## 4 Laboratory SSCB Model Test under the Lateral No-Flow Condition
### Diversion Length on the SSCB Model Test  
The existing diversion length (L\({}_{\text{D}}\)) under the lateral flow (drained) condition was derived, like in past studies, by using Equation (2) of [PERSON] et al. (2018). Each parameter related to Equation (2) was determined as follows. First, \(b\) was derived from the equation of the unsaturated hydraulic conductivity: the unsaturated permeability coefficient (\(k_{w}\)) of sand was approximated by applying the exponential function of the negative pressure head (\(h\)), as shown in Equations (7) and (8). In particular, since the seepage (i.e., the breakthrough) into the gravel layer from the sandy layer in the CB system occurs when the sand layer is close to saturation, the value of \(b\) for this condition in this study was applied. Thus, the value of \(b\) in this study was determined from the relationship of the instantaneous change in the permeability coefficient for increased suction from the AEV point. Based on the exponential approximation of the relationship between the unsaturated permeability coefficient and the negative pressure head for Toyoura sand (Figure 10), the value of \(b\) was estimated as 0.13. \[k_{w}=k_{sat} h<h_{ae} \tag{7}\] \[k_{w}=k_{sat}\exp[-b(h-h_{ae})] h>h_{ae} \tag{8}\]  
Notably, the saturation permeability coefficient of Toyoura sand (\(k_{sat}\)) was 1.45 \(\times\) 10\({}^{-2}\) cm/s, its slope angle (\(\Phi\)) 10\({}^{\circ}\), and its AEV (\(h_{ae}\)) 24.5 cm; moreover, the air expulsion value of silica sand No. 1 (\(h_{ae}\)) was 0.5 cm. For each case, we applied the following the \(\kappa\) values (= \(q_{v}/k_{sat}\)) and rainfall intensities: Case 1: \(q_{v}\) = 20 mm/h \(\rightarrow\)\(\kappa\) = 0.038, Case 2: \(q_{v}\) = 50 mm/h \(\rightarrow\)\(\kappa\) = 0.096, and Case 3: \(q_{v}\) = 100 mm/h \(\rightarrow\)\(\kappa\) = 0.19. All the determined parameters (shown in Table 6) were substituted into Equation (2) for each case, deriving the following diversion lengths (L\({}_{\text{D}}\)) under the lateral flow (drained) condition: 145.5 cm (Case 1), 56.8 cm (Case 2), and 27.7 cm (Case 3). As such, the diversion length estimated by Equation (2) can be understood as the maximum theoretical water-shielding capability of the CB system. As explained earlier, it can be seen that the length of the diversion length is significantly affected by the rainfall intensity. Figure 10: Exponential approximation of the relationship between the unsaturated permeability coefficient and the negative pressure head for Toyoura sand. Meanwhile, the lengths of \(\Delta\)L\({}_{1}\) and \(\Delta\)L\({}_{2}\) were used to identify the wet areas of the gravel layer that formed due to breakthrough (Figure 8a). By considering \(\Delta\)L\({}_{1}\) = L\({}_{\text{st}}\) - L\({}_{\text{UDI}}\) and \(\Delta\)L\({}_{2}\) = L\({}_{\text{st}}\) - L\({}_{\text{UD2}}\), the following values were derived for \(\Delta\)L\({}_{1}\) and \(\Delta\)L\({}_{2}\): 31.3 and 20.0 cm (Case 1), 33.1 and 33.6 cm (Case 2), and 44.8 and 44.8 cm (Case 3), respectively. We indicated with L\({}_{\text{st}}\) the horizontal length of the soil tank when considering the slope angle (i.e., L\({}_{\text{st}}\) = 45.5 \(\times\) cos\(\Phi\), in cm). The effective diversion lengths (L\({}_{\text{ED1}}\) and L\({}_{\text{ED2}}\)) became smaller under the lateral no-flow condition: L\({}_{\text{ED1}}\) = L\({}_{\text{D}}\) - \(\Delta\)L\({}_{1}\) and L\({}_{\text{ED2}}\) = L\({}_{\text{D}}\) - \(\Delta\)L\({}_{2}\) (Figure 8a). L\({}_{\text{ED1}}\) represented the diversion length in the SSCB model test after it reached the level of the interface between the sand and gravel layers, while L\({}_{\text{ED2}}\) represented the diversion length in the CB model test after it reached the sensor installation level in the gravel layer. We obtained the following values for L\({}_{\text{ED1}}\) and L\({}_{\text{ED2}}\): 114.2 and 125.5 cm (Case 1), 23.7 and 23.1 cm (Case 2), and \(-\)17.1 and \(-\)17.1 cm (Case 1), respectively.