Quantifying the Impact of Additional Laboratory Tests on the Quality of a Geomechanical Model

Abstract

In an open-pit mine operation, the design of safe and economically viable slopes can be significantly influenced by the quality and quantity of collected geomechanical data. In several mining jurisdictions, codes and standards are available for reporting exploration data, but similar codes or guidelines are not formally available or enforced for geotechnical design. Current recommendations suggest a target level of confidence in the rock mass properties used for slope design. As these guidelines are qualitative and somewhat subjective, questions arise regarding the minimum number of tests to perform in order to reach the proposed level of confidence. This paper investigates the impact of defining a priori the required number of laboratory tests to conduct on rock core samples based on the geomechanical database of an operating open-pit mine in South Africa. In this review, to illustrate the process, the focus is on uniaxial compressive strength properties. Available strength data for 2 project stages were analysed using the small-sampling theory and the confidence interval approach. The results showed that the number of specimens was too low to obtain a reliable strength value for some geotechnical domains even if more specimens than the minimum proposed by the ISRM suggested methods were tested. Furthermore, the testing sequence used has an impact on the minimum number of specimens required. Current best practice cannot capture all possibilities regarding the geomechanical property distributions, and there is a demonstrated need for a method to determine the minimum number of specimens required while minimising the influence of the testing sequence.

Appendices

Appendix 1: Minimum Sample Size for Normally Distributed Data

The equations used to derive the minimum number of specimen for normally distributed data are from Hines et al. (2003). Equation 5 presents the two-sided confidence interval on the population mean:

$$ \bar{X} - t_{{\frac{\alpha }{2},\upsilon }} \frac{s}{\sqrt n } \le \mu \le \bar{X} + t_{\alpha /2,\upsilon } \frac{s}{\sqrt n } $$

(5)

where: μ = the population arithmetic mean, $\overline{X}$ = the arithmetic sample mean, s = the sample standard deviation, $t_{\alpha /2,\nu }$ = the confidence coefficient obtained from the Student t distribution for a two-sided confidence on μ, α = a parameter determined by the required confidence level, ν = the number of degrees of freedom, n = the number of specimens in the sample

In Equation 5, $\overline{X}$ is the arithmetic sample mean and is defined by Equation 6, and s is the sample standard deviation and is defined by Equation 7.

$$ \bar{X} = \mathop \sum \limits_{i = 1}^{n} \frac{{X_{i} }}{n} $$

(6)

$$ s = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {X_{i} - \bar{X}} \right)^{2} }}{n - 1}} $$

(7)

where: $X_{i}$ the values observed on each of the n specimens in the sample

The precision index p, which is the ratio of the upper and lower limits of the population mean interval is given by Equation 8:

$$ p = \frac{{\bar{X} + t_{\alpha /2,\nu } \frac{s}{\sqrt n }}}{{\bar{X} - t_{\alpha /2,\nu } \frac{s}{\sqrt n }}}\quad {\text{where}}\quad p \ge \, 1 $$

(8)

By combining Eqs. 5 and 8, the true mean interval can also be defined using the precision index p (Eq. 9).

$$ \left( {\frac{2}{p + 1}} \right)\bar{X} \le \mu \le \left( {\frac{2p}{p + 1}} \right)\bar{X} $$

(9)

The left-hand terms of Eqs. 5 and 9 allow one to define Eq. 10.

$$ \left( {\frac{2}{p + 1}} \right) = 1 - t_{\alpha /2,\upsilon } \frac{cv}{\sqrt n } $$

(10)

The coefficient of variation cv in Eq. 10 is defined by:

$$ cv = \frac{s}{{\bar{X}}} $$

(11)

The minimum number of specimens in a sample n can then be determined by expressing Eq. 10 in terms of n.

$$ n = \left[ {\left( {\frac{p + 1}{p - 1}} \right)t_{\alpha /2,\upsilon } \cdot cv} \right]^{2} $$

(12)

Appendix 2: Minimum Sample Size for Lognormally Distributed Data

The equations used to derive the minimum number of specimen for normally distributed data are from Olsson (2005). Equation 13 presents the two-sided confidence interval on the natural logarithm of the population mean of the lognormal distribution based on the Cox modified method.

$$ \bar{Y} + \frac{{S_{Y}^{2} }}{2} - t_{{\frac{\alpha }{2},\upsilon }} s\sqrt {\frac{{S_{Y}^{2} }}{n} + \frac{{S_{Y}^{4} }}{{2\left( {n - 1} \right)}}} \le \ln \left( {\mu_{X} } \right) \le \bar{Y} + \frac{{S_{Y}^{2} }}{2} + t_{{\frac{\alpha }{2},\upsilon }} \sqrt {\frac{{S_{Y}^{2} }}{n} + \frac{{S_{Y}^{4} }}{{2\left( {n - 1} \right)}}} $$

(13)

where: $X$ = the lognormally distributed random variable, $Y$ = the normally distributed variable Y = ln(X), $\overline{Y}$ = the arithmetic sample mean of variable Y = ln(X), $S_{Y}^{2}$ = the variance of variable Y = ln(X), and $S_{Y}$ = is the standard deviation of variable Y, $t_{{\frac{\alpha }{2},\upsilon }}$ = the confidence coefficient obtained from the Student t distribution for a two-sided confidence on $\mu_{X}$ = a parameter determined by the required confidence level, ν = the number of degrees of freedom and ν = n − 1, n = the number of specimens in the sample, µ _x  = Mean of the lognormally distributed variable X

The mean of the lognormally distributed variable X is given by Eq. 14.

$$ \mu_{X} = e^{{\mu_{Y} + \frac{{\sigma_{Y}^{2} }}{2}}} $$

(14)

where: µ _Y  = Mean of the normally distributed variable Y = ln(X). $\sigma_{Y}^{2}$ = Variance of the population of the normally distributed variable Y = ln(X) and $\sigma_{Y}$ = is the standard deviation of variable Y

The sample variance of the normally distributed variable Y = ln(X) is given by Eq. 15.

$$ S_{Y}^{2} = \mathop \sum \limits_{i = 1}^{n} \frac{{\left( {Y_{i} - \overline{Y} } \right)^{2} }}{n - 1} $$

(15)

where: Y _i  = the values observed on each of the n specimens in the sample, $n, \overline{Y}$ as described previously.

The precision index p is the ratio of the upper and lower limits of the population mean interval and is determined by Eq. 16.

$$ p = \frac{{\left( {\bar{Y} + \frac{{S_{Y}^{2} }}{2}} \right) + t_{{\frac{\alpha }{2},\upsilon }} \sqrt {\frac{{S_{Y}^{2} }}{n} + \frac{{S_{Y}^{4} }}{{2\left( {n - 1} \right)}}} }}{{\left( {\bar{Y} + \frac{{S_{Y}^{2} }}{2}} \right) - t_{{\frac{\alpha }{2},\upsilon }} \sqrt {\frac{{S_{Y}^{2} }}{n} + \frac{{S_{Y}^{4} }}{{2\left( {n - 1} \right)}}} }}\quad {\text{where}}\quad p \ge 1 $$

(16)

Based on Eq. 16, the equation for the confidence interval on ${ \ln }\left( {\mu_{X} } \right)$ can be rewritten as Eq. 17.

$$ \left( {\frac{2}{p + 1}} \right)\left( {\bar{Y} + \frac{{S_{Y}^{2} }}{2}} \right) \le \ln \left( {\mu_{X} } \right) \le \left( {\frac{2p}{p + 1}} \right)\left( {\bar{Y} + \frac{{S_{Y}^{2} }}{2}} \right) $$

(17)

The left term in Eq. 17 is then equal to the left term in Eq. 13.

$$ \left( {\frac{2}{p + 1}} \right)\left( {\bar{Y} + \frac{{S_{Y}^{2} }}{2}} \right) = \bar Y + ~\frac{{S_Y^2}}{2} - {t_{\frac{\alpha }{2},\upsilon }}\sqrt {\frac{{S_Y^2}}{n} + \frac{{S_Y^4}}{{2\left( {n - 1} \right)}}} $$

(18)

The minimum number of specimens can be determined by isolating n in Eq. 18:

$$ n = \left[ {\frac{{S_{Y}^{2} n}}{{2\left( {n - 1} \right)}} + 1} \right] \cdot \left[ {\left( {\frac{p + 1}{p - 1}} \right) \cdot cv_{Y} \cdot \frac{{t_{{\frac{\alpha }{2},\upsilon }} }}{{1 + \frac{{S_{Y}^{2} }}{{2\bar{Y}}}}}} \right]^{2} $$

(19)

where the coefficient of variation of the normally distributed variable Y is defined by:

$$ cv_{Y} = \frac{{S_{Y} }}{{\bar{Y}}}.$$

(20)