Rock Mechanics and Rock Engineering

, Volume 51, Issue 7, pp 2153–2174 | Cite as

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

  • Bhardwaj Pandit
  • Gaurav Tiwari
  • Gali Madhavi Latha
  • G. L. Sivakumar Babu
Original Paper


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.


Rock slope Spatial variation Random field Reliability index 

List of symbols


Rock quality designation


Rock mass rating


Young’s modulus


Poisson’s ratio


Tensile strength


Unit weight


Uniaxial compressive strength


Factor of safety




Coefficient of variation


Monte Carlo


Hoek–Brown strength parameter for intact rock


Geological strength index


Hoek–Brown strength parameter for rock mass


Hoek–Brown strength parameter for rock mass


Deformation modulus


Radial basis function


Latin hypercube simulation


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


Number of random input vectors obtained from Latin hypercube simulation


Probability of failure


Observed value of FOS obtained from FLAC analysis


Simulated values of FOS obtained from response surface


Nash–Stucliffe efficiency


Percent bias


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


Probability density function


Reliability index

\(\varPhi^{ - 1}\)

Standard normal inverse


Horizontal and vertical coordinates of 2D slope model


Gaussian random field function


Mean of w(xz)


Variance of w(xz)

\(\Delta x,\Delta z\)

Horizontal and vertical distances of a point from (x0z0)


Autocorrelation function

\(\rho_{\text{w}} \left( {\Delta x,\Delta {\text{z}}} \right)\)

Analytical form of ACF




Scale of fluctuation

δx, δz

Horizontal and vertical scale of fluctuations

\(\tau_{x} , \tau_{z}\)

Lag in horizontal and vertical directions

Dx, Dz

Rectangular zone size in FLAC model


Spatial average function of random field w(xz) over zone of size D x , D z


Variance reduction factor


Expected value


Variance value

\(Y_{1} , Y_{2} , Y_{3} \ldots Y_{p}\)

Discrete random variables (\(p\) in number)


Geometric mean of discrete random variables


General 1D random field


Element length in 1D


Geometric average of X over D


Spatial coordinate in 1D


Local average subdivision

xe, ze

Centroid of the FLAC2D zone


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

Lx, Lz

Length and width of rectangular region in which random field is generated


Real part of complex number

\(m, n\)

Summation indices of Fourier series

amn, bmn

Zero mean independent Gaussian random variables


Variance of a mn , b mn


Lognormal random field


Mean value of lognormal random field


COV of lognormal random field


Mean FOS




Radial basis function


Radius of domain of compact support of RBF


Coefficients for ith 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 \(\varvec{Z}_{\varvec{i}}\)


Dimension of input vector


d + 1


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


Zero matrix


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  • Bhardwaj Pandit
    • 1
  • Gaurav Tiwari
    • 1
  • Gali Madhavi Latha
    • 1
  • G. L. Sivakumar Babu
    • 1
  1. 1.Department of Civil EngineeringIndian Institute of ScienceBangaloreIndia

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