) model. Rousseva (1997) investigated the effects of textural classes on the performances of the power-law (1) and power-law (2) models and concluded that these models could be applied to represent the PSDs of fine and coarse textured soils, respectively. In the cluster analysis, SL and ONL models were classified in the same class and they had not a significant (P < 0.05) difference in terms of RMSE and R2 criteria. The simple Jaky oneparameter model performed better (significantly, P < 0.05) than the SL model according to the evaluation criteria. Buchan et al. (1993) analyzed 71 PSD data sets collected from two regions of New Zealand and by comparing five soil PSD models of the Jaky (1944) oneparameter, the SL, the ORL, the ONL and a bimodal lognormal model of Shiozawa and Campbell, and reported that based on the R2 and CP1 (Hocking, 1976) tests, the ORL model performed better than the other models for the majority (82%) of the soils studied. The Gompertz and Kravchenko and Zhang (1998) models had a significantly (P < 0.05) less accurate estimation in terms of the RMSE and R2 criteria (with the exception of Nesbitt and Breytenbach (2006) model), in comparison with the other models. Even the difference between these two models was significant (P < 0.05) in terms of RMSE criterion and Gompertz model accuracy was lower than the Kravchenko and Zhang (1998) model. The Gompertz model was placed in a separate class in the cluster analysis. Also, the Kravchenko and Zhang (1998) model had a lower accuracy (significantly, P < 0.05) in comparison with

1

Mallows' Cp has been shown to be equivalent to Akaike information criterion in the special case of Gaussian linear regression.

19

the other models, in terms of R2 criterion. The Gompertz and Kravchenko and Zhang (1998) models are inflexible models with overlap of the effect of their parameters on the shape and position of the PSD curve. Therefore, they could not describe the PSD curve accurately. Hwang et al. (2002) by comparing seven PSD models, including five lognormal models (Jaky, SL, ORL, ONL, Shiozawa and Campbell), the Gompertz and the Fredlund models reported that the R2 values of the Gompertz four-parameter model were not much higher than the other models with fewer parameters, indicating that increasing the number of parameters does not always guarantee a better fit. Hwang (2004) by studying 1385 soils of Korea reported that the Kravchenko and Zhang (1998) fractal model showed poorer performance in most texture classes. According to researchers (e.g. Hwang et al. (2002), Hwang (2004) and this study), the fitting results of the PSD models depend on the quality and quantity of data. Data quality refers to the reliability and efficiency of the data (e.g. the number of readings (points) for each soil or distribution quality of soil samples in different texture classes). The quantity of data refers to the mean or variance of the sand, silt and clay contents. One reason for the difference between the results of this study and those of other researchers may be the differences in the quality and quantity of data. Consequently, all of the results reported in the literature and in this study can be correct and, their differences may be attributed to the different characteristics of the datasets that were used. However, the UNSODA database covers an acceptable quality and quantity of soil PSD data then, the results of this study may be more reliable. The results (Table 6) showed that there is not any relationship between the accuracy of the models (RMSE, R2 and AICc) and function types. It means that the accuracy of the models was not related to the type of the model.

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Table 6. Fig. 12. Fig. 13 shows the percentage of the soil samples in which the PSD models performed the best fit on the PSD curve. The hyperbolic, ORL, Fredlund unimodal, S-curve and Gompertz models performed better than other models on 46, 11, 8, 7 and 7 percent of the soil samples, respectively. The order of models in Table 6 and Fig. 13 are different. This disagreement may be due to the different standard deviations of the RMSE of fitting PSD models on the PSD data. S-curve and Gompertz models have greater standard deviations of RMSE in comparison with the other models (Table 6), indicating a high degree of variability in fitting accuracy using the S-curve and Gompertz models. This means that these two models could fit the PSD of some soil samples very well or very bad. In fact, they fitted most of the soil samples in a wide range of accuracy. The Fredlund unimodal model was flexible over the wide range of particle sizes. Nonetheless, the main disadvantage of the Fredlund models is the large number of parameters they use, and the resulting difficulties to fit them. Although the fractal models have a mathematical concept, they could not describe the soil PSD curves more accurately than other model types. Fig. 13. Fig. 14 shows the distribution of fitted versus measured relative cumulative mass fraction of the particles for the first three classes and the last class of cluster analysis. The models from the first three classes provided the best fit to the experimental data in all ranges of particle sizes whereas the only one model from the last class of cluster analysis (Gompertz model) showed the poorest performance. Fig. 14.

21

7

Advantages and disadvantages of the PSD models

The advantages and disadvantages of the PSD models were described in Table 7 based on the following criteria. The first part (from left) is based on examining the behavior of the fitted parameters of the PSD models according to the following four factors: a) Flexibility and high accuracy (or inflexibility and low accuracy), which show that a PSD model has high accuracy and flexibility in fitting to different soils, and its parameters could describe the PSD curve adequately. b) Overlap (or non-overlap), which shows that the effect of the parameters of a PSD model on the shape and position of the PSD curve is not distinguishable. c) Over-prediction, which shows the over-prediction of cumulative mass fraction at the coarse end of the PSD curve by a PSD model. d) Complex equation form and difficult fitting procedure. The second part (from left) is based on the analysis of the PSD models for UNSODA database according to the cluster analysis. a) Excellent: for the models in clusters 1 and 2. b) Good: for the models in clusters 3 and 4. c) Not so good: for the models in cluster 5. d) Bad: for the models in clusters 6, 7 and 8. Overall classification of the PSD models in the three classes of good, moderate and bad was based on the following criteria: a) Good: for the models in clusters 1 and 2, if having at least two advantages of the behavior of the fitted parameters for the PSD models. Otherwise, the model was placed in the moderate class. b) Moderate: for the models that are neither in the good nor in the bad class. c) Bad: for the models in clusters 6, 7 and 8 if having at least two disadvantages of the behavior of the fitted parameters for the PSD models. Otherwise, the model was placed in the moderate class. 22

In the analysis of advantages and disadvantages of the PSD models, the BEST, Rosin and Rammler (1933) (1) and the general expression of the Rosin and Rammler (1933) (2), Fredlund’s unimodal, ORL, two and three parameter Weibull and lognormal models were classified in the good class (Table 7). The parameters of these models have physical meaning. They described the particle size distribution curve well, and these models can be used for different soils with different textures. In these models, the effect of each parameter on the PSD curve can readily be seen. In other words, the effect of one parameter on the shape and position of the PSD curve can be distinguished from the effect of the other parameters. Therefore, the fitting procedure would be easy for these models, and description of the parameters would not be complex. The general expression of the Rosin and Rammler (1933) (2) model is very similar to the two-parameter Weibull model. Consequently, their fitting accuracies were the same. Among the advantages of these models are their simple form with fewer parameters, and higher fitting accuracy along with the physical meaning of their parameters. Brown and Wohletz (1995) pointed out that the Weibull model has a physical basis as the Rosin and Rammler (1933) is the integral form of the Weibull. On the other hand, the Weibull model is very flexible and can be applicable to a wide field of problems (Weibull, 1951). Also, Zobeck et al. (1999) confirmed that the Weibull model described the dust distribution very well. The Fredlund et al. (2000) model was flexible over the wide range of particle size distributions. Nevertheless, the main disadvantage of the Fredlund et al. (2000) models is the large number of parameters they use, and the resulting difficulties to fit them. The Nesbitt and Breytenbach (2006), power law (1 and 2), logarithm, SL, ONL, Gompertz, Perrier and Bird (2002, 2003), and Kravchenko and Zhang (1998) models were classified in the bad class (Table 7). These models lack flexibility and are unable to show the uniformity

23

or non-uniformity of the PSD curve. Therefore, they are not suitable models for describing the PSD curve. The disadvantage of the SL, ONL, and Nesbitt and Breytenbach (2006) models is the difficulty of fitting on the PSD data due to their complex fitting function. In the logarithm model, nonetheless, the effect of one parameter can be distinguished from the effect of the other parameter, and the equation has a simple form containing two parameters. Still, it does not seem to be a flexible model since the soil PSD in the nature does not follow the log-linear trend described by the logarithm model (Fig. 7). This might be a reason for its poor performance that was reported by Hwang (2004). A poor performance of the power law (2) model in most textural classes was reported by Hwang (2004). The Gompertz (with four parameters) and ONL (with three parameters) models utilize a relatively large number of parameters, yet they perform relatively weak. Most of the fractal models were classified in the moderate class. The fitting investigations of fractal models (Table 6) showed that they have low accuracy and flexibility in different soils, and the parameters of these models do not describe the particle size distribution curve well. The over-prediction of relative mass fraction (more than 1) and overlap of the effect of their parameters on the PSD curve were two disadvantages of these models. The advantages of these models are their physical basis and simple form with a fewer parameters. However, the results showed that these models could not describe the particle size distribution curve better than the non-fractal models. The number of parameters and the equation type of the models did not affect their fitting accuracy. Eventually, given the small number of model parameters used by the BEST, Rosin and Rammler (1933) (1 and 2), and two-parameter Weibull models makes them easy to use in soil physics and mechanics research. .

24

Table 7. 8

Conclusions and outlook

The particle size distribution of a soil is its fundamental physical property. A better representation of soil texture could be gained by describing the PSD using mathematical models. Many options have been proposed to choose a model to characterize PSD. Mathematical representation of the PSD provides many benefits. Therefore, it would be very useful to use a mathematical model to give a description of the soil PSD. Abundant different models have been suggested to give details of soil PSD data, but their characteristics, accuracy, and the effect of their parameters on the shape and the position of PSD curve have not been investigated. Our studies showed that the BEST, Weibull (two and three parameters), Rosin and Rammler (1933) (1 and 2), hyperbolic and ORL models had a high fitting accuracy over a range of particle sizes. Also, our studies showed that the BEST, Weibull (two and three parameters), Rosin and Rammler (1933) (1 and 2), unimodal and bimodal Fredlund, and van Genuchten models had adequate flexibility over a range of particle sizes. Therefore, these models’ parameters could fit particle size distribution curve in the range of sand, silt, and clay, very well. The BEST, Weibull (two and three parameters), Rosin and Rammler (1933) (1 and 2), hyperbolic and ORL models can be used successfully in soil physics and soil mechanics research due to their simplicity and acceptable accuracy. Among the fractal models, Millan model showed high fitting accuracy and more flexibility rather than the other fractal models. Although the fractal models have the physical and mathematical basis, our review showed that these models do not have the adequate flexibility to give a description of the PSD curve. The GGS and Nesbitt and Breytenbach (2006) models, which have one parameter, did not show a good description of the PSD curve. However, that might be due to the low number of model parameters. In the contrary, the Jaky and Gaudin-Melloy models with one parameter were able to have a better description of the

25

PSD curve. Some models, including power law (2), SL, ONL, logarithm and Gompertz with more than one parameter, did not have a good description of the PSD curve. Therefore, increasing model parameters is not an insuring factor for a model to be favored.

Acknowledgements The authors are deeply grateful to two anonymous reviewers and the editor for their helpful comments on the manuscript.

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35

Fig. 1. An overview of the properties and importance of particle size distribution. Fig. 2. An overview of the mathematical models of soil particle size distribution. Fig. 3. Continuation of the mathematical models of soil particle size distribution. Fig. 4. Continuation of the mathematical models of soil particle size distribution. Fig. 5. A plot of BEST model while (A) n and m constant and dg varying, (B) dg and m constant and n varying, and (C) dg and n constant and m varying. Fig. 6. A plot of Rosin and Rammler (1933) (1) model while (A) k2 constant and k1 varying; and (B) k1 constant and k2 varying. Fig. 7. A plot of logarithm model while (A) b constant and a varying; and (B) a constant and b varying. Fig. 8. A plot of hyperbolic model while (A) c constant and A varying; and (B) A constant and c varying. Fig. 9. A plot of the simple Jaky (1944) model while p varying. Fig. 10. A plot of segmented model while (A) c1, k2, c2, D1 and D2 constant and k1 varying; and (B) k1, k2, c2, D1 and D2 constant and c1 varying; and (C) k1, k2, c1, c2 and D2 constant and D1 varying; and (D) k1, c1, c2, D1 and D2 constant and k2 varying; and (E) k1, k2, c1, D1 and D2 constant and c2 varying; and (F) k1, k2, c1, c2 and D1 constant and D2 varying. Fig. 11. Textural distribution of the soil samples on the USDA textural triangle. Fig. 12. Cluster analysis diagram for the PSD models, based on the RMSE criterion. The horizontal dashed line shows the separation of the groups. Fig. 13. Percent of the soil samples in which the PSD models performed the best fit on the PSD curve, based on the RMSE criterion. Fig. 14. Distribution of fitted versus measured relative cumulative mass fraction of the particles for the first three classes and the last class of cluster analysis. 36

Dynamic light scattering

Pipette (based on Stokes law) Laser diffraction

Elutriation

Imaging particle

Hydrometer (based on Stokes law) Wiegner sedimentatio n cylinder

Sieve (for coarse soil )

So =

D75 D 25

Sk =

Coefficient of sorting

Test tube shaking

D 25 × D75 M d2

Coefficient of skewness

1. Method of particle size analysis

M=

D16 +D50 +D84 3

SD=

D84 -D16 D95 -D5 + 4 6.6

Standard deviation

Mean

3. Describing PSD statistics

Particle size distribution (PSD) 4. Mathematical models of PSD

Exponential -power equations

Hyperbolic equations

Power law equations

Statistical distribution

Logarithm equations

Textural classification systems

2. Soil gradation Well graded

Logarithmexponential equations

Poorly graded Uniformlygraded

USCS

5. Application of the PSD in soil science

Estimation of soil hydraulic properties Estimating soil water retention curve

USDA

Gap-graded

Fractal equations

Regression models (3)

Mechanical properties

Saturated as well as unsaturated hydraulic conductivities (1)

Physico-empirical models (2)

Atterberg limits

Tensile strength (TS)

AASHTO ANN (4) Well graded

Uniform graded

Australian Standards

GMDH (4)

Decision trees and support vector machines (5)

British Standards

Gap graded

Note: Note: D Dxxis is the the diameter, diameter, in in millimeters, millimeters, at at which which xx percent percent of of materials, materials, by by weight, weight, are are finer. finer. USDA: USDA: United United states states department department of of agriculture. agriculture. USCS: USCS: Unified Unified soil soil classification classification system. system. AASHTO: AASHTO: American American association of state highway and association of state highway and transportation transportation officials. officials. ANN: ANN: Artificial Artificial neural neural network. network. GMDH: GMDH: Group Group method method of of data data handling. handling. (1): (1): Vereecken Vereecken et et al., al., (2010), (2010), (2): (2): Arya Arya and and Paris Paris (1981), (1981), (3):Gupta (3):Gupta and and Larson Larson (1979), (1979), (4):Bayat (4):Bayat et et al. al. (2011) (2011) and and (5): (5): Lamorski Lamorski et et al. al. (2014) (2014)

Fig. 1. An overview of the properties and importance of particle size distribution.

37

4. Mathematical models of PSD

4.1. Power law equations van Genuchten or Haverkamp and Parlange (1986) (HP)

Beerkan estimation of soil transfer (BEST) model

P(d) m

1 ( 1

dg d

4.4. Hyperbolic equations

Gates–Gaudin– Schuhman (GGS) (1940) model

Hyperbolic

Gaudin- Melloy (Harris, 1968)

m

)n

P(d)

2 n

[1 (

k1 k 2 ) ] d

1 1 k2

P(d)

Parameters: k1 and k2

[

d d max

P(d)

]a

1 (1 (

d d max

))

P 100

a

(0.5) 1 tgnh( sin h(x) cos h(x)

tanh(x)

Parameter: a and dmax=2 mm

Parameter: a and dmax=2 mm

Power law (1)

(

P(d)

10 (

1

d ) ( )a a 1 2

1

(0.1

1

d ) ( ) a 1 2

1 1 0.1 a 1

P(d)

)

c d

Power law (2)

P(d)

Parameters: c and β

k ( 1 ) d (1 1 k2

ex ex

A c e e

) x x

Parameters: A and c

Parameters: dg , n and m Nesbitt and Breytenbach (2006)

d

4.3. Logarithm equations Logarithm

k2 )

P(d)

Parameters: k1 and k2

Parameter: a

a (ln d)

b

Parameters: a and b

The GGS equation is a fractal model and the a parameter equals 3 – D (Ahmed and Drzymala, 2005; (Turcotte, 1986; Tyler and Wheatcraft, 1992; Yang et al., 1993) 4.2. Exponential-power equations

Rosin-Rammler (1933) model

(1) P(d)

1 exp[ k1 d k 2 ] Parameters: k1 and k2

(2) P(d)

Exponential power

P(d)

d 1 exp[ ( ) m ] L

exp(

1

d 2)

Parameters: β1 and β2

Parameters: L and m

Note: Note: where where P(d) P(d) is is the the cumulative cumulative mass mass of of particles particles with with equivalent equivalent diameter diameter << d, d, dd is is the the particle particle size size (mm). (mm).

The general expression of the Rosin and Rammler (1933) model

Fig. 2. An overview of the mathematical models of soil particle size distribution.

38

4. Mathematical models of PSD

4.5. Statistical distribution Two-parameter log-normal

Weibull Weibull three parameter

P(d)

a

exp

Parameters: a , b and c

d b

Weibull two parameter c

P(d)

1 e

P(d)

d ( )c b

Parameters: b and c

Gompertz model

2 ) e(

1/ (a

a

b exp{ exp[ c(D

d)]}

Parameters: a, b, c and d

Offset-renormalized lognormal (ORL)

Lognormal distribution

P(d)

probnorm(

log d a ) b

probnorm

exp(

x2 )dx 2

Offset-nonrenormalized lognormal (ONL) model

(log(d) b) 2 /2 a 2 )

G(X)

Parameters: a and b

x

P(D)

Simple lognormal (SL)

(1

) F(X)

Parameter: ε

F(X) F(X) erf (X)

1 X {1 erf ( )} 2 2 1 X {1 erf ( )} 2 2 2

x 2

e t dt 0

Parameters: σ and μ

Parameters: a and b

F(X)

c

Parameter: c

X X

Note: Note: where where P(d) P(d) is is the the cumulative cumulative mass mass of of particles particles with with equivalent equivalent diameter diameter << d, d, dd is is the the particle particle size size (mm). (mm). μμ is is the the mean mean of of X X and and σσ is is the the standard standard deviation deviation of of X. X. G(X) G(X) is is the the modified modified cumulative cumulative function. function. F(X), F(X), cumulative cumulative function function as as defined defined by by the the SL. SL. erf erf (X) (X) is is the the error error function. function. X=Ln(d) X=Ln(d)

Fig. 3. Continuation of the mathematical models of soil particle size distribution.

39

G(X)

4. Mathematical models of PSD 4.6. Logarithm-exponential equations S-Curve

p 100 x

e

Logarithm-exponential

P(d)

nx

k ln(

d d min

4.7. Fractal equations

A exp(B log d)

P(d)

1 d ln( ) p2 d Max

exp

Parameters: A and B

d d

)

Ms (d

di )

[k 2

c2 di 3

c1di3

[k1 D2

Millan et al. (2003) model

](di

D1

](di

dc ) Ms (d

[c2di3

dc )

di ) D2

](di

[c1di3

D1

](di

dc )

dc )

where D1 and D2 are fractal dimensions of the first

Fredlund’s bimodal

and the second domain, respectively, and dc is the

Fredlund’s unimodal

P

Ln 1

1 (

a gr d

1 )

n gr

mgr

Ln 1

d rgr d d rgr

7

(

a bi n bi ) d

accounting to smaller scales and c is a composite

m bi

scaling constant defined in Perrier et al. (1999) and Bird et al. (2000).

1

(1 w) ln exp(1)

dm ln(1 1 (

Parameters: agr, ngr, mgr and drgr. dm= 0.001

cut-off of the whole domain. k is a constant

1

w ln exp(1)

L n exp(1)

Segmented fractal model

2

Parameter: p and dmax=2 mm

Parameters: n, k, dα and dmin=0.002 mm

P(d)

Two-segment fractal models

Jaky (1944)

ln(1

j ( bi ) k bi d

lbi

d rbi ) d )7 d rbi ) dm

Parameters: abi, nbi, mbi , w, jbi, kbi, lbi and dm= 0.0001 and drbi= 0.001

Single-segment fractal models Mandelbort (1983)

Kravchenko and Zhang (1998) 2

P(d) Cd Note: Note: where where P(d) P(d) is is the the cumulative cumulative mass mass of of particles particles with with equivalent equivalent diameter diameter << d, d, dd is is the the particle particle size size (mm). (mm).

D

Parameters: C and D ( fractal dimension)

P(d) exp{lnk2 (

Perrier et al. (1999)

3D 13D 14 1) lnd} D2 5D 4

Parameters : k2 and D

3 D

P(d) k d

Parameters: k and D

Bird et al. (2000)

Perrier and Bird (2002, 2003)

3 D

P(d) cd

P(d)

Parameters: c and D

Fig. 4. Continuation of the mathematical models of soil particle size distribution.

40

d (a D 3 ) ( )3 L

Parameters: a, L and D

D

1.2

A BEST model n=3.62 m=0.121

1 0.8

Cumulative mass fraction

Cumulative mass fraction

1.2

0.6 dg=0.04 0.4

dg=0.13 dg=0.4

0.2

1

B BEST model dg=0.121 m=0.121

0.8 0.6

n=0.45 0.4

n=1.5 n=2.5

0.2

n=3.6

dg=0.8

0 0.001

0.01

0.1

1

10

0 0.001

0.01

Particle size, mm

Cumulative mass fraction

1.2 1

0.1

1

10

Particle size, mm

C BEST model dg=0.121 n=3.62

0.8 0.6 m=0.02 0.4

m=0.12 m=0.64

0.2

m=0.9 0 0.001

0.01

0.1

1

10

Particle size, mm

Fig. 5. A plot of BEST model while (A) n and m constant and dg varying, (B) dg and m constant and n varying, and (C) dg and n constant and m varying.

41

1.2

1

A Rossin-Rammler (1933) model k2=0.633

Cumulative mass fraction

Cumulative mass fraction

1.2

0.8 0.6 k1=1.37 0.4

k1=3.37 k1=5.37

0.2 0 0.001

k1=7.36 0.01

0.1 Particle size, mm

1

1

B Rossin-Rammler (1933) model k1=3.37

0.8 0.6

k2=0.48 0.2 0 0.001

10

k2=0.32

0.4

k2=0.63 k2=0.78 0.01

0.1 Particle size, mm

1

10

Fig. 6. A plot of Rosin and Rammler (1933) (1) model while (A) k2 constant and k1 varying; and (B) k1 constant and k2 varying.

42

1

1.2 A Logarithm model b=0.896

Cumulative mass fraction

Cumulative mass fraction

1.2

0.8 0.6

a=0.06

0.4

a=0.08 a=0.11

0.2 0 0.001

a=0.13 0.01

0.1

1

1 0.8

B Logarithm model a=0.06

0.6 b=0.75 b=0.8 b=0.85 b=0.9

0.4 0.2 0 0.001

10

Particle size, mm

0.01

0.1 Particle size, mm

1

10

Fig. 7. A plot of logarithm model while (A) b constant and a varying; and (B) a constant and b varying.

43

1 0.8

1.2

A Hyperbolic model c=0.245 A=0.03

0.6

Cumulative mass fraction

Cumulative mass fraction

1.2

A=0.08 A=0.1 A=0.2

0.4

0.2 0 0.001

0.01

0.1 Particle size, mm

1

0.8

c=0.4

c=0.6

0.6

c=0.8

0.4 0.2

0 0.001

10

B Hyperbolic model c=0.2 A=0.045

1

0.01

0.1 1 Particle size, mm

10

Fig. 8. A plot of hyperbolic model while (A) c constant and A varying; and (B) A constant and c varying.

44

Cumulative mass fraction

1.2 1

Jaky model

0.8

0.6

p=2.5 p=4

0.4

p=5.3 0.2 0 0.001

p=7 0.01

0.1 Particle size, mm

1

10

Fig. 9. A plot of the simple Jaky (1944) model while p varying.

45

1.2

1.5

Cumulative mass fraction

1 0.8 0.6

Cumulative mass fraction

A Segmented model c1=1.05 D1=2.79 k2=-0.001 c2=-0.001 D2=2.43 k1=-0.1

0.4

k1=-0.2 0.2

B Segmented model k1=-0.01 D1=2.79 k2=-0.01 c2=-0.1 D2=2.43

1

c1=0.83

0.5

c1=0.92

k1=-0.25

c1=1

k1=-0.3 0 0.001

0.01

0.1

1

c1=1.15

0 0.001

10

Particle size, mm

Cumulative mass fraction

1 0.8

1

0.6 0.4

D1=2.79 D1=2.83

0.2

0.1 Particle size, mm

1

0.6

0.6

k2=-0.01

0.4

k2=-0.08

10

k2=-0.11 k2=-0.15 0.01

0.1 Particle size, mm

1

10

1.4 E Segmented model k1=-0.01 c1=1.05 D1=2.758 k2=-0.1 D2=2.43

1.2 Cumulative mass fraction

Cumulative mass fractiontle

0.8

10

D Segmented model k1=-0.01 c1=1.05 D1=2.758 c2=-0.1 D2=2.43

0 0.001

1.2 1

0.8

D1=2.9 0.01

1

0.2

D1=2.87

0 0.001

0.1 Particle size, mm

1.2

C Segmented model k1=-0.01 c1=1.05 k2=-0.2 c2=-0.01 D2=2.43

Cumulative mass fraction

1.2

0.01

0.8

c2=-0.001

0.4

c2=-0.07

0.2

0.01

0.1 Particle size, mm

1

0.6

D2=2.4

0.4

D2=2.6 D2=2.8

0.2

c2=-0.15 c2=-0.22

0 0.001

1

F Segmented model k1=-0.01 c1=1.05 d1=2.758 k2=-0.1 c2=-0.1

0 0.001

10

D2=3 0.01

0.1 Particle size, mm

1

10

Fig. 10. A plot of segmented model while (A) c1, k2, c2, D1 and D2 constant and k1 varying; and (B) k1, k2, c2, D1 and D2 constant and c1 varying; and (C) k1, k2, c1, c2 and D2 constant and D1 varying; and (D) k1, c1, c2, D1 and D2 constant and k2 varying; and (E) k1, k2, c1, D1 and D2 constant and c2 varying; and (F) k1, k2, c1, c2 and D1 constant and D2 varying. 46

Fig. 11. Textural distribution of the soil samples on the USDA textural triangle.

47

Fig. 12. Cluster analysis diagram for the PSD models, based on the RMSE criterion. The horizontal dashed line shows the separation of the groups.

48

Percentage of siol samples Two parameter Weibull 2% Rossin Rammler (1) 3%

Other models

BEST 3%

Three parameter Weibull 3%

Fredlund bimodal 4% Hyperbolic 47%

Gompertz 7%

S-Curve 7%

Fredlund unimodal 8% ORL 11%

Fig. 13. Percentage of soil samples in which the PSD models performed the best fit on the PSD curve, based on the RMSE criterion.

49

Fitted cumulative mass fraction

0.8

0.8

0.8

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2 0

0 0

0.5

1

Two parameter Weibull RMSE=0.052 R2=0.968

1

0.8

Three parameter Weibull RMSE=0.040 R2=0.982

1

0.6

0

Fitted cumulative mass fraction

ORL RMSE=0.039 2 R =0.920

1

Hyperbolic RMSE=0.024 R2=0.989

1

0

0.5 Rossin Rammler (2) RMSE=0.052 R2=0.968

1

0

1

0.5

1

Fredlund bimodal RMSE=0.060 R2=0.990

1

0.6

0.5

0.5

0.4 0.2

Fitted cumulative mass fraction

0

0.5

1

Fredlund unimodal RMSE=0.0615 R2=0.962

1

0

0.5

0

1

1

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0.5

1

Lognormal RMSE=0.068 R2=0.968

1

0.8

0

0.5

0.8

0.8

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.5 1 Measured cumulative mass fraction

1

0

0 0

0.5

Gaudian-Melloy RMSE=0.091 R2=0.882

1

Jaky RMSE=0.088 2 R =0.900

1

0

1

0.6

0

1

0

0

0

0.5

BEST model RMSE=0.0622 R2=0.965

1

Rossin Rammler (1) RMSE=0.0619 0.8 R2=0.907

0.8

Fitted cumulative mass fraction

0

0

0

0

0.5

1

Measured cumulative mass fraction

50

0

0.5

1

Measured cumulative mass fraction

Fitted cumulative mass fraction

Millan RMSE=0.094 R2=0.946

1

0.8

Gompertz (last model) 1 RMSE= 0.267 and R2= 0.464

1 Exponential power RMSE=0.095 R2=0.895

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0 0

0.5 1 Measured cumulative mass fraction

0 0

0.5 1 Measured cumulative mass fraction

0

0.5 1 Measured cumulative mass fraction

Fig. 14. Distribution of fitted versus measured relative cumulative mass fraction of the particles for the first three classes and the last class of cluster analysis.

51

Fig. 2. An overview of the mathematical models of soil particle size distribution.

52

Fig. 3. Continuation of the mathematical models of soil particle size distribution.

53

Fig. 4. Continuation of the mathematical models of soil particle size distribution.

54

100 90 80

70

Clay

60

clay 50 silty clay

40

sandy clay

silty clay loam

30

clay loam sandy clay loam

loam

20 silt loam 10

sandy loam

silt 0 0

10

20

30

40

50

60

70

loamy sand 80

sand 90

100

Sand

Fig. 11. Textural distribution of the soil samples on the USDA textural triangle.

55

Table 1. Models that were developed to estimate tensile strength (TS) and Atterberg limits. Soil PTF$ property

Ln Ts  4.3707  0.0209 (clay  silt)  0.0596 (Organic Matter)  0.0015 ((clay  silt)  organic matter )

Tensile strength

Ln Ts  0.021  0.0015 (organic matter)  (clay  silt) Ln Ts  0.021  0.0015 (clay  silt)  (organic matter ) Ln Ts  2.6413  0.469 Fe oxides  0.0148 (organic matter )

Atterberg limits $

Plastic limit  25.73  0.035sand  0.027  cation exchange capacity   0.2 (organic matter) Liquid limit  51.39  0.029sand  0.03  cation exchange capacity  Linear Shrinkage  2.79  0.96(AL oxides)  0.058 clay

. In accordance with the USDA classification system, the clay, silt and sand are the particles

in the ranges between 0 and 0.002 mm, 0.002 and 0.05 mm and 0.05 and 2.0 mm, respectively. Organic matter is a matter composed of organic compounds coming from the remains of organisms such as plants and animals, and their waste products in the environment (Senesi et al., 2009).

56

Table 2. Models that were developed to estimate saturated hydraulic conductivity (Ks). Name Campbell and Shiozawa (1992) Cosby et al. (1984)

$

PTF$

Ks (mm/h)=54 exp(-0.007 sand-0.167 clay)

Ks (mm / h)

25.4 10(

0.6 0.012 sand-0.0064×clay)

Dane and Puckett (1992)

Ks (mm/h)=303.84 exp(-0.144 clay)

Puckett et al. (1985)

Ks (mm/h)=156.96 exp(-0.001975 clay)

. In accordance with the USDA classification system, the clay, silt and sand are the particles

in the ranges between 0 and 0.002 mm, 0.002 and 0.05 mm and 0.05 and 2.0 mm, respectively.

57

Table 3. Second derivations of particle size distribution models. Models

Inflection point

GGS (Turcotte (1986), Tyler and Wheatcraft (1992) and Yang et al. (1993),) Power law (1)

0

0

Power law (2)

0

Gompertz Kravchenko and Zhang (1998), Perrier et al. (1999) and Bird et al. (2000)

d

0

25 b 2 exp(a ) 98 25 b 2 exp(a ) 98 25 b 2 exp(a ) 98

SL

ONL

ORL

a

Hyperbolic

log(10) 2 a

b sqrt log(10) 2 b 2

2 b

exp

4

b 2 log(10)

4

b 2 log(10)

2

Log-normal

log(10) 2 a

2 b

b sqrt log(10) 2 b 2

exp

2

Gaudin-Melloy (Harris, 1968)

2

Mandelbrot (1983b)

0

10

b

10

b

a log(10) sqrt log(10) 2 a 2 exp

Two parameter lognormal

exp

4

a 2 log(10) 2 2

2 a 2 log(10) 2 2

58

a log(10) sqrt log(10) 2 a 2 2

4

Table 4. The statistics of the clay (0-0.002mm), silt (0.002–0.05mm) and sand (0.05–2.0mm) contents (n=713). Textural fraction Clay (%) Silt (%) Sand (%)

Mean 13.1 30.3 56.6

Standard deviation 14.4 20.9 30.0

59

Min 0.0 0.0 2.4

Max 64.8 85.1 100.0

Table 5. Representation of the distribution of the 713 soil samples of the UNSODA database in 12 USDA soil textural classes. Textural class Number of soil samples Percentage of soil samples, % a.

Sa 195

LS 73

SL 83

L 136

SiL 96

Si 3

SCL 21

SiCL 10

CL 40

SC 2

SiC 8

C 46

Total 713

27.35

10.24

11.64

19.07

13.46

0.42

2.95

1.40

5.61

0.28

1.12

6.45

100

S sand, LS loamy sand, SL sandy loam, L loam, SiL silt loam, Si silt, SCL sandy clay loam,

SiCL silty clay loam, CL clay loam, SC sandy clay, SiC silty clay, C clay.

60

Table 6. Statistical analysis of the accuracy of the fitting particle size distribution models on the 712 PSD experimental data of the UNSODA database. Row 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 $

Model Hyperbolic ORL Three parameter Weibull Two parameter Weibull Rossin and Rammler (1933) (2) Fredlund bimodal Fredlund unimodal Rossin and Rammler (1933) (1) BEST Log-Normal Jaky (1944) Gaudian-Melloy Millan Exponential power S-Curve Two parameter lognormal Segmanted Power law (1) Mandelbort (1983) Bird et al. (2000) van Genuchten GGS Logarithm-exponential Perrier et al. (1999) Power law (2) SL ONL Perrier & Bird (2002, 2003) Logarithm Nesbitt & Breytenbach (2006) Kravchenko & Zhang (1998) Gompertz

Equation Hyperbolic Statistical distribution Statistical distribution Statistical distribution Exponential power Logarithm-Exponential Logarithm-Exponential Exponential power Power Law Statistical distribution Logarithm-Exponential Power Law Two-segment Exponential power Logarithm-Exponential Statistical distribution Two-segment Power Law Single-segment Single-segment Power Law Power Law Logarithm-Exponential Single-segment Power Law Statistical distribution Statistical distribution Single-segment Logarithm Power Law Single-segment Statistical distribution

Cluster 1 1 1 2 2 2 2 2 2 2 3 3 3 3 4 4 4 5 5 5 5 5 5 5 5 6 6 7 7 7 7 8

RMSE$ Mean SD 0.024a 0.026 0.039b 0.060 0.040b 0.027 0.052bc 0.051 0.052bc 0.051 0.060cd 0.048 0.062cd 0.067 0.062cd 0.110 0.062cd 0.069 0.068d 0.039 0.088e 0.061 0.091e 0.057 0.094e 0.050 e 0.095 0.081 0.115f 0.136 0.116f 0.079 fg 0.119 0.071 0.130gh 0.077 0.130gh 0.077 0.132gh 0.078 0.135gh 0.214 0.136hi 0.076 0.139hi 0.080 0.142hi 0.073 0.143hi 0.092 0.149ij 0.151 0.156j 0.138 0.172k 0.060 0.181kl 0.121 l 0.190 0.167 0.206m 0.197 0.267n 0.220

R2 Mean 0.989a 0.920abcdef 0.982ab 0.968abc 0.968abc 0.990a 0.962abc 0.907bcdef 0.965abc 0.968abc 0.900cdefg 0.882defg 0.946abcde 0.895cdefg 0.862fgh 0.888cdefg 0.959abcd 0.878efg 0.878efg 0.873efg 0.550k 0.826gh 0.859fgh 0.870efg 0.790hi 0.672j 0.725ij 0.878efg 0.665j 0.451l 0.340m 0.464l

SD 0.028 0.177 0.027 0.076 0.076 0.015 0.096 0.404 0.091 0.032 0.413 0.218 0.052 0.274 0.314 0.175 0.072 0.100 0.100 0.102 1.068 0.131 0.122 0.101 0.651 0.546 0.358 0.100 1.032 1.547 2.100 0.452

AICc Mean -50.28a -40.91b -38.16c -27.04ef -27.04ef 43.57m -23.63gh -28.20e -32.44d -18.09i -23.66gh -24.33fgh -13.93j -25.75efg -25.78efg -11.39jkl 42.33m -10.21klm -10.12klm -9.96klmn -17.46i -18.10i -9.25lmn -8.96lmn -9.17lmn -13.11jk -22.54h -13.02jk -7.21mn -18.04i -6.94n -12.13jkl

SD 24.98 23.10 22.37 27.54 27.54 57.55 22.08 28.59 19.37 22.94 14.91 17.42 18.94 23.34 22.96 19.41 42.23 18.87 17.88 17.77 25.16 10.63 17.64 21.65 17.76 27.38 27.88 13.85 21.65 16.32 18.81 37.00

. RMSE, root mean square error; R2, coefficient of determination; AI c, Akaike’s

information criterion.

61

Table 7. The advantages and disadvantages of the PSD models.

power law

ExponentialPower logarithm LogarithmExponential Hyperbolic

Statistical distribution

Two-segment fractal Singlesegment fractal a

BEST$ van Genuchten$ Gaudin-Melloy Nesbitt and Breytenbach (2006)$ power law (1) power law (2) GGS Rosin and Rammler (1933) (1)$ and the general expression of the Rosin-Rammler (1933) (2)$ Exponential-power logarithm Fredlund’s unimodal$ Fredlund’s bimodal$ Jaky (1944) S-Curve Logarithm-Exponential Hyperbolic ORL$ Two parameter Weibull$ Three parameter Weibull$ Lognormal$ Two- parameter log-normal$ Simple lognormal (SL)$ ONL$ Gompertz$ Millan et al. (2003)$ Segmented$ Mandelbrot (1983)$ Kravchenko and Zhang (1998)$ Perrier et al (1999)$ Bird et al (2000)$ Perrier and Bird (2002, 2003)$

2 2 1

*

1

*

2 2 1

* * *

* *

* * *

* *

* *

* * *

* * * *

* *

*

*

* *

*

*

* * *

* * * * * * *

* *

* * * * * * * *

*

* *

* * * * * * *

Bad

Moderate

Good

Bad (for clusters 6, 7 and 8)

Not so good (for cluster 5)

Good (for clusters 3 and 4)

* *

* *

*

* * *

*

* *

* *

* * * * * * * *

* * * * * *

* * * * * * * *

* * * * * *

*

* * * * * * * * * *

*

* * *

* *

* * *

The quality of fitting

*

*

* *

Overall classific ation of the models d

*

*

* *

* * *

*

*

* * * * * * * * * * *

Non-overlap Simple form and easy fitting procedure.

* *

2 2 2 4 7 1 3 2 2 3 2 3 2 2 2 3 4 4 6 2 2 2 2 2

Flexible and high accuracy

Advantag es

Complex form and difficult fitting procedure.

Over-prediction c

Overlap b

Inflexible and low accuracy a

Number of parameters

Model

Equation

Disadvantages

Analysis of the PSD models for UNSODA database (based on the cluster analysis). The quality of fitting

Excellent (for clusters1 and 2)

Examining the behavior of the fitted parameters for the PSD models

* * * * *

* * * * * *

* * * *

* * *

*

This model has low accuracy and flexibility in different soils and its parameters could not describe the PSD curve adequately.

62

*

b

The effect of parameters of this model on the shape and position of the PSD curve is not distinguishable.

c

The over-prediction of cumulative mass fraction at the coarse end of the PSD curve was one of the disadvantages of this model. In fact this model predicted the values more than 1 for relative cumulative mass fraction that did not have a physical meaning.

d

On the basis of examining the behavior of the fitted parameters for the PSD models and fitting accuracy of the PSD models for UNSODA database.

$

Some parameters of these models have a physical concept.

63

Highlights 

Important step in using soil PSD is determining accurate and continuous curve for it



The BEST, Weibull, Rossin-Rammler and Fredlund models had adequate flexibility



BEST and Hyperbolic models had a high fitting accuracy over the entire PSD curve



Fractal models have the physical basis but, have no adequate flexibility



Segmented and Millan fractal models had more flexibility than the others

64

Particle size distribution models, their characteristics and fitting capability Hossein Bayata*, Mostafa Rastgob, Moharram Mansouri Zadehc, Harry Vereeckend,

a

Hossein Bayat*: Assistant Professor (Ph. D.), Department of Soil Science, Faculty of

Agriculture, Bu Ali Sina University, Hamedan, Iran. Postal Address: Department of Soil Science, Faculty of Agriculture, Bu Ali Sina University, Hamedan, Iran. E-mail: [email protected] Other e-mail: [email protected]. Office phone: +98-81-34424189, Mobile phone: +98-918-8188378. Fax: +98-81-34424189. b

Mostafa Rastgo: Ph. D. Student of Soil Science, Bu Ali Sina University, Faculty of

Agriculture, Department of Soil Science, Hamedan, Iran. E-mail: [email protected], c

Moharram Mansouri Zadeh: Assistant Professor (Ph. D.), Department of Computer,

Faculty

of

Engineering,

Bu

Ali

Sina

University,

Hamedan,

Iran.

E-mail:

[email protected] d

Harry Vereecken: Prof. Agrosphere Institute, ICG-4, Forschungszentrum Jülich GmbH, D-

52425 Jülich, Germany. E-mail: [email protected]

*Corresponding author ([email protected]).

65