# Prediction of Soil-Water Retention Curves of Road Base Aggregate with Various Clay Fine Contents
## 4 SWRC Model
By putting \(\eta_{f}\cdot\frac{\theta_{\text{sf}}}{\theta_{s}}=k\cdot FC\) and \(\eta_{c}\cdot\frac{\theta_{\text{sc}}}{\theta_{s}}=1-k\cdot FC\) into Equation (9), \(S_{e}(r)\) can be rewritten as:  
\[S_{e}(r)=(1-k\cdot FC)\cdot S_{\text{ec}}(r\cdot\chi_{c})+k\cdot FC\cdot S_{ \text{ef}}(r) \tag{11}\]  
In the present study, the van Genuchten model [39] was adopted to express the relationship between effective saturation, \(S_{e}\) and matrix suction, \(s\):  
\[S_{e}(s)=\left[1+(\alpha s)^{n}\right]^{-m} \tag{12}\]  
By introducing Equation (12) into Equation (11), \(S_{e}(r)\) turns into:  
\[S_{e}(r)=(1-k\cdot FC)\cdot S_{\text{ec}}(r\cdot\chi_{c})+(k\cdot FC)\Big{[}1 +\left(\alpha_{f}s\right)^{n_{f}}\Big{]}^{-m_{f}} \tag{13}\]  
where \(\alpha_{f}\), \(n_{f}\) and \(m_{f}\) are the fitting parameters of the SWRC formed by intra-aggregate pores. For the SWRC of inter-aggregate pores, the formation of the SWRC is much more complicated, due to the PSD of inter-aggregate pores being shifted by the addition of fines. The SWRC considering the shifting factor needs to be determined from the integration of PSD function as Equation (2), due to the change in PSD. The PSD function is defined as follows [23; 36; 37]:  
\[f(lnr)=-(\theta_{s}-\theta_{r})\frac{\partial S_{e}}{\partial s}\frac{C}{r} \tag{14}\]  
By introducing the van Genuchten model [39] (Equation (12)) and Laplace equation (Equation (1)) into Equation (14), the PSD function can be obtained as:  
\[f(lnr)=\theta_{s}mn\left[1+\left(\frac{\alpha C}{r}\right)^{n}\right]^{-m-1} \left(\frac{\alpha C}{r}\right)^{n} \tag{15}\]  
By putting Equation (5) into Equation (15), the PSD function of the inter-aggregate pores accounting for the variation of \(FC\) can be expressed as:  
\[ \begin{array}{ll}f_{c}(lnr)&=\eta_{c}f_{c}[ln(\chi r)]\\ &=\eta_{c}(\theta_{\text{sc}}-\theta_{\text{cr}})m_{c}n_{c}\Big{[}1+\left( \frac{\alpha_{c}C}{\chi_{c}r}\right)^{n}\Big{]}^{-m_{c}-1}\left(\frac{\alpha _{c}C}{\chi_{c}r}\right)^{n_{c}}\end{array}  \tag{16}\]  
where \(m_{c}\), \(n_{c}\), and \(\alpha_{c}\) are the fitting parameters for the inter-aggregate pores and \(\chi_{c}\) and \(\eta_{c}\) are the shifting parameter and scaling parameter for inter-aggregate pores. \(\theta_{\text{sc}}\) is the saturated volumetric water content of inter-aggregate pores. It can be observed from Figure 3a that AEV increases with the increase in \(FC\), which implies that \(\chi\) increases with the increase in \(FC\). An exponential relationship was assumed to exist between \(\chi\) and \(FC\) as:  
\[ \begin{array}{ll}&ln(\chi)=\beta\cdot ln(FC)\\ &\chi=\left(FC\right)^{\beta}\end{array}  \tag{17}\]  
where \(\beta\) is a fitting parameter. By incorporating Equation (17) into (16), \(f_{c}(lnr)\) can be rewritten as:  
\[ \begin{array}{ll}f_{c}(lnr)&=\eta_{c}f_{c}[ln(\chi r)]\\ &=\eta_{c}(\theta_{\text{sc}}-\theta_{\text{cr}})m_{c}n_{c}\Big{[}1+\left( \frac{\alpha_{c}C}{\left(FC\right)^{\beta_{r}}}\right)^{n}\Big{]}^{-m_{c}-1} \left(\frac{\alpha_{c}C}{\left(FC\right)^{\beta_{r}}}\right)^{n_{c}}\end{array}  \tag{18}\]  
Thus, at a given matric suction, the saturation of the SWRC formed by inter-aggregate pores is:  
\[ \begin{array}{ll}S_{\text{ec}}(r\cdot\chi_{c})&=\frac{f^{\prime}_{min}f_{c }\left[ln\left(r\cdot(FC)^{\beta}\right)\right]}{f^{\prime\prime}_{min}f_{c} \left[ln\left(r\cdot(FC)^{\beta}\right)\right]}dr\\ &=\frac{\left(\theta_{\text{sc}}-\theta_{\text{cr}}\right)\left[1+\left(\frac{ \alpha_{c}C}{\left(FC\right)^{\beta}}\right)^{n_{c}}\right]^{-m_{c}}\right|^{ \prime}}{\left(\theta_{\text{sc}}-\theta_{\text{cr}}\right)\left[1+\left( \frac{\alpha_{c}}{\left(FC\right)^{\beta}}\right)^{n_{c}}\right]^{-m_{c}}} \Bigg{|}_{0}^{+\infty}\\ &=\left[1+\left(\frac{\alpha_{c}r}{\left(FC\right)^{\beta}}\right)^{n_{c}} \right]^{-m_{c}}\end{array}