## 3 Soil Samples and Water Retention Characteristics
### Water Retention Characteristics  
The water retention characteristics of soils can be understood based on their respective SWCCs, representing the relationship between soil water and matric suction. In order to obtain the SWCCs of Toyoura sand and silica sand No. 1, we performed water retention tests through the continuous pressurization method [46]. Cylindrical specimens of 50 mm in diameter and 50 mm in height, with dry densities of 1.50 g/cm\({}^{3}\) and 1.64 g/cm\({}^{3}\), were used for these tests. The correspondent results were fitted by Equation (3) [34]. Figure 3a shows the fitted SWCCs resulting from the drying path of Toyoura sand and the wetting path of silica sand No. 1, related to the CB system. Specifically, in the CB system, since water infiltrates from the sand layer to the gravel layer below, the water retention characteristics of the sand and gravel layers correspond to the drying path where water is drained and the wetting path where water permeates, respectively. The fitting parameters of the SWCCs are summarized in Table 2. An AEV of 2.30 kPa and an air expulsion value of 0.05 kPa were derived for Toyoura sand and silica sand No. 1, respectively. By comparing the SWCCs of the two soils, it was observed that there is a difference in the suction value for the volumetric water content between Toyoura sand and silica sand No. 1. As previously mentioned for the CB system, these different suction values indicate that the soils have distinct water retention capacities (of the water that infiltrates the sand layer) [12; 16; 41; 47]. \[\theta=\ C(\psi)\left\{\frac{\theta_{s}}{\ln[e+\left(\psi/a\right)^{n}]}\right\} ^{m} \tag{3}\]  
where \(\psi\) is the total soil suction, \(e\) is the natural logarithmic constant (2.71828), \(a\) is a soil parameter related to the AEV of the soil (kPa), \(n\) a soil parameter that controls the slope at the inflection point along the soil-water retention curve, \(m\) a soil parameter related to \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample** & **Toyoura Sand** & **Silica Sand No. 1** \\ \hline Soil particle density, \(\rho_{\mathrm{s}}\) (g/cm\({}^{3}\)) & 2.64 & 2.65 \\ Mean particle size, \(D_{50}\) (cm) & 1.69 & 4.65 \\ Maximum dry density, \(\rho_{d\mathrm{\ max}}\) (g/cm\({}^{3}\)) & 1.64 & 1.67 \\ Minimum dry density, \(\rho_{d\mathrm{\ min}}\) (g/cm\({}^{3}\)) & 1.37 & 1.45 \\ Uniformity coefficient, \(\mathrm{C_{u}}\) & 1.63 & 2.24 \\ Curvature coefficient, \(\mathrm{C_{c}}\) & 0.97 & 0.84 \\ Relative density, \(D_{\mathrm{r}}\) (\%) in the CB model test & 53.8 & 87.9 \\ Saturated hydraulic conductivity, \(k_{sat}\) (m/s) & \(1.45\times 10^{-4}\) & \(2.44\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical properties and hydraulic conductivities of the soil samples. Figure 2: Grain size distribution curves of Toyoura sand and silica sand No. 1.  
the residual water content of the soil, and \(C(\psi)\) is the correction function that forces the soil-water retention curve through a suction of 1,000,000 kPa and zero water content. The correction function is given as follows:  
\[\text{C}(\psi)=\left[1-\frac{\ln(1+\psi/\psi_{\text{r}})}{\ln(1+10^{6}/\psi_{ \text{r}})}\right] \tag{4}\]  
where \(\psi_{\text{r}}\) is the suction value corresponding to the residual water content. Since the water-shielding performance of the CB system is affected by the difference in hydraulic conductivity functions and soil-water characteristic curves between sand and gravel, it is very important to grasp the hydraulic conductivities (\(k_{w}\)) of the two samples. The relative hydraulic conductivities (\(k_{r}\)) of the Toyoura sand and silica sand No. 1 under the unsaturated condition in this study were estimated based on Equation (5) proposed by [PERSON] et al. (2018). Here, Equation (5) for the relative hydraulic conductivity function was proposed by the combining Equation (3) with the statistical pore size distribution model of [PERSON] (2018).