# Estimating the Optimal Overall Slope Angle of Open-Pit Mines with Probabilistic Analysis
## 1 Introduction  
Stability is of the utmost importance for the success of any mining operation (e.g., surface or/and underground). It is crucial to minimize the overall mining costs while maintaining safety for better productivity. Eventually the amount of waste rock (e.g., barren dirt), which has to be removed to extract the ore, should be minimized (e.g., reducing stripping ratio). In open-pit mines, slope stability is a major concern, and its failure has negative consequences on the economy and safety of personnel and equipment. Therefore, it is necessarily required to determine the factor of safety associated with the rock slope and its possible failure pattern. Slope stability is governed by geological conditions and rockmass characterization, which are unique to each site [1]. Estimating rock slope stability is sometimes difficult due to rock mass variability/heterogeneity (e.g., existence of joint sets, faults, anisotropy, etc.) [2; 3; 4; 5]. Thus, the rock slope of open-pit mines, quarries and other kinds of embankments (e.g., mine tailing dams, stockpiles, waste dumps and mine tailing storage facilities, etc.) should be analyzed as \"geotechnical structures\". Therefore, their stability should be evaluated according to the economic (e.g., design/select a reasonable slope angle which affects the stripping ratio and overall cost), environmental (e.g., long-term stability) and safety issues to personnel and machinery [6; 7; 8]. There are several factors controlling the rock slope stability, such as rock mass properties, the geometry of the slope (e.g., height and ultimate slope angle), the presence of groundwater and discontinuities (e.g., joints, faults, fissures). Geological structures determine the slope failure mechanism associated with rock mass and its pattern (e.g., planar, wedge, toppling and/or circular failure) [4; 9; 10; 11]. Sometimes the dimensions of open pits may extend to hundreds of meters (e.g., South Africa Sandsloot, located at an elevation of 1100 m, is the largest open-pit mining platform in the world, about 1500 m long, 800 m wide and projected depth of 325 m). Consequently, millions of dollars may be lost if an inappropriate design of the overall slope angle is achieved. Slope angle affects, more or less, the stripping ratio (e.g., the ratio of tonnage or volume of overburden to be removed to tonnage or volume of ore to be extracted) and consequently the mining profitability [9; 11]. Alternatively, the higher the stripping ratio, the more expensive the mining and the less profit is then gained. The reduction of stripping ratio (e.g., less waste rock to be removed) requires maintaining the overall slope angle as steep as possible (e.g., increase the ore recovery). To do so, successful pit planning is essential and requires accurate information about geology and site characteristics (e.g., slope geometry, rock properties, groundwater conditions and associated discontinuities). The final design of the optimal slope angle is controlled not only by ore grade distribution and operational cost but also by overall rock mass properties. Therefore, it is recommended that the potential for failure be incorporated into the ultimate open pit design. It helps, in advance, to know temporally (i.e., when) and spatially (i.e., where) actions have to be taken [12]. Characteristics of rock mass are crucial design input parameters. These parameters are never known precisely. There are always uncertainties associated with them. Some of these uncertainties exist due to limited data, errors in testing, random data collection and lack of knowledge. Herein, probabilistic methods, as introduced in the next section, are used to tackle the inherent uncertainty associated with rock mass properties. Probabilistic methods are used to deal with inherent heterogeneity/variability associated with rock mass properties. They are applied to estimate the likelihood of failure at various slope angles and take measures to reduce the risk to an acceptable level.