# Prediction of Soil-Water Retention Curves of Road Base Aggregate with Various Clay Fine Contents
## 4 SWRC Model  
Since the inclusion of fines leads to the different evolution patterns of inter-aggregate pores and intra-aggregate pores, two PSD functions were employed to account for the change in PSD. This method was adopted to capture the bimodal feature of the SWRC [27; 28; 31]:  
\[f(lnr)=f_{c}(lnr)+f_{f}(lnr) \tag{4}\]  
where \(f_{c}(lnr)\) represents the PSD of inter-aggregate pores and \(f_{f}(lnr)\) represents the PSD of intra-aggregate pores. For inter-aggregate pores, the PSD function considering the variation of \(FC\) can be expressed as:  
\[f_{c}(lnr)\xrightarrow{FC\text{ increases}} \begin{array}{l}\eta_{i}\cdot f_{c}( lnr+ln\chi_{i})\\ =\eta_{i}\cdot f_{c}[ln(r\chi_{i})]\end{array}  \tag{5}\]  
Figure 5: A conceptual model considering the change in PSD as \(FC\) increases. For intra-aggregate pores, the PSD function considering the variation of \(FC\) can be expressed as:  
\[f_{f}(lnr)\xrightarrow{\text{$FC$ increases}}\eta_{i}\cdot f_{f}(lnr+ln\chi_{i}) \tag{6}\]  
By incorporating Equations (5) and (6) into Equation (4), the PSD function changes to:  
\[f(lnr)=\eta_{c}\cdot f_{c}[ln(r\cdot\chi_{c})]+\eta_{f}\cdot f_{f}[ln(r)] \tag{7}\]  
where \(\eta_{c}\) and \(\chi_{c}\) refer to the scaling and shifting factor for inter-aggregate pores; \(\eta_{f}\) refers to the scaling factor for intra-aggregate pores. By putting Equation (7) into Equation (2), the volumetric water content of soil can be rewritten as:  
\[ \begin{array}{ll}\theta(r)&=\int_{R_{min}}^{R}f(lnr)dlnr\\ &=\int_{R_{min}}^{R}\eta_{c}\cdot f_{c}[ln(r\cdot\chi_{c})]dlnr+\int_{R_{min}}^ {R}\eta_{f}\cdot f_{f}[ln(r)]dlnr\\ &=\eta_{c}\cdot\theta_{\text{sc}}S_{\text{ec}}(r\cdot\chi_{c})+\theta_{\text{ sf}}\cdot\eta_{f}S_{\text{ef}}(r)\end{array}  \tag{8}\]  
where \(S_{\text{ec}}(r\cdot\chi_{c})\) refers to the effective saturation of inter-aggregate pore and \(S_{\text{ef}}(r)\) refers to the effective saturation of intra-aggregate pore; \(\theta_{\text{sc}}\) and \(\theta_{\text{sf}}\) refers to the saturated volumetric water content of the inter-aggregate pores and intra-aggregate pores. The effective saturation is calculated by \(S_{e}=\frac{\theta(r)-\theta_{s}}{\theta_{s}-\theta_{r}}\), where \(\theta_{s}\) is the volumetric water content at the saturated state and \(\theta_{r}\) is the volumetric water content at the residual state, which is assumed to be zero [28; 31]. Thus, \(\theta(r)\) can be rewritten as:  
\[\theta(r)=\eta_{c}\cdot\theta_{\text{sc}}\cdot S_{\text{ec}}(r\cdot\chi_{c})+ \theta_{\text{sf}}\cdot\eta_{f}\cdot S_{\text{ef}}(r) \tag{9}\]  
Furthermore, the saturation of the bimodal SWRC at a given pore radius/suction can be obtained as:  
\[S_{e}(r)=\frac{\theta(r)}{\theta_{s}}=\eta_{c}\cdot\frac{\theta_{\text{sc}}}{ \theta_{s}}\cdot S_{\text{ec}}(r\cdot\chi_{c})+\eta_{f}\cdot\frac{\theta_{ \text{sf}}}{\theta_{s}}\cdot S_{\text{ef}}(r) \tag{10}\]  
where \(\theta_{s}\) is the saturated volumetric water content of the soils; \(\theta_{s}=\theta_{\text{sc}}+\theta_{\text{sf}}\). For the SWRC with the bimodal feature, \(\eta\) affects the volumetric water content in the saturated state. For a given suction located in \"stage c\", \(S_{\text{ec}}(r\cdot\chi_{c})\) is assumed to be zero and \(S_{\text{ef}}(r)\) is assumed to be one; thus, the saturation of the bimodal SWRC is equal to \(\eta_{f}\cdot\frac{\theta_{\text{sf}}}{\theta_{s}}\). Since the \(S_{e}(r)\) increases linearly with the increase in \(FC\) (the saturation at \"Stage c\" is 24% for \(FC=4\%\) and 47% for \(FC=8\%\)), \(\eta_{f}\cdot\frac{\theta_{\text{sf}}}{\theta_{s}}=k\cdot FC\) is assumed, where \(k\) is a fitting parameter. Furthermore, if \(r\) approaches infinity (s approaches zero), \(S_{e}(r)\), \(S_{\text{ec}}(r\cdot\chi_{c})\) and \(S_{\text{ef}}(r)\) approaches one, then \(\eta_{c}\cdot\frac{\theta_{\text{sc}}}{\theta_{s}}=1-k\cdot FC\).