# A Simple Water Retention Model Based on Grain Size Distribution
## 4 Materials and Methods
### Validation of the Proposed WRC Prediction Model
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & \begin{tabular}{c} **Input** \\ **Parameters** \\ \end{tabular} & \begin{tabular}{c} **Equation Related to Volumetric** \\ **Water Content** \\ \end{tabular} & \begin{tabular}{c} **Equation for Section** \\ **or Suction Head** \\ \end{tabular} & **Outputs** & **Source** \\ \hline Proposed & \begin{tabular}{c} Sand, silt and clay \\ fraction, \(\theta_{\text{s}}\), GSD \\ \end{tabular} & \(\theta_{\text{ik}}\)\(=\theta_{\text{i}}\frac{\Delta_{\text{k}}}{\Delta_{\text{i}}}\) & \(\text{s}_{\text{ik}}\)\(=\frac{2\gamma\cos(\alpha_{\text{s}})}{\tau_{\text{k}}}\) & \begin{tabular}{c} A PSD and its \\ theoretical WRC \\ \end{tabular} & N/A \\ \hline MV & e, \(\theta_{\text{s}}\), GSD & \(\theta_{\text{i}}\)\(=\theta_{\text{s}}\sum\limits_{\text{j=1}}^{\text{j=i}}w_{\text{j}}\) & \(\text{h}_{\text{i}}\)\(=\frac{0.543\times 10^{-4}}{\text{K}_{\text{i}}}\xi\) & \begin{tabular}{c} Volumetric water \\ content and \\ suction head for \\ each GSD bin \\ \end{tabular} & [16] \\ \hline AP & e, \(\text{G}_{\text{s}}\), \(\alpha\), GSD & \(\theta_{\text{vi}}^{*}\)\(=(\theta_{\text{vi}}+\theta_{\text{vi}+1})/2\) & \(\psi_{\text{i}}\)\(=2\gamma\cos(\Theta)/\varrho_{\text{w}}\text{g}\)r\({}_{\text{i}}\) & \begin{tabular}{c} Volumetric water \\ content and \\ suction head for \\ each GSD bin \\ \end{tabular} & [11] \\ \hline CCQ & \begin{tabular}{c} Clay fraction, \(\beta\), \(\theta_{\text{s}}\), \\ GSD \\ \end{tabular} & \(\theta_{\text{i}}\)\(=\theta_{\text{s}}\sum\limits_{\text{j=1}}^{\text{j=i}}\text{v}_{\text{j}}\) & \(\psi_{\text{i}}\)\(=\frac{3000}{\text{d}_{\text{i}}}\) & \begin{tabular}{c} Volumetric water \\ content and \\ suction head for a \\ range of GSD bins \\ \end{tabular} & [12] \\ \hline NSMC & e, \(\text{G}_{\text{s}}\), \(\theta_{\text{s}}\), GSD & \(\theta_{\text{i}}\)\(=\theta_{\text{s}}\sum\limits_{\text{j=1}}^{\text{j=i}}\text{m}_{\text{g}\text{j}}\) & \(\psi_{\text{i}}\)\(=\frac{3000}{\text{D}_{\text{D}}[(\tau_{\text{i}}\text{e}+\tau_{\text{i}}) ^{1/3}-1]}\) & \begin{tabular}{c} Volumetric water \\ content and \\ spot density particle \\ size are water filled \\ \end{tabular} & [20] \\ \hline MK & \begin{tabular}{c} D\({}_{10}\), D\({}_{60}\), \(\text{a}_{\text{c}}\), LL, \\ \(\text{G}_{\text{s}}\), m, e, \(\lambda\) \\ \end{tabular} & \(\text{S}_{\tau}\)\(=\frac{\theta}{\text{n}}\)\(=\text{S}_{\text{c}}\)\(+\)S\({}_{\text{a}}^{*}(1-\text{S}_{\text{c}})\) & N/A & \begin{tabular}{c} Saturation degree \\ at an arbitrary \\ suction \\ \end{tabular} & [70] \\ \hline IMP & \
\begin{tabular}{c} \(\theta_{\text{s}}\), \(\hat{\varepsilon}_{\text{1}}\), \(\text{P}\), \(\beta_{\text{1}}\), \(\hat{\beta}_{\text{2}}\), \\ \(\psi_{\text{max}}\), \(\hat{\theta}_{\text{r},\text{max}}\), \(\alpha_{\text{1}}\), \\ \(\alpha_{\text{2}}\), \(\lambda\), GSD \\ \end{tabular} & \(\theta_{\text{i}}\)\(=(\theta_{\text{s}}-\theta_{\text{r PSD}})\)\(\frac{\sum\
olimits_{\text{j=1}}^{\text{j=i}}w_{\text{j}}^{\text{j=i}}}{\sum\
olimits_{\text{j=1}}^{\text{j=i}}w_{ \text{j}}^{\text{j=i}}}\)\(+\)\(\theta_{\text{r PSD}}\)\(\psi_{\text{i}}\)\(=\psi_{\text{max}}\)\(\left(\frac{\frac{1}{\text{k}}-\frac{1}{\text{k}_{\text{max}}}}{\sum\
olimits_{\text{min}}^{ \text{k}_{\text{max}}}}\right)^{\lambda}\) &
\begin{tabular}{c} Volumetric water \\ content and \\ suction for each \\ GSD bin \\ \end{tabular} & [71] \\ \hline \hline \end{tabular}  
Input parameter definitions: e = void ratio; \(\theta_{\text{s}}\) = saturated volumetric water content; GSD = particle size bins from grain size distribution; \(\text{G}_{\text{s}}\) = specific gravity; clay fraction = mass fraction of clay sized particles; D10 and D60 are the particle diameters at which respectively, 10% or 60% mass fraction of the soil is smaller (from GSD); \(\alpha\), \(\beta\), \(\text{a}_{\text{c}}\), \(\text{m}\), \(\lambda\) (MK), \(\(\lambda_{\text{1}}\), \(\text{p}\), \(\beta_{\text{1}}\), \(\hat{\beta}_{\text{2}}\), \(\psi_{\text{max}}\), \(\text{p}_{\text{max}}\), \(\alpha_{\text{1}}\), \(\alpha_{\text{2}}\), \(\lambda\) (IMP) are model specific input parameters that are either assumed or adjusted to fit WRC data.  
\end{table}