Probabilistic block theory analysis for a rock slope at an open pit mine in USA
Introduction
Open-pit mining is one of the most important technologies for extracting mineral resources from the earth’s crust. Production rates of open pits have progressively grown over the last 100 years and they will continue to grow in the future [1]. The ultimate slopes of an open pit mine are generally excavated to the steepest possible angles because the economic consequences of the excavation angles are significant [2]. For large scale open pits, changes in slope angle by approximately 2–3° may correspond to hundreds of millions of dollars in project value [3]. However, steeper slope angles result in higher probabilities of slope failure. Therefore, it is critical to calculate the maximum safe slope angles (MSSA) and also to estimate probabilities of slope failure with respect to different cut slope dip angles.
The open pit mine slope stability is one of the geotechnical subjects dominated by variability and uncertainty because the slopes are composed of natural heterogeneous materials containing a large number of discontinuities. In discontinuous hard rock masses, the variability and uncertainty of rock slope stability analyses mainly arise from discontinuity geometry and discontinuity strength. Application of probabilistic analysis has provided an objective tool to quantify and model variability and uncertainty [4], [5]. The first two types of probabilistic analyses to rock slope stability have been applied using either kinematic or limit equilibrium analysis [6], [7], [8], [9], [10].
The block theory, introduced by Shi [11], [12] and Goodman and Shi [13], is a very useful technique to investigate possible failure modes of rock blocks and to determine MSSA for rock slopes [14], [15]. The block theory considers block formation by discontinuity planes along with a cut slope, without assuming presence of lateral release planes as in kinematic analysis. Therefore, the results coming from the block theory analysis can be considered to be closer to the reality than that coming from kinematic analysis [14], [15]. However, the block theory is usually used to perform deterministic analysis using single fixed values (typically mean values) to represent orientation of discontinuity sets and strength parameters [14], [15], [16], [17], [18], [19]. The deterministic block theory analysis is usually conducted using the stereographic projection method. In performing analysis for a field rock slope stability problem, it is tedious and very time consuming to use the stereographic projection method to perform a probabilistic block theory analysis because a probabilistic analysis requires changing the discontinuity orientation combinations as well as changing the discontinuity strength. The vector based method is more suitable to perform probabilistic block theory analysis.
A few studies have been conducted to estimate key block formation probabilities and likelihood of key block failure [20], [21], [22]. In the paper by Hatzor [20], first a joint combination probability is calculated using joint set normal vectors and corresponding linear frequencies. Then a block failure likelihood probability is calculated using the product of the following three independent parameters: (a) the aforementioned joint combination probability, (b) a block instability parameter which is a function of net sliding force and (c) a shape parameter. The methodology has been applied to joint orientation data sampled from side walls of two pilot tunnels. Joint friction angle has been treated as a deterministic parameter in the calculations. Successes and failures of the approach obtained from the results are discussed in the paper. Chen et al. [21] have applied first-order second-moment technique along with the reliability index to calculate removability of a rock block and its failure probability. Beta and Fisher distributions have been used to represent the variability of the dip angle and dip direction of joints. The dip angle and dip direction have been treated as uncorrelated parameters (even though theoretically not justifiable) to simplify the analysis in using the beta distribution to represent each of the two parameters. Joint friction angle is considered as a deterministic parameter in the study. Monte-Carlo simulations also have been conducted to compare the results with the aforementioned approach. A hanging wall case study had shown that with the same mean values of the random variables the block removable probabilities differ significantly if the dispersions of the random variables are different. Chen [22] in another study has performed a probabilistic key block analysis to evaluate stability of a mine ventilation shaft constructed in a granitic rock mass. The variability of the dip angle, dip direction and friction angle has been modeled using a beta distribution. Key block failure probabilities, the probabilistic distribution of factor of safety and the probabilistic distribution of potential maximum key block volumes have been calculated in the study. The results have indicated that although the safety factor calculated based on the mean values of the variables was above 1.0 for the stability of the most critical key block, the block had a considerable probability of failure with a significant rock volume due to variations in discontinuity orientations and rock properties.
In this paper an entirely different type of probabilistic block theory analysis, compared to the ones discussed above, is presented for a part of the rock mass (see Fig. 1) that exists in an open pit mine in USA. In the analysis, the joint orientation has been considered as a bivariate random variable including the correlation that exists between the joint dip angle and dip direction angle. Also the variability of the discontinuity shear strength is incorporated in the analysis. To accomplish the task, first the data available on geology and discontinuity orientations measured through manual mapping were obtained from the mining company. Remote fracture mapping were conducted in the field by the research group from the University of Arizona to obtain the discontinuity geometry that exists in these formations. Also laboratory direct shear tests were conducted on the discontinuities at the University of Arizona geo-mechanics laboratory to estimate the friction angle of the discontinuities and its possible variability.
Section snippets
Brief report on geology and site investigations
Based on the geological information received from the mining company, the considered section of the slope can be divided into two major lithological formations: (a) Devonian Rodeo Creek (DRC) unit and (b) Devonian Popovich (DP) unit. The DRC formation rocks consist of argillite, sandstone and interbedded siltstone, and argillite-mudstone. The DP formation rocks consist of laminated limy to dolomitic mudstone and micritic limestone. Block samples and rock cores for both DRC and DP rocks were
Orientations of discontinuity sets
For the conducted study, eight hundred and four discontinuities were mapped in the DRC formation and four hundred and eighty-one discontinuities were mapped in the DP formation. In each domain, the discontinuities can be divided into three sets as shown in Fig. 3. The VX spatial station shown in Fig. 2 was used to perform remote fracture mapping. This instrument has the capability to function as a total station, a camera and a laser scanner. Fig. 3 shows the typical discontinuity system of both
Discontinuity strength properties
Based on the direct shear tests conducted in the laboratory, the discontinuity friction angle (ϕ) for all the three discontinuity sets belonging to DRC rocks varies between 19° and 34° from 16 test results. For DP rocks ϕ varies between 20° and 32° from 9 test results. Because the discontinuities are open with no filling material the cohesion value is zero for the discontinuities. The number of ϕ data available for each domain is not sufficient to establish realistic probability distributions
Development of the probabilistic formulation
As stated in Section 3, each discontinuity combination is formed by taking one mean orientation value applicable for a particular cell from each set. As stated in Section 2, eight values are selected for CSDD to perform probabilistic block theory analysis. Mid values of ϕ intervals shown in Fig. 5 (6 values for DRC and 5 values for DP), are selected for ϕ to perform probabilistic block theory analysis. This means for the DRC rocks 48 different combinations exist based on the different CSDD and ϕ
Summary and conclusions
A new formulation is given to conduct a probabilistic block theory analysis. A new computer code (PBTAC) was developed to perform both deterministic and probabilistic block theory analysis. The variability of discontinuity orientation and shear strength is incorporated in the probabilistic block theory analysis. Discontinuity orientation is treated as a bivariate random variable including the correlation that exists between the dip angle and dip direction. PBTAC code was applied to perform both
Acknowledgments
The support provided by the mining company through providing geological data, rock core and block samples and allowing access to the mine to perform field investigations is very much appreciated. The work was partially funded by the Centers for Disease Control and Prevention of USA (Contract No. 200-2011-39886), National Basic Research Program of China (973 Program No. 2010CB732005) and the National Natural Science Foundation Project of China (No. 51079093). The first author is grateful to the
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