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
  • 158 Downloads

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

Ei

Young’s modulus

\(\nu\)

Poisson’s ratio

σt

Tensile strength

γ

Unit weight

UCS

Uniaxial compressive strength

FOS

Factor of safety

COV

Covariance

CV

Coefficient of variation

MC

Monte Carlo

mi

Hoek–Brown strength parameter for intact rock

GSI

Geological strength index

mb

Hoek–Brown strength parameter for rock mass

sb

Hoek–Brown strength parameter for rock mass

Em

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

Pf

Probability of failure

FOSobs

Observed value of FOS obtained from FLAC analysis

FOSsim

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

PDF

Probability density function

R

Reliability index

\(\varPhi^{ - 1}\)

Standard normal inverse

xz

Horizontal and vertical coordinates of 2D slope model

w(xz)

Gaussian random field function

μw

Mean of w(xz)

σw2

Variance of w(xz)

\(\Delta x,\Delta z\)

Horizontal and vertical distances of a point from (x0z0)

ACF

Autocorrelation function

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

Analytical form of ACF

VAR

Variance

SOF

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

wD(xz)

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

γ(DxDz)

Variance reduction factor

E[…]

Expected value

Var[…]

Variance value

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

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

YGM

Geometric mean of discrete random variables

X

General 1D random field

D

Element length in 1D

XGM

Geometric average of X over D

ξ

Spatial coordinate in 1D

LAS

Local average subdivision

xe, ze

Centroid of the FLAC2D zone

wD(xeze)

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

Re(…)

Real part of complex number

\(m, n\)

Summation indices of Fourier series

amn, bmn

Zero mean independent Gaussian random variables

σmn2

Variance of a mn , b mn

q(xz)

Lognormal random field

μq

Mean value of lognormal random field

vq

COV of lognormal random field

μFOS

Mean FOS

VFOS

COV of FOS

ψ(r)

Radial basis function

r0

Radius of domain of compact support of RBF

λi

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 \(\varvec{Z}_{\varvec{i}}\)

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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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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