# Prediction of Soil-Water Retention Curves of Road Base Aggregate with Various Clay Fine Contents
## 3 Effect of Fine Content on Soil Water Retention Curve and Pore Size Distribution  
Figure 3a presents the SWRCs with different \(F\)Cs. The SWRC shifts upward with the increase in \(F\)C, indicating that the water retention properties increase with the increase in \(F\)C. Additionally, the increase in \(F\)C results in SWRC changing from unimodal (\(F\)C = 0%) to bimodal (\(F\)C = 4% and 8%). Bimodal SWRCs are caused by the discontinuity of pore size distribution (PSD) between inter-aggregate pores (due to coarse grain) and intra-aggregate pores (formed by fines) [11; 29]. Furthermore, the bimodal SWRC can be divided into four stages (Figure 3b), which are \"stage a\", \"stage b\", \"stage c\", and \"stage d\". In stage a, the pores stay saturated, with no water exchange; in stage b, the bulk water is drained from the inter-aggregate pores; in stage c, the meniscus water is drained from the intra-aggregate pores, and a large increase in matric suction leads to minor variations in Sr; in \"stage d\", the water is drained from the intra-aggregate pores [27; 29]. Figure 2: A schematic view of the LSSTA. To investigate the effects of \(FC\) on the SWRCs formed by inter-aggregate pores and intra-aggregate pores, the SWRCs formed by coarse-grain and fine-grain soil need to be distinguished first. This separation is based on the characteristics of \"stage c\", in which a large increase in matrix suction results in a minor variation in Sr; the average value of the \(S_{\rm r}\) at the start and end points of \"stage c\" is deemed as the end point of the SWRC formed by inter-aggregate pores and the start point of the SWRC formed by intra-aggregate pores [27,29]. The SWRCs formed by the coarse-grain soil and fine-grain soil are presented in Figure 4a,b, respectively. It can be observed that an increase in \(FC\) leads to the water retention properties of the SWRC formed by inter-aggregate pores increased and the increase in the air entry values (the boundary between \"stage a\" and \"stage b\"). Moreover, the SWRC formed by intra-aggregate pores seemed to be affected by the addition of fines. Before analysing the evolution of PSD with the increase in \(FC\), two basic assumptions should be made. The first one is that the pores in soils are a bunch of connecting tubes that are randomly distributed with different diameters and heights [27,36,37]; thus, the relationship between matric suction and the radius of the pore tube can be established by the Laplace equation:  
\[s=\frac{2T_{\rm s}cos\alpha}{r} \tag{1}\]  
where \(T_{\rm s}\) is the surface tension of water and soil and \(\alpha\) is the contact angle. The second assumption is for a given soil, the pores with a radius smaller than \(2T_{\rm s}cos\alpha/r\) are potentially occupied by water and the rest by air, which is termed as the local equilibrium assumption [23,36,38]. The volumetric water content of soil can be obtained:  
\[\theta(r)=\int_{C/s_{\rm mr}}^{C/s}f(lnr)dlnr \tag{2}\]  
Figure 4: (**a**) SWRCs of inter-aggregate pores at different \(F\)Cs; (**b**) SWRCs of intra-aggregate pores at different \(F\)Cs. Figure 3: (**a**) The SWRCs at different fine contents; (**b**) physical model for bimodal SWRC. where \(f(lnr)\) represents the PSD function. For the evolution of PSD, a shifting value \(ln\chi_{i}\) and a scaling factor \(\eta_{i}\) were employed [23]. The scaling factor is unable to change the feature of the SWRC, due to \(V_{\text{v}}\) and \(V_{\text{w}}\) scales at identical times. In comparison, the shifting factor can change the feature of SWRC. The shifting value \(ln\chi_{i}\) and scaling factor \(\eta_{i}\) at a given \(FC\) can be calculated from:  
\[ \begin{array}{c}ln\chi_{i}=ln\overline{R}_{i}-ln\overline{R}_{0}\\ \eta_{i}=\frac{f_{i}^{\prime}}{f_{0}^{\prime}}\\ i=1,2,3,4,\cdots\end{array}  \tag{3}\]  
where \(\overline{R}_{i}\) and \(\overline{R}_{0}\) are the mean pore radius at a target \(FC\) and \(FC\) = 0%, respectively.