On the Spatial Variability of Rock Properties

V. N. Pulyaev, E. A. Pisarev

(Presented by the Council of the Research Institute of Hydrogeology and Engineering Geology)

The study of spatial patterns in the variability of the physicomechanical properties of rocks serves as the basis for the most appropriate engineering-geological sampling of rocks and therefore assumes special importance in the analysis of factual material from engineering-geological investigations. At present, theoretical results and methodological recommendations have been set forth in the works of N. V. Kolomenskii, in which the following types of spatial variability of rock properties are distinguished: nearly functional, stepwise-regular, and stepwise-irregular. However, this author did not provide practical procedures for correctly distinguishing the noted types of variability, while a visual study of spatial variables does not provide a reliable basis for the corresponding conclusions. The present article, to some extent, eliminates this gap.

As is known, when solving a number of engineering-geological problems—sampling of rocks, selection of design indices, etc.—from a practical point of view it is of great importance to distinguish the stepwise-regular or stepwise-irregular types of variability, whereas the nearly functional type, as N. V. Kolomenskii notes, usually does not occur in nature. Thus, the problem consists in establishing the presence or absence, within the limits of some engineering-geological element, of orderliness in the space of particular values of indices of rock properties. In this connection, each time it becomes necessary to test the null hypothesis, which consists in the assertion that an increase or decrease in the values of a characteristic is random in nature. This hypothesis is easily tested by applying the so-called “sign test,” which is based on the binomial distribution law and belongs to the nonparametric methods of mathematical statistics [1, 2].

Suppose that we have \(N\) values of some index of a rock of the same type, \(x_1, x_2, \ldots, x_k\), arranged in order of increase of some spatial coordinate. Next we find the median \(Me\) and form the differences \(x_1 - Me\), \(x_2 - Me\), ...... Suppose that for these differences we have obtained the following sequence of signs:

\[ - \; - \; + \; - \; + \; + \; + \; - \; - \]

Any sequence consisting of identical signs will be called a run. The probability of obtaining \(m\) runs in a se-

of the sequence is expressed by the function \(h(m)\), which, for even

\(m = 2j\), i.e. \(j = \dfrac{m}{2}\), is equal to

\[ h(m)=\frac{C_{N_1-1}^{j-1}\cdot C_{N_2-1}^{j-1}}{C_{N_1+N_2}^{N_1}} . \tag{3} \]

For odd \(m = 2j+1\), where \(j=\dfrac{m-1}{2}\),

\[ h(m)=\frac{1}{C_{N_1+N_2}^{N_1}}\cdot \left(C_{N_1-2}^{j}\cdot C_{N_2-1}^{j-1} + C_{N_1-1}^{j-1}\cdot C_{N_2-1}^{j}\right). \tag{4} \]

In formulas (3) and (4), \(N_1\) is the number of “plus” signs, \(N_2\) is the number of “minus” signs; for example,

\[ C_{N_1-1}^{j}=\frac{(N_1-1)!}{j!(N_1-1-j)!} \]

and so on.

For \(N_1=N_2=N^*\) and \(m=2j\),

\[ h(m)=2\cdot \frac{C_{N^*-1}^{j-1}}{C_{2N^*}^{N^*}}, \tag{5} \]

and for \(m=2j+1\),

\[ h(m)=2\cdot \frac{C_{N^*-1}^{j}\cdot C_{N^*-1}^{j-1}}{C_{2N^*}^{N^*}} . \tag{6} \]

The probability that the number of series \(m\) will be equal to or less than some number \(d_0\), obtained from the factual data, takes the form

\[ P(m\leqslant d_0)=\sum_{m=2}^{m=d_0} h(m). \tag{7} \]

The nature of fluctuations of particular indicators may be considered random if

\[ P(m\leqslant d_0)>\gamma, \tag{8} \]

where \(\gamma\) is the significance level. Otherwise (i.e. for \(P(m\leqslant d_0)\leqslant \gamma\)) there is a systematic change in the characteristics of the properties in space, i.e. the hypothesis of their independence is rejected. If \(N_1\geqslant 10\) and \(N_2\geqslant 10\), then the form of the normal distribution may be used, i.e.

\[ P(m\leqslant d_0)=0.5-\Phi(t), \tag{9} \]

where

\[ t= \frac{\dfrac{2N_1\cdot N_2}{N_1+N_2}-d_0} {2\cdot \sqrt{N_1+N_2}\cdot \dfrac{N_1\cdot N_2}{(N_1+N_2)^2}} . \tag{10} \]

The value of the computed \(\Phi(t)\) is taken from the table given in [1]. In accordance with the recommendations of I. Naisk [2], for testing the hypothesis of the random nature of fluctuations of indicators of rock properties with depth, one may take \(\gamma=0.05\).

With application of the “sign test” considered here, the patterns of variability of particular indicators of properties with depth were investigated using the example of Mesozoic semi-rocky rocks of the Tataurovo brown-coal-

Cohesion \(C\), t/m\(^2\)

Depth \(z\), m

Series

---- | + | - | + | - | + | - | + | -- | +++ | - | ++++ | -----
  1    2   3   4   5   6   7   8    9    10   11    12     13

+  + | - | +  + | - | + | -- | + | - | + | -- | +++ | -
14 15  16 17 18 19 20 21 22 23 24 25

\[ N=43,\qquad M_e=5.83;\qquad N_1=21;\qquad N_2=22;\qquad d_0=25 \]

\[ \rho(m\le d_0)=0.5-\Phi\left( \frac{\dfrac{2\cdot22\cdot21}{43}-25} {2\sqrt{43}\cdot\dfrac{21\cdot22}{43^2}} \right)=0.3573 \]

\[ p(m\le 25)=0.3573 \gg \gamma=0.05\ [5] \]

Fig. 1. Distribution of the cohesion coefficients \((c)\) of sandstones of the Tataurow brown-coal deposit with depth \((z)\).

of the deposit, located in the large Chita–Ingoda depression. The results are given in Table 1, and the computational procedure is illustrated for the cohesion index of sandstones in Fig. 1. The calculations showed that the properties of sandstones, siltstones, and argillites are not ordered

Table 1

Note. The significance level \(\gamma\) was taken equal to 0.05 [2].

by depth, and the observed scatter of particular characteristics is random and can be analyzed using the formulas of the mathematical statistics of independent random variables.

REFERENCES

1. V. V. Nalimov. Application of Mathematical Statistics in the Analysis of Substances. Fizmatgiz, 1960.

2. I. Hauck. Dei Auswertung der äußeren Kontolle chemischer Analysen mit Hilfe des Zeichen—Tests und des Zeichen—Rangsummen—Tests. «Bergbautechnik», 4, 1966.