# Influence of Rock Slide Geometry on Stability Behavior during Reservoir Impounding
## 3 Rock Slide Characterization and Dam Reservoir-Related Mechanisms
### Block Model
The shear force for block 3, including the resultants of water pressure forces, is calculated by  
\[S_{3}=W_{3}\cdot\sin(\varepsilon)+Z_{2}\cdot\cos(\varepsilon-\beta/2)+Q_{2} \cdot\sin(\varepsilon-\beta/2)-Z_{3}\cdot\cos(\varepsilon/2)-Q_{3}\cdot\sin( \varepsilon/2)-P_{w13}\cdot\cos(\varepsilon/2). \tag{4}\]  
The water pressure \(p\) at the base of the block can be expressed as  
\[p=\rho_{w}\cdot g\cdot h_{w3}=\gamma_{w}\cdot h_{w3}, \tag{5}\]  
with \(\rho_{w}\) being the density of water and \(\gamma_{w}\) the specific weight of water. The hydrostatic normal force acting at the sliding plane \(P_{w3}\) of block 3 is determined as  
\[P_{w3}=\frac{\gamma_{w}\cdot h_{w3}^{2}}{2\cdot\sin(\varepsilon)}, \tag{6}\]  
and the hydrostatic normal force at the lateral boundary of block 3, labeled as \(P_{wi3}\), is determined by  
\[P_{wi3}=\frac{\gamma_{w}\cdot{h_{w3}}^{2}}{2\cdot\cos(\varepsilon/2)}. \tag{7}\]  
For block 4, the reservoir water load affecting the exposed rock slide surface has to be considered, which leads to the shear force, \(S_{4}\). Figure 2:0  
\(P_{wi4}\) and \(P_{4}\) are the water pressure forces acting on the rock slide surface and are given by  
Figure 2:1  
and, respectively,  
Figure 2:2  
In order to calculate the FoS of the rock slide, the forces induced by the weight of the blocks, i.e., \(W_{1}\) to \(W_{4}\), and the interslice forces must be calculated starting from block 1, followed by a successive calculation of blocks 2, 3 and 4, resulting in \(Q_{1}\) to \(Q_{3}\) and \(Z_{1}\) to \(Z_{3}\), respectively. Forces induced by the water pressures at the block boundaries, i.e., \(P_{w3}\), \(P_{wi3}\), \(P_{wi4}\), \(P_{wi4}\), \(P_{wi4}\), and \(P_{4}\), are determined for block 3 and 4. Based on this, the resulting normal (effective) forces \(N_{1}\), \(N_{2}\), \(N_{3}^{\prime}\), and \(N_{4}^{\prime}\) were calculated to obtain the shear resistance, T\({}_{1}\) to T\({}_{4}\)of the basal plane of each block. Combining the shear resistances with acting shear forces at the basal plane provides the FoS of the slide. A detailed summary of the equations relevant for the calculation of the four-block model are shown in Appendix A. Furthermore, FoS calculations were carried out by means of the software Slide2 [18] to verify the calculations conducted by hand. A factor of safety of FoS = 1 is assumed for the rock slide model without a reservoir. In doing so, the FoS reduces to values below one if the water pressure from the reservoir is included in the stability calculations. Figure 3 shows the calculated FoS values in relation with the reservoir water level. Regardless of the applied calculation method, the stability of the slope decreases by increasing the level of the reservoir until a certain threshold water level is reached. From this turning point, the FoS increases with further impounding. A similar FoS to reservoir level behavior was also obtained by previous studies (e.g., [12; 30]). The Fellerius method is considered the simplest of the methods of slices since the interslice forces are neglected [19]. In this study, the FoS calculations based on the Fellerius method provides the most conservative results, with an FoS lower than 3-5%. The method of [PERSON] and [PERSON] [19] gave similar results, only varying by less than 1%, at least at lower reservoir elevations. The FoS is slightly more conservative with lower values than those obtained by the method of [PERSON] and [PERSON]. The four-block model fits the [PERSON] and [PERSON] calculations reasonably well, especially in the section below the threshold water level. When the water level is further elevated, the deviation between both curves becomes larger, most likely influenced by the different calculation methods (Figure 3). The block model assumes a constant density of the rock slide mass, regardless of whether the mass is partially or completely saturated with water. Furthermore, the orientation and the shear properties of the arbitrarily chosen contact planes have an effect on the calculated FoS.