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

- 424 Downloads
- 1 Citations

## 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

## References

- Al-Bittar T, Soubra AH (2016) Bearing capacity of spatially random rock masses obeying Hoek–Brown failure criterion. Georisk Assess Manag Risk Eng Syst Geohazards 11(2):215–229CrossRefGoogle Scholar
- Ang AHS, Tang WH (1975) Probability concepts in engineering planning and design—basic principles. Wiley, New YorkGoogle Scholar
- Bhasin R, Kaynia AM (2004) Static and dynamic simulation of a 700 m high rock slope in western Norway. Eng Geol 71(3–4):213–226CrossRefGoogle Scholar
- Ching J, Hu YG, Yang ZY, Shiau JQ, Chen JC, Li YS (2011) Reliability-based design for allowable bearing capacity of footings on rock masses by considering angle of distortion. Int J Rock Mech Min Sci 48(5):728–740CrossRefGoogle Scholar
- Duzgun HSB, Bhasin RK (2009) Probabilistic stability evaluation of Oppstadhornet rock slope Norway. Rock Mech Rock Eng 42(5):729–749CrossRefGoogle Scholar
- Duzgun HSB, Yucemen MS, Karpuz C (2002) A probabilistic model for the assessment of uncertainties in the shear strength of rock discontinuities. Int J Rock Mech Min Sci 39(6):743–754CrossRefGoogle Scholar
- Fenton GA, Griffiths DV (2008) Risk assessment in geotechnical engineering. Wiley, New YorkCrossRefGoogle Scholar
- Fenton GA, Vanmarcke EH (1990) Simulation of random fields via local average subdivision. J Eng Mech 116(8):1733–1749CrossRefGoogle Scholar
- Fenton GA, Griffiths DV, Williams MB (2005) Reliability of traditional retaining wall design. Geotechnique 55(1):55–62CrossRefGoogle Scholar
- Griffiths DV, Fenton GA (2000) Influence of soil strength spatial variability on the stability of an undrained clay slope by finite elements. Proceedings of ASCE Geo-Denver 2000:184–193Google Scholar
- Griffiths DV, Fenton GA (2004) Probabilistic slope stability analysis by finite elements. J Geotech Geoenviron Eng 130(5):507–518CrossRefGoogle Scholar
- Griffiths DV, Fenton GA, Denavit MD (2007) Traditional and advanced probabilistic slope stability analysis. In: Proceedings of ASCE Geo-Denver 2000Google Scholar
- Griffiths DV, Huang J, Fenton GA (2009) Influence of spatial variability on slope reliability using 2-D random fields. J Geotech Geoenviron Eng 135(10):1367–1378CrossRefGoogle Scholar
- Hoek E, Diederichs MS (2006) Empirical estimation of rock mass modulus. Int J Rock Mech Min Sci 43(2):203–215CrossRefGoogle Scholar
- Hoek E, Carranza-Torres C, Corkum B (2002) Hoek–Brown failure criterion—2002 edition. In: Proceedings of the 5th North American rock mechanics symposium, Toronto, Canada, pp 267–273Google Scholar
- Hsu SC, Nelson PP (2006) Material spatial variability and slope stability of weak rock masses. J Geotech Geoenviron Eng 132(2):183–193CrossRefGoogle Scholar
- Huang J, Griffiths DV (2015) Determining an appropriate finite element size for modelling the strength of undrained random soils. Comput Geotech 69:506–513CrossRefGoogle Scholar
- ISRM (1981) Rock characterization, testing and monitoring. ISRM suggested methods. Pergamon Press, New YorkGoogle Scholar
- Jha SK, Ching J (2013) Simulating spatial averages of stationary random field using Fourier series method. J Eng Mech 139(5):594–605CrossRefGoogle Scholar
- Krishnamurthy T (2003) Response surface approximation with augmented and compactly supported radial basis functions. In: Proceedings of 44th aiaa/asme/asce/ahs/asc structures, structural dynamics, and materials conference, VirginiaGoogle Scholar
- Li DQ, Chen YF, Lu WB, Zhou CB (2011) Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables. Comput Geotech 38(1):58–68CrossRefGoogle Scholar
- Montgomery DC (2001) Design and analysis of experiments. Wiley, New YorkGoogle Scholar
- Moriasi DN, Arnold JG, Van LMW, Bingner RL, Harmel RD, Veith TL (2007) Model evaluation guidelines for systematic quantification of accuracy in watershed simulations. Trans ASABE 50(3):885–900CrossRefGoogle Scholar
- Pal S, Kaynia AM, Bhasin RK, Paul DK (2012) Earthquake stability analysis of rock slopes: a case study. Rock Mech Rock Eng 45(2):205–215CrossRefGoogle Scholar
- Park HJ, West TR, Woo I (2005) Probabilistic analysis of rock slope stability and random properties of discontinuity parameters, Interstate Highway 40. Eng Geol 79(3–4):230–250CrossRefGoogle Scholar
- Phoon KK, Kulhawy FH (1999) Characterisation of geotechnical variability. Can Geotech J 36(4):612–624CrossRefGoogle Scholar
- Sow D, Carvajal C, Breul P, Peyras L, Rivard P, Bacconnet C, Ballivy G (2017) Modeling the spatial variability of the shear strength of discontinuities of rock masses: application to a dam rock mass. Eng Geol 220:133–143CrossRefGoogle Scholar
- Srivastava A (2012) Spatial variability modelling of geotechnical parameters and stability of highly weathered rock slope. Ind Geotech J 42(3):179–185CrossRefGoogle Scholar
- SRK (2012) Updated mineral resource estimate and preliminary assessment of the economic potential of the Ganajur Main Gold Project, Karnataka, India. Report Prepared for Deccan Gold Mines Ltd.Google Scholar
- Suchomel R, Mašín D (2010) Comparison of different probabilistic methods for predicting stability of a slope in spatially variable c–φ soil. Comput Geotech 37(1–2):132–140CrossRefGoogle Scholar
- Tiwari G, Latha GM (2016) Design of rock slope reinforcement: an Himalayan case study. Rock Mech Rock Eng 49(6):2075–2097CrossRefGoogle Scholar
- U.S. Army Corps of Engineers (1999) Risk-based analysis in geotechnical engineering for support of planning studies, engineering and design. Department of Army, WashingtonGoogle Scholar
- Vanmarcke EH (1983) Random fields: analysis and synthesis. MIT Press, CambridgeGoogle Scholar
- Wu Z (1995) Compactly supported positive definite radial function. Adv Comput Math 4:283–292CrossRefGoogle Scholar