Table 2.

Particle-size distribution models.

Name	Model†	Parameters
Anderson (AD [8])	$F(d)=f_{\mathrm{0}}+b\arctan \left(c\log \frac{d}{d_{\mathrm{0}}}\right)$	b, c, f 0, d 0
Fredlund4P (F4P [11])	$F(d)=\frac{\mathrm{1}}{{\left\{\ln \left[\exp (\mathrm{1})+{\left(a/d\right)}^n\right]\right\}}^m}\left\{\mathrm{1}-{\left[\frac{\ln \left(\mathrm{1}+d_f/d\right)}{\ln \left(\mathrm{1}+d_f/d_m\right)}\right]}^{\mathrm{7}}\right\}$	a, n, m, d f   (d m = 0.0001 mm)
Fredlund3P (F3P [11])	$F(d)=\frac{\mathrm{1}}{{\left\{\ln \left[\exp (\mathrm{1})+{\left(a/d\right)}^n\right]\right\}}^m}\left\{\mathrm{1}-{\left[\frac{\ln \left(\mathrm{1}+\mathrm{0.001}/d\right)}{\ln \left(\mathrm{1}+\mathrm{0.001}/d_m\right)}\right]}^{\mathrm{7}}\right\}$	a, n, m  (d m = 0.0001 mm)
Modified logistic growth (ML [43])	$F(d)=\frac{\mathrm{1}}{\left[\mathrm{1}+a\exp \left(-b{d}^c\right)\right]}$	a, b, c
Offset-nonrenormalized lognormal (ONL [9])	G(X) = F(X) + c,  where X = In⁡(d)   ${F(X)=\frac{{\mathrm{1}+\mathrm{erf}\left[(X-\mu )/(\sigma \sqrt{\mathrm{2}})\right]}^{‡}}{\mathrm{2}\left(X\ge \mu \right)}}^{}$  $F(X)=\frac{\mathrm{1}-\mathrm{erf}\left[(X-\mu )/(\sigma \sqrt{\mathrm{2}})\right]}{\mathrm{2}\left(X<\mu \right)}\mathrm{\hspace{0.17em}\hspace{0.17em}}$	μ, σ, c
Offset-renormalized lognormal (ORL [9])	G(X) = (1 − ε)F(X) + ε  [F(X) defined in ONL model]	μ, σ, ε
Skaggs (S [12])	$F(d)=\mathrm{\hspace{0.17em}\hspace{0.17em}}{\left\{\mathrm{1}+\left(\frac{\mathrm{1}}{F\left(d_{\mathrm{0}}\right)}-\mathrm{1}\right)\exp \left[-u{D}^c\right]\right\}}^{-\mathrm{1}}$,  where D = (d − d 0)/d 0    d 0 = 0.002 mm for T1; d 0 = 0.001 mm for T2,  T3; F(d 0): fraction < d 0	u, c
van Genuchten type (VG [7])	F(d) = [1+(d g/d)n]  −m,   where m = 1 − 1/n	d g, n
van Genuchten type modified (VGM [34])	F(d) = 1 − (1 − F min⁡)[1+(ad)n]−m,  where m = 1 − 1/n;  F min⁡, fraction of minimum particle size	a, n
Weibull (W [10])	F(d) = c + (1 − c){1 − exp⁡(−aD b)},  where D = (d − d min⁡)/(d max⁡ − d min⁡)   d max⁡ = 2 mm, d min⁡ = 0.002 mm for T1;   d max⁡ = 1 mm, d min⁡ = 0.001 mm for T2, T3	a, b, c,

^† d: particle diameter in mm.

^‡ erf⁡[]: error function.