# Stability Analysis of a Large Gold Mine Open-Pit Slope Using Advanced Probabilistic Method

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## Abstract

A large gold reserve was recently discovered at Haveri district of Karnataka state of India where open-pit mining was planned to extract these deposits. Stability analysis for open-pit mine slope at this site is presented in the article. Extensive geological investigations and laboratory testing suggested high variability in geological features of discontinuities, rock mass quality and intact rock properties. Hence, it was decided to perform stability analysis of the rock slope using probabilistic approach along with deterministic approach. Deterministic analysis was carried out with average properties of rock, and reliability analysis of the rock slope was carried out using both traditional and advanced probabilistic methods. In traditional probabilistic method, rock mass strength properties were treated as random variables without considering spatial variation of rock properties and reliability index was evaluated by Monte Carlo (MC) simulation on augmented radial basis function-based response surface. In advanced probabilistic analysis, spatial variability of rock mass strength properties was considered by generating anisotropic random field using Fourier series method with spatial averaging over finite difference zones. Reliability index was then estimated by performing MC simulation using random finite difference method. A comparison was provided between the results of stability analysis of slope from all these approaches. Rock slope was found to be stable in both deterministic and probabilistic approaches; however, the degree of stability predicted was different for both methods. Deterministic approach was found to be inappropriate to analyse the stability of slope having rock mass with variable properties. Further, reliability index and expected performance level of slope were highly underestimated by traditional probabilistic method as compared to advanced probabilistic method.

## Keywords

Rock slope Spatial variation Random field Reliability index## List of symbols

- RQD
Rock quality designation

- RMR
Rock mass rating

*E*_{i}Young’s modulus

- \(\nu\)
Poisson’s ratio

*σ*_{t}Tensile strength

*γ*Unit weight

- UCS
Uniaxial compressive strength

- FOS
Factor of safety

- COV
Covariance

- CV
Coefficient of variation

- MC
Monte Carlo

*m*_{i}Hoek–Brown strength parameter for intact rock

- GSI
Geological strength index

*m*_{b}Hoek–Brown strength parameter for rock mass

*s*_{b}Hoek–Brown strength parameter for rock mass

*E*_{m}Deformation modulus

- RBF
Radial basis function

- LHS
Latin hypercube simulation

- \(\varvec{Z}\)
Input vector for a general response surface

- \(\varvec{Z}_{1} ,\varvec{Z}_{2} , \ldots \varvec{ },\varvec{ Z}_{\varvec{k}}\)
Input vectors obtained from Latin hypercube simulation

*k*Number of random input vectors obtained from Latin hypercube simulation

*P*_{f}Probability of failure

- FOS
^{obs} Observed value of FOS obtained from FLAC analysis

- FOS
^{sim} Simulated values of FOS obtained from response surface

- NSE
Nash–Stucliffe efficiency

- PBIAS
Percent bias

- RSR
Ratio of root-mean-square error to standard deviation of observed data

Probability density function

- R
Reliability index

- \(\varPhi^{ - 1}\)
Standard normal inverse

*x*,*z*Horizontal and vertical coordinates of 2D slope model

*w*(*x*,*z*)Gaussian random field function

*μ*_{w}Mean of

*w*(*x*,*z*)*σ*_{w}^{2}Variance of

*w*(*x*,*z*)- \(\Delta x,\Delta z\)
Horizontal and vertical distances of a point from (

*x*_{0},*z*_{0})- ACF
Autocorrelation function

- \(\rho_{\text{w}} \left( {\Delta x,\Delta {\text{z}}} \right)\)
Analytical form of ACF

- VAR
Variance

- SOF
Scale of fluctuation

*δ*_{x},*δ*_{z}Horizontal and vertical scale of fluctuations

- \(\tau_{x} , \tau_{z}\)
Lag in horizontal and vertical directions

*D*_{x},*D*_{z}Rectangular zone size in FLAC model

*w*_{D}(*x*,*z*)Spatial average function of random field

*w*(*x*,*z*) over zone of size*D*_{ x },*D*_{ z }*γ*(*D*_{x},*D*_{z})Variance reduction factor

*E*[…]Expected value

- Var[…]
Variance value

- \(Y_{1} , Y_{2} , Y_{3} \ldots Y_{p}\)
Discrete random variables (\(p\) in number)

*Y*_{GM}Geometric mean of discrete random variables

*X*General 1D random field

*D*Element length in 1D

*X*_{GM}Geometric average of

*X*over*D**ξ*Spatial coordinate in 1D

- LAS
Local average subdivision

*x*_{e},*z*_{e}Centroid of the FLAC2D zone

*w*_{D}(*x*_{e},*z*_{e})Averaged rock property over the rectangular zone defined by \([ {x_{\text{e}} - \frac{\Delta x}{2},x_{\text{e}} + \frac{\Delta x}{2}} ]\) and \([ {z_{\text{e}} - \frac{\Delta z}{2}, z_{\text{e}} + \frac{\Delta z}{2}} ]\)

*L*_{x},*L*_{z}Length and width of rectangular region in which random field is generated

*Re*(…)Real part of complex number

- \(m, n\)
Summation indices of Fourier series

*a*_{mn},*b*_{mn}Zero mean independent Gaussian random variables

*σ*_{mn}^{2}Variance of

*a*_{ mn },*b*_{ mn }*q*(*x*,*z*)Lognormal random field

*μ*_{q}Mean value of lognormal random field

*v*_{q}COV of lognormal random field

*μ*_{FOS}Mean FOS

*V*_{FOS}COV of FOS

*ψ*(*r*)Radial basis function

*r*_{0}Radius of domain of compact support of RBF

*λ*_{i}Coefficients for

*i*th RBF- \(g\left( \varvec{Z} \right)\)
FEM/FDM model output with vector \(\varvec{Z}\) as input

- \(\parallel \varvec{Z} - \varvec{Z}_{\varvec{i}} \parallel\)
Euclidean norm (distance) of vector \(\varvec{Z}\) from \(\varvec{Z}_{\varvec{i}}\)

*d*Dimension of input vector

*l**d*+ 1- \(\varvec{b}\)
*l*Constants in RBF approximation- \(P\left( \varvec{Z} \right)\)
Linear polynomial augmented to RBF

- \(\varvec{g}_{{\varvec{n} \times 1}}\)
Output vector obtained by solving \(g\left( \varvec{Z} \right)\) at Latin hypercube samples

- \(\varvec{A}_{{\varvec{n} \times \varvec{n}}}\), \(\varvec{B}_{{\varvec{n} \times \varvec{m}}}\)
Matrices involved in construction of RBF response surface

- 0
Zero matrix

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