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

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.

Appendix

Monte Carlo method to determine first order and total effect Sobol indices (Saltelli 2002).

• 1. A quasi-random number was generated in form of matrix of size (k, 2d), where k is called base sample and d is the dimension of the input vector. k in this study was taken as 10⁵. Quasi-random numbers were generated in the Matlab 2016. Further, two matrices A and B are defined having half of the sample.

  $$A=\left[ {\begin{array}{*{20}{l}} {X_{1}^{{(1)}}}&{X_{2}^{{(1)}}}& \cdots &{X_{i}^{{(1)}}}& \cdots &{X_{d}^{{(1)}}} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ {X_{1}^{{(k)}}}&{X_{2}^{{(k)}}}& \cdots &{X_{i}^{{(k)}}}& \cdots &{X_{d}^{{(k)}}} \end{array}} \right]$$

  (17)
  $$B=\left[ {\begin{array}{*{20}{l}} {X_{{d+1}}^{{(1)}}}&{X_{{d+2}}^{{(1)}}}& \cdots &{X_{{d+i}}^{{(1)}}}& \cdots &{X_{{2d}}^{{(1)}}} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ {X_{{d+1}}^{{(k)}}}&{X_{{d+2}}^{{(k)}}}& \cdots &{X_{{d+i}}^{{(k)}}}& \cdots &{X_{{2d}}^{{(k)}}} \end{array}} \right].$$

  (18)

• 2. Another matrix C_i is defined which contains all elements of B, except the ith column, which is taken from A.

  $${C_i}=\left[ {\begin{array}{*{20}{c}} {X_{{d+1}}^{{(1)}}}&{X_{{d+2}}^{{(1)}}}& \cdots &{X_{i}^{{(1)}}}& \cdots &{X_{{2d}}^{{(1)}}} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ {X_{{d+1}}^{{(k)}}}&{X_{{d+2}}^{{(k)}}}& \cdots &{X_{i}^{{(k)}}}& \cdots &{X_{{2d}}^{{(k)}}} \end{array}} \right].$$

  (19)

• 3. Compute the output from of A, B and C_i matrices to obtain column matrix of outputs Y_A, Y_B and ${Y_{{C_i}}}$.

• 4. Now, the first-order sensitivity index S_i and total effects ${S_{{T_i}}}$ are calculated via Eqs. (20) and (21):

  $${S_i}=\frac{{\left( {\frac{1}{k}} \right)\mathop \sum \nolimits_{{j=1}}^{k} y_{A}^{{(j)}}y_{{{C_i}}}^{{(j)}} - f_{0}^{2}}}{{\left( {\frac{1}{k}} \right)\mathop \sum \nolimits_{{j=1}}^{k} {{\left( {y_{A}^{{(j)}}} \right)}^2} - f_{0}^{2}}},$$

  (20)
  $${S_{{T_i}}}=1 - \frac{{\left( {\frac{1}{k}} \right)\mathop \sum \nolimits_{{j=1}}^{k} y_{B}^{{(j)}}y_{{{C_i}}}^{{(j)}} - f_{0}^{2}}}{{\left( {\frac{1}{k}} \right)\mathop \sum \nolimits_{{j=1}}^{k} {{\left( {y_{A}^{{(j)}}} \right)}^2} - f_{0}^{2}}},$$

  (21)

  where $y_{A}^{{(j)}}$, $y_{B}^{{(j)}}$ and $y_{{{C_i}}}^{{(j)}}$ are the jth element of column vectors Y_A, Y_B and ${Y_{{C_i}}}$, and

  $$f_{0}^{2}=~{\left( {\frac{1}{k}\mathop \sum \limits_{{j=1}}^{k} y_{A}^{{(i)}}} \right)^2}.$$

  (22)