CORRELATION DEPENDENCES OF CHANGES IN THE PHYSICO-MECHANICAL PROPERTIES OF JURASSIC ROCKS OF THE KANSK-ACHINSK BASIN

N. S. Rogova, G. L. Kalinichenko

(Presented by the scientific seminar of the Department of Hydrogeology and Engineering Geology)

The study of regularities in changes in the indices of the physico-mechanical properties of rocks is of substantial importance in assessing engineering-geological conditions and makes it possible to approach the assessment of the stability of quarry slopes on a more substantiated basis.

A detailed study of the physico-mechanical properties of Jurassic semirocky rocks from several deposits of the Kansk-Achinsk Basin made it possible to identify a number of regularities in their variation.

To assess the nature and degree of variability of physico-mechanical properties with depth, as well as the dependence of some indices on others, the method of correlation analysis was used.

The main task of correlation analysis is to establish the form and closeness of the correlation relationship between variables. As indices of the closeness of the relationship, the correlation ratio and the correlation coefficient are used.

The correlation ratio is a more general measure of relationship and is used as a preliminary estimate of the closeness of the relationship between variables. It is defined as

\[ \eta_{x/y}=\frac{\sigma_{xy}}{\sigma_x}; \qquad \eta_{y/x}=\frac{\sigma_{yx}}{\sigma_y}, \]

where

\[ \eta \]
is the correlation ratio;

\[ \sigma_{yx},\ \sigma_{xy} \]
are the root-mean-square deviations of the mean values of \(y\), calculated for individual intervals of variation of the quantity \(x\), and of the mean values of \(x\), calculated for individual intervals of variation of the quantity \(y\);

\[ \sigma_y,\ \sigma_x \]
are the root-mean-square deviations of the individual values of the quantities \(y\) and \(x\).

It is known that, for large \(n\), the sample value of the correlation ratio has a distribution law close to normal, with mathematical expectation \(\eta\) and standard deviation:

\[ \sigma_{\eta}=\frac{1-\eta^2}{\sqrt{N-1}}, \]

where \(N\) is the number of observation points.

To assess the closeness of a relationship, the value of the ratio \(\frac{\eta}{\sigma_{\eta}}\) is used.

The relationship between the indices is entirely reliable if

\[ \frac{\eta}{\sigma_{\eta}} \geq 3. \]

For a more precise assessment of the closeness of the relationship between indices, when the number of determinations is \(>20\), the correlation ratio is used, reduced by a value equal to three times the error.

If

\[ \eta - 3\sigma_{\eta} < 0 \]

— there is no relationship;

\[ 0 < \eta - 3\sigma_{\eta} < 0.5 \]

— very weak;

\[ 0.5 < \eta - 3\sigma_{\eta} < 0.7 \]

— weak;

\[ 0.7 < \eta - 3\sigma_{\eta} < 0.9 \]

— close;

\[ \eta - 3\sigma_{\eta} < 0.9 \]

— very close.

The correlation coefficient is determined by the formula

\[ r = \frac{\sum (x-\bar{x})(y-\bar{y})}{(N-1)\cdot \sigma_x \cdot \sigma_y} = \frac{\sum (x-\bar{x})(y-\bar{y})} {\sqrt{\sum (x-\bar{x})^2}\cdot \sqrt{\sum (y-\bar{y})^2}}, \]

where \(\bar{x}, \bar{y}\) are the mean values of the variables, determined by the formulas

\[ \bar{x}=\frac{\sum x}{N}; \qquad \bar{y}=\frac{\sum y}{N}, \]

\(N\) is the number of determinations;

\(\sigma_x, \sigma_y\) are the mean-square deviations of the individual values of the quantities \(x\) and \(y\), found from the formulas:

\[ \sigma_y=\sqrt{\frac{\sum (y-\bar{y})^2}{N-1}}; \qquad \sigma_x=\sqrt{\frac{\sum (x-\bar{x})^2}{N-1}}. \]

The mean-square deviation of the correlation coefficient is determined as

\[ \sigma_r=\frac{1-r^2}{\sqrt{N-1}}. \]

If one approaches the matter strictly, then in practice, with a number of determinations equal to or greater than 30, the true value of the correlation coefficient, with reliability 0.99, must lie in the interval \((r-3\sigma_r,\ r+3\sigma_r)\). In our case a reliability of 0.90 is quite sufficient; then the true value of the correlation coefficient will lie within the limits \(r-1.6\sigma_r\) and \(r+1.6\sigma_r\) (Smirnov, Dunin-Barkovskii, 1959). Special tables and graphs may also be used to obtain confidence intervals of the correlation coefficient.

Thus, when processing factual material in the case of a linear relationship between variables, the true correlation coefficient must lie in the confidence interval \(r \pm 1.6\sigma_r\).

If the value of the lower limit is very small \((r-1.6\sigma_r \leq 0)\), this means that there is no linear correlation relationship between the variables; however, this does not exclude the possibility of a curvilinear relationship. To verify this proposition, one may use a comparison of the values of the correlation ratio and the limiting values of the correlation coefficient. If the values \(\eta_{x/y}\) and \(\eta_{y/x}\) lie within the limiting values of the correlation coefficient, the relationship between the variables is considered linear. If the lower limit of the correlation coefficient is very small and the value of the correlation ratio exceeds it, then the relationship is nonlinear.

A more stringent criterion for testing the hypothesis of a linear form of dependence is based on the fact that the quantity

\[ W^{2}=\frac{(n-k)\left(\hat{\eta}^{2}_{y/x}-r^{2}\right)}{(k-2)\left(1-\hat{\eta}^{2}_{y/x}\right)} \]

is distributed according to the law \(F(m,l)\) with the number of degrees of freedom: numerator \(m=K-2\) and denominator \(l=n-k\), where \(n\) is the sample size and \(k\) is the number of intervals into which the possible values of \(x\) are divided.

The determination of the form of dependence between the quantities under study is carried out by constructing an empirical regression line from the mean values \(\bar{y}\), calculated for each value of \(x\). The form of the relationship may be linear or curvilinear.

To determine the closeness of the dependence and the form of the relationship between indicators of the physico-mechanical properties of rocks and depth, on the one hand, and the interdependence among some of them, on the other, we first systematized all the factual material.

The results of each observation were marked in a coordinate system by a point. Having plotted all observation points, we obtained a correlation field. In our case, 33 correlation fields were constructed. The necessity of such a construction lies in the fact that, with their aid, correlation tables are compiled, which are needed for calculating the correlation coefficient, the correlation ratio, and for constructing theoretical regression curves of \(X\) on \(Y\) and \(Y\) on \(X\). We then define a system of grouping intervals according to the attributes \(X\) and \(Y\), and, in accordance with it, superimpose a coordinate grid on the correlation field. In each cell of the coordinate grid we count the number of points and enter it into the correlation table (see table). We calculate the mean values for all rows of the distribution \(Y\) corresponding to the given values of \(X\). The calculations are made by the formula:

\[ \overline{Y_i}=\frac{1}{n}\sum_{j=1}^{m_i}Y_{ij}. \]

We plot the results on the correlation field. Connecting them successively by straight lines, we obtain the empirical regression line; the parameters of the theoretical regression line are found by the method of least squares.

All calculations for determining the correlation coefficient and the correlation ratio were performed according to the methodology given above, and were also programmed and carried out on a “Minsk-1” electronic computer.

Processing the available factual material using correlation analysis made it possible to establish a qualitative and quantitative dependence between the following indicators.

1. Between rock density and the depth of their occurrence

At the Berezovsk deposit, the relationship between rock density and the depth of their occurrence is nonlinear. This is confirmed by the very low values of the correlation coefficients and their lower limits. Thus, the correlation coefficient for siltstones is 0.14. The confidence limits of the correlation coefficient are \(0.04<r<0.28\). The values of the correlation ratios are \(\eta_{y/x}=0.26\), \(\eta_{x/y}=0.25\).

Correlation table for argillites
Attribute: skeletal bulk density — moisture content

Table 1

For the rocks of the Nazarovo and Aban deposits this relationship is linear. For the sandstones of the Aban deposit, the correlation coefficient is \(r=0.50\). The true value of the correlation coefficient lies within \(0.34<r<0.66\). The correlation ratios are \(\eta_{x/y}=0.67\) and \(\eta_{y/x}=0.58\), that is, both values of the correlation ratios lie within the confidence limits of the correlation coefficient. Thus, the values \(r\), \(\eta_{x/y}\), \(\eta_{y/x}\) indicate a weak linear dependence between the bulk density of the skeleton and the depth of occurrence of the sandstones of the Aban deposit. For the siltstones of the Aban deposit, this relationship is characterized by the value \(r=0.40\), which lies within \(0.27<r<0.53\). The correlation ratios are \(\eta_{x/y}=0.47\), \(\eta_{y/x}=0.48\) and lie within the confidence limits of the correlation coefficient.

For the siltstones of the Nazarovo deposit, the relationship of density with depth is characterized by \(r=0.58\), \(\eta_{x/y}=0.83\), \(\eta_{y/x}=0.81\). The confidence limits of the correlation coefficient are \(0.34<r<0.82\). Thus, the values \(r\), \(\eta_{x/y}\), \(\eta_{y/x}\) and their ratio indicate a weak linear relationship between density and depth of occurrence of the siltstones of this deposit.

2. Between the density of rocks and their moisture content

At the Berezovo deposit, the dependence between sandstone density and moisture content is expressed by the following formula:

\[ \delta=1.95-0.014W \quad \text{at } r=0.49. \]

For the sandstones of the Aban deposit this dependence has the form

\[ \delta=1.71-0.0052W \quad \text{at } r=0.12, \]

and for the Nazarovo deposit—

\[ \delta=1.93-0.01W \quad \text{at } r=0.48, \]

where \(\delta\) is the bulk density of the skeleton, \(W\) is moisture content.

However, the closeness of the relationship between moisture content and density is not the same for different types of rocks. In sandstones this dependence is of a very weak linear character.

A closer dependence between moisture content and density has been established for clayey rocks. The confidence limits of the correlation coefficient in this case are \(0.25<r<0.97\). The values of the correlation ratios are \(\eta_{x/y}=0.81\), \(\eta_{y/x}=0.74\), that is, both lie within the confidence limits of the correlation coefficient. Thus, \(r\), \(\eta_{x/y}\), and \(\eta_{y/x}\), and their ratio, indicate a close linear relationship between moisture content and density of clayey rocks. The equation of the linear dependence between density and moisture content of clayey rocks has the following form:

Berezovo deposit: siltstones—

\[ \delta=1.99-0.013W \quad \text{at } r=0.64; \]

argillites—

\[ \delta=2.19-0.25W \quad \text{at } r=0.77. \]

For the siltstones of the Aban deposit,

\[ \delta=2.03-0.02W \quad \text{at } r=0.71; \]

for the siltstones of the Nazarovo deposit—

\[ \delta=2.06-0.02W \quad \text{at } r=0.80. \]

3. Between rock cohesion and their moisture content

For the sandstones of the Berezovo deposit this dependence has the form \(C=9.52-0.31W\); for siltstones—\(C=8.40-0.20W\); for argillites—\(C=11.66-0.34W\).

Under the conditions of the Aban deposit, these equations have the following mathematical expression: for sandstones—$C = 11.33 - 0.39W$; for siltstones—$C = 15.69 - 0.31W$, where $C$ is cohesion, t/m$^2$; $W$ is moisture content in %.

4. Between the cohesion and density of clayey rocks

It has been established that, with increasing density, cohesion increases. The mathematical expression of the dependence under the conditions of the Berezovsk deposit has the following form. For siltstones $C = 0.9\delta - 1.0$ at $r = 0.30$; for argillites $C = 12.4\delta - 16.1$ at $r = 0.37$.

5. Between the temporary compressive strength and moisture content

In determining the temporary compressive strength of rocks of the Berezovsk deposit, it was established that rocks with high values of $\sigma_{\mathrm{cж}}$ are characterized, as a rule, by low moisture-content values. The dependence between these variables is curvilinear in form.

For sandstones it is expressed as

\[ \sigma_{\mathrm{cж}} = \frac{3668}{W} - 168, \]

for siltstones

\[ \sigma_{\mathrm{cж}} = \frac{6226}{W} - 204.4; \]

for argillites

\[ \sigma_{\mathrm{cж}} = \frac{5500}{W} + 50, \]

where $\sigma_{\mathrm{cж}}$ is the temporary compressive strength of rocks, t/m$^2$; $W$ is moisture content in %.

Thus, using correlation analysis, we obtained a qualitative and quantitative understanding of the nature and closeness of the relationships between individual indicators.

REFERENCES

N. V. Smirnov, I. V. Dunin-Barkovskii. A Short Course of Mathematical Statistics for Technical Applications. Fizmatgiz, 1959.