# Estimating the Optimal Overall Slope Angle of Open-Pit Mines with Probabilistic Analysis
## 4 Discussion
#### Case Study  
The open pit case study was situated in the centre of the Andes mountains of Peru at an elevation of 5000 m. Limestone is the primary lithology, whereas the ore deposit is a polymetallic (e.g., \(\mathrm{zn}\), \(\mathrm{pb}\) and \(\mathrm{Ag}\)). Skarn rock comes into contact with an ore deposit at the bottom. The host rock (limestone) has a length of 172 m, width of 202 m and projects 50 m downward. The dimensions of orebody are 170 m \(\times\) 100 m (L \(\times\) W), and it extends 75 m to the bottom. The skarn is located at the base of the ore deposit. Figure 13a,b show a three-dimensional perspective view and a two-dimensional sectional plan. The geomechanical properties of the open pit rock masses are listed in Table 5. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Rock Mass**} & **Unit Weight,** & **Compressive** & **Elastic** & **Poisson’s** & **Cohesion,** & **Friction Angle, Deg.** \\  & **kN/m\({}^{3}\)** & **Strength, Mpa** & **Modulus, GPa** & **Ratio** & **Mpa** & **Friction Angle, Deg.** \\ \hline Limestone & 27 & 80 & 55 & 0.25 & 4.8 & 40 \\ Ore & 25 & 50 & 57 & 0.20 & 2.5 & 25 \\ Skarn & 30 & 85 & 65 & 0.18 & 4.6 & 35 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Geomechanical properties of rock masses of open pit case study [57]. Figure 13: (a) 3D perspective view of open pit case study. (b) 2D plan section. Figure 14 depicts the contours of the factor of safety (**FOS**) for the open pit case study. There are three failure surfaces visible; two of them failed (e.g., **FOS** < 1.5). The minimum **FOS** of deterministic analysis is 1.414. The groundwater effect was taken into account during the analysis. For example, the pore fluid unit weight is 9.81 \(\mathrm{kN/m^{3}}\) and the rapid drawdown method is [PERSON], [PERSON], [PERSON] (1990) [65]. Figure 15 depicts the contours of slip surfaces after accounting for the groundwater effect. Three slip surfaces were noticed, with a minimum **FOS** of 1.265. The effect of a pseudostatic seismic load on the minimum safety factor was also investigated in this analysis. The critical seismic coefficient (ky) that results in an unstable slope with **FOS** = 1.5 has been calculated for all failure surfaces. The critical slip surface is depicted in Figure 16 along with the critical seismic coefficient (ky = 0.154). Figure 14: Contours of multi-material slip surfaces (deterministic analysis). Figure 15: The contours of multi-material slip surfaces due to the effect of groundwater. The probabilistic analysis was carried out in order to address the inherent uncertainty associated with rock mass. Figure 17 shows one slip surface with an **FOS** of 1.246 and a failure probability of 16%. Figure 18 depicts the factor of safety (**FOS**) distribution for Monte Carlo simulations. Based on this, the probability of pit slope failure P[**FOS** < 1.5] is 16%. According to Table 3 [56], the calculated Probability of Pit Slope Failure (PoF) is unlikely, but it could happen at any time. Figure 16: The critical seismic coefficient contours for multi-material slip surfaces. Figure 17: Contours of failure surface (probabilistic analysis). The overburden volume is 1.74 \(\times\) 10\({}^{6}\) m\({}^{3}\) (4.7 \(\times\) 10\({}^{6}\) ton), while the ore deposit volume is 1.28 \(\times\) 10\({}^{6}\) m\({}^{3}\) (3.2 \(\times\) 10\({}^{6}\) ton). Assume that the cost of removing one ton of overburden and recovering one ton of ore is $5/ ton and $100/ton, respectively. As a result, the total costs of overburden removal and ore recovery are $23.5 \(\times\) 10\({}^{6}\) and $320 \(\times\) 10\({}^{6}\), respectively. It is important to note that the total operating cost includes both the cost of ore extraction (e.g., $320 \(\times\) 10\({}^{6}\)) and the cost of overburden removal (e.g., $23.5 \(\times\) 10\({}^{6}\)). As a result, the total operating cost is 343.5 \(\times\) 10\({}^{6}\). If the selling price of one ton of ore is $200, then the total selling price will be $6.4 \(\times\) 10\({}^{9}\).