# Probabilistic Characterization of Rock Mass from Limited Laboratory Tests and Field Data: Associated Reliability Analysis and Its Interpretation

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## Abstract

Probabilistic methods are the most efficient methods to account for different types of uncertainties encountered in the estimated rock properties required for the stability analysis of rock slopes and tunnels. These methods require estimation of various parameters of probability distributions like mean, standard deviation (SD) and distributions types of rock properties, which requires large amount of data from laboratory and field investigations. However, in rock mechanics, the data available on rock properties for a project are often limited since the extents of projects are usually large and the test data are minimal due to cost constraints. Due to the unavailability of adequate test data, parameters (mean and SD) of probability distributions of rock properties themselves contain uncertainties. Since traditional reliability analysis uses these uncertain parameters (mean and SD) of probability distributions of rock properties, they may give incorrect estimation of the reliability of rock slope stability. This paper presents a method to overcome this limitation of traditional reliability analysis and outlines a new approach of rock mass characterization for the cases with limited data. This approach uses Sobol’s global sensitivity analysis and bootstrap method coupled with augmented radial basis function based response surface. This method is capable of handling the uncertainties in the parameters (mean and SD) of probability distributions of rock properties and can include their effect in the stability estimates of rock slopes. The proposed method is more practical and efficient, since it considers uncertainty in the statistical parameters of most commonly and easily available rock properties, i.e. uniaxial compressive strength and Geological Strength Index. Further, computational effort involved in the reliability analysis of rock slopes of large dimensions is comparatively smaller in this method. Present study also demonstrates this method through reliability analysis of a large rock slope of an open pit gold mine in Karnataka region of India. Results are compared with the results from traditional reliability analysis to highlight the advantages of the proposed method. It is observed that uncertainties in probability distribution type and its parameters (mean and SD) of rock properties have considerable effect on the estimated reliability index of the rock slope and hence traditional reliability methods based on the parameters of probability distributions estimated using limited data can make incorrect estimation of rock slope stability. Further, stability of the rock slope determined from proposed approach based on bootstrap method is represented by confidence interval of reliability index instead of a fixed value of reliability index as in traditional methods, providing more realistic estimates of rock slope stability.

## Keywords

Rock slopes Rock mass characterization Probabilistic approach Response surface Bootstrap sampling## List of Symbols

- SD
Standard deviation

- UCS
Uniaxial compressive strength

- GSI
Geological Strength Index

- RBF
Radial basis function

- FOS
Factor of safety

- CCDF
Complimentary cumulative distribution function

- \({X_1},~{X_2},~{X_3}, \ldots ,{X_N}\)
Original data set with

*N*observations- \({\bar {X}_N}\)
Mean of original data set

- \({S_N}\)
SD of original data set

- \({{\varvec{B}}_{\varvec{j}}}\)
*j*th bootstrap sample set of input parameter*X*- \({N_{\text{s}}}\)
Total number of bootstraps

- \(k\)
No. of quasi-random samples for estimation of Sobol indices

- AIC
Akaike Information Criterion

- AIC
_{D} Akaike Information Criterion value associated with distribution

*D*- \({L_D}\)
Maximum likelihood estimator of the data set associated with the distribution

*D*- \({K_D}\)
Number of parameters required to fully characterize the distribution

*D*- \({\mu _{{\text{AI}}{{\text{C}}_D}}}\)
Mean AIC value of distribution

*D*- \({\sigma _{{\text{AI}}{{\text{C}}_D}}}\)
SD of AIC value of distribution

*D*- \({\bar {B}_i}\)
Mean of

*i*th bootstrap sample- \({S_i}\)
SD of

*i*th bootstrap sample- \({\left[ {{{\bar {X}}_{{N_{\text{s}}}}}} \right]_{{\text{mean}}}}\)
Mean of

*N*_{s}bootstrap sample means- \({\sigma _{{{\bar {X}}_{{N_{\text{s}}}}}}}\)
SD of

*N*_{s}bootstrap sample means- \({\left[ {{S_{{N_{\text{s}}}}}} \right]_{{\text{mean}}}}\)
Mean of

*N*_{s}bootstrap sample SDs- \({\sigma _{{S_{{N_{\text{s}}}}}}}\)
SD of

*N*_{s}bootstrap sample SDs- MC
Monte Carlo

Probability density function

- FEM
Finite element method

- FDM
Finite difference method

*JCond*_{89}Joint condition factor of RMR 89

- LH
Latin hypercube

- \(g({\varvec{X}})\)
Performance function with

vector as input**X**- \(\phi (r)\)
Radial basis function

- \({P_j}({\varvec{X}})\)
Monomial terms of augmented polynomial

*P*(*x*)- \({\lambda _i}\)
Unknown constants associated with

*i*th RBF- \({b_j}\)
Unknown coefficients

- \(r\)
Euclidean norm (distance) of vector

from**X****X**_{i}- \({r_0}\)
Radius of compact support of RBF

- \(t\)
*r*/*r*_{0}- LOOCV
Leave-one-out cross-validation error

- \({y_i}\)
Output obtained from \(g({{\varvec{X}}_{\varvec{i}}})\)

*n*Number of LH samples drawn from input space

- HDMR
High dimensional model representation

- \({S_i}\)
First-order Sobol index for

*i*th input parameter- \({S_{{T_i}}}\)
Total effects Sobol index for

*i*th input parameter- \({{\varvec{X}}_{\sim i}}\)
Input vector having all components except the

*i*th component- RQD
Rock quality designation

- RMR
Rock mass rating

- \({E_i}\)
Young’s modulus

- \(\nu\)
Poisson’s ratio

- \({\sigma _t}\)
Tensile strength

- \(\gamma\)
Unit weight of rock mass

- SSR
Shear strength reduction

- \({m_i}\)
Hoek–Brown constant for intact rock

- NSE
Nash–Sutcliffe efficiency

- PBIAS
Percent bias

- RSR
Ratio of root-mean-square error to SD of observed data

*R*Reliability index

- \({\mu _{{\text{FOS}}}}\)
Mean FOS for obtained from single bootstrap sample as input

- \({V_{{\text{FOS}}}}\)
Coefficient of variation of FOS for obtained from single bootstrap sample as input

## Notes

## References

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