A Priori Information on the Distribution Laws of Indicators of the Engineering-Geological Properties of Rocks

E. A. Pisarev

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

This article examines all the basic a priori information currently available on the types of distribution laws for specific, most commonly used engineering-geological indicators of rock properties.

Granulometric characteristics. In engineering geology, as a rule, the distribution laws of granulometric characteristics of rocks are not studied, although this question is of fundamental importance—for example, from the standpoint of making the most effective classification constructions. Evidently, this circumstance has led to the fact that at the present stage nothing definite can be said about the types of distribution laws for the percentage contents of particular granulometric fractions of rocks. The situation is no better with the clarification of the types of distribution laws for metric generalized characteristics of granulometric composition (mean grain sizes, measures of dispersion, etc.).

Water-physical properties of rocks. Here it should be emphasized that, when the action of elements of effective heterogeneity on a given class of properties proceeds according to the scheme A—H—P (after M. V. Ratsu, 1968), then, other conditions being equal, one may expect that the distribution of indicators of these properties will obey the normal law. This theoretically important proposition for loose sandy-clayey rocks has been confirmed by numerous experimental checks carried out, for example, by E. N. Kolomenskii, I. N. Ivanova, and others. For semi-rocky rocks, the conclusion formulated above and the normality of the distribution of indicators of their water-physical properties are demonstrated by our extensive data, which are presented in a number of scientific reports and articles [3, 4, 5]. Somewhat different conclusions may be drawn about the types of distribution laws for characteristics of the physical properties of rocky rocks; to do this, let us turn to the work of K. S. Turitsyn [7], which presents the results of testing assumptions about the distribution laws of the specific and volumetric weights of various lithological-petrographic types of rocky rocks in the south of the Siberian Platform. According to K. S. Turitsyn’s data, the hypothesis of a normal distribution law for specific and volumetric weights does not contradict the experimental data (and the scheme of action A—H—P) if the rock (sandstones, anhydrites, rock salt, gypsum, pegmatites, crystalline schists, granites) does not contain mineral inclusions that are sharply anomalous with respect to the indicated properties.

At the same time, the specific gravity of carbonate rocks of the Lower Cambrian does not fit within the framework of the normal law. As analysis shows, the distribution of the specific gravity of limestones is affected by dolomitization and their anhydritization, while for dolomites it is affected by the combined influence of clayization, calcitization, and salinization. There is much in common with what has been considered in the laws of distribution of unit weight. As in the case of specific gravity, the dominant influence of impurities is quite clearly manifested. Thus, the presence of spodumene in pegmatites \((Pt_2—S_n)\) of the White River basin, and of various dark-colored and ore minerals in granitic gneisses of the same area, completely disrupts the normality of the distribution of the densities of these rocks. At the same time, distributions of unit weights and specific gravities of rocks that deviate substantially from the normal law agree rather well with generalized lognormal models using transformations of the initial data of the form \(\log(ax+b)\). This model of the distribution of characteristics of rock properties corresponds to the A—B—P scheme of action of the elements of effective heterogeneity on these properties [6].

The legitimacy of applying the normal or any of the lognormal models should be checked in each particular case by known statistical criteria. In this connection, let us emphasize that increasing the variety of types of theoretical distribution laws makes sense not only from the point of view of more effective compression of engineering-geological information on the water-physical properties of rocks, but also from the point of view of more reliable testing by statistical methods of various engineering-geological hypotheses¹. The use of statistical criteria without taking into account the type of distribution functions of the indicators being studied may lead to erroneous engineering-geological conclusions.

Strength characteristics of rocks. The study of the distribution laws of this class of indicators is of especially great importance from the engineering-geological point of view. To clarify these laws, let us use certain results of the statistical theory of strength.

W. Weibull, G. A. Kontorova, and Ya. I. Frenkel considered the statistical aspects of the brittle strength of solids. In recent years these same questions have also been developed in the works of V. V. Bolotin, S. D. Volkov, B. B. Chechulin, L. G. Sedrakyan, and others. The theory of brittle strength is based on the following assumptions:

a) the source of failure is microcracks (defects), uniformly distributed in the volume of the body, i.e., the number of defects in a certain small volume is proportional to the magnitude of the volume and does not depend on its position within the body;

b) in the process of loading, up to the onset of failure, microcracks do not change their characteristics;

c) a single microcrack of critical dimensions is sufficient to cause rapidly propagating failure of a specimen, irrespective of its dimensions;

d) the appearance of a microcrack corresponds to the critical stress at which failure begins;

e) each material corresponds to a definite distribution function of critical stresses for microcracks;

f) a priori, the decisive role of the most dangerous one may with equal probability be assigned to any of the defects present in the body.

Taking into account the assumptions introduced, and also on the basis of using—

of the probabilistic model of the distribution of brittle-strength characteristics of solids [2], for specimens of brittle rocks of volume \(V\) the distribution function \(F_{(V)}(R)\) of the strength characteristics \(R\) can be written in the form

\[ F_{(V)}(R)=1-\exp\left[-\frac{V}{V_0}\left(\frac{R-S_0}{S_c}\right)^\alpha\right]. \tag{1} \]

Then the mean value \(R_{(V)}\) can be obtained from the expression

\[ \overline{R}_{(V)}=S_0+S_c\left(\frac{V_0}{V}\right)^{1/\alpha}\Gamma(1+1/\alpha), . \tag{2} \]

and the coefficient of variation \(Var_{(V)}\) is determined as

\[ Var_{(V)}= \frac{ S_c\left(\frac{V_0}{V}\right)^{1/\alpha}\cdot \sqrt{\Gamma(1+2/\alpha)-\Gamma^2(1+1/\alpha)} }{ S_0+S_c\left(\frac{V_0}{V}\right)^{1/\alpha}\Gamma(1+1/\alpha) }. \tag{3} \]

In formulas (1, 2, 3), \(S_c\), \(\alpha\) are the parameters of distribution (1) (some positive numbers); \(S_0\) is the minimum value of strength; \(V_0\) is the volume of the reference rock specimen.

In solving engineering-geological problems, the apparatus of the statistical theory of brittle fracture was used by Ruppeneit (1964), Chirkov (1964), M. V. Ratsem (1968), and I. Kontek (1965). In this, satisfactory agreement was obtained between calculated data and the actual strength of coal pillars. In addition, L. A. Aronova and M. V. Ratsem (1967), using the statistical theory of brittle fracture in Bolotin’s variant, processed data of G. L. Fissiko and Yu. I. Turin­tsev obtained in field investigations of the shear resistance of large prisms of fractured rocks. The processing results showed satisfactory agreement between theory and experiment [1]. Similar processing of the results of field and laboratory investigations of the shear resistance of semirocky rocks of the Itat brown-coal deposit was also carried out by the author of the present work. The data of the corresponding calculations are summarized in Table 1 and emphasize the possibility of estimating the strength of a rock mass by extrapolating the results of laboratory investigations of strength properties.

Table 1

The reasoning we have presented above was based on the assumption that, in the process of failure of brittle rocks, there is no redistribution of stresses, and therefore the quantities \(R_1, R_2, \ldots, R_k\) were regarded as independent, i.e., each “weak point” is in this case as it were an independent element participating in the overall strength of the rock. However, by virtue of known circumstances, such assumptions do not hold for viscous and elastoviscous rocks, since such rocks usually resist external loads as a single whole, and the isolation of

of individual “weak spots” in this case is hardly justified.

In the destruction scheme, the failure of viscous and elastic-viscous rocks can be represented as a process of gradual accumulation of unit damages, owing to an increase in the specified stresses acting on the rock. When the number of such damages is sufficiently large and equal to some number \(M\), destruction of the rock occurs. Let us denote by \(R^{(\kappa)}\) the stress required for the accumulation of \(\kappa\) unit damages. It is assumed that the latter arise at random moments of time, and that, by virtue of this, the quantities \(R^{(\kappa)}\) are random variables that will be connected with one another by the following recurrence relation:

\[ R^{(\kappa)} = R^{(\kappa-1)} + \xi_\kappa , \]

\(\xi_\kappa\) is some additional stress required for the transition from \(\kappa - 1\) damages to \(\kappa\) damages.

The stress \(R^{(M)}\), required for the accumulation of \(M\) damages, corresponds to the stress up to the final destruction of the rocks under study. The distribution of \(R^{(M)}\) depends essentially on the nature of the quantities \(\xi_\kappa\), the interpretation of which may be given as follows.

Let \(\xi_\kappa = \eta_\kappa \cdot h(R^{(\kappa-1)})\), where \(\eta_\kappa\) is some random variable indicating the intensity of damage accumulation, and \(h(R^{(\kappa-1)})\) is a specified function of \(R^{(\kappa-1)}\). If we now consider the process of rock destruction in all its stages, then under the assumptions made above one may write the system of equalities

\[ R^{(1)} = R^{(0)} + \eta_1 h(R^{(0)}); \]

\[ R^{(2)} = R^{(1)} + \eta_2 h(R^{(1)}); \]

\[ \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \quad \cdot \]

\[ R^{(M)} = R^{(M-1)} + \eta_M h(R^{(M-1)}). \]

Let us next find the increment \(R^{(i)} - R^{(i-1)}\) in the form of a function of \(\eta_1, \eta_2, \ldots, \eta_\ast\), i.e.,

\[ \eta_i = \frac{R^{(i)} - R^{(i-1)}}{h(R^{(i-1)})} \]

or, after summation,

\[ \sum_{i=1}^{M} \eta_i = \sum_{i=1}^{M} \frac{R^{(i)} - R^{(i-1)}}{h(R^{(i-1)})}. \]

Assuming that the causes affecting the transition from \(\kappa\) damages to \(\kappa + 1\) damages vary greatly, while the number of such causes is large and each of them is small, we obtain

\[ \sum_{i=1}^{M} \eta_i = \sum_{i=1}^{M} \frac{R^{(i)} - R^{(i-1)}}{h(R^{(i-1)})} \approx \int_{R^{(0)}}^{R^{(M)}} \frac{dR}{h(R)} . \tag{4} \]

Since we have assumed that \(M\) is large, each of the summands is small, and their variances are probably bounded, it apparently may be considered that this process occurs under the conditions of the central limit theorem. In this case the right-hand side of (4) will be normally distributed, and the characteristic of the elastic-viscous strength of rocks will have a distribution depending on the function \(h(R)\). In addition, it is obvious that the parameters of the resulting distribution depend on the initial conditions of the process, i.e., in the present case on \(R^{(0)}\), which should be considered constant. In other words, it is necessary to take into account the additional condition \(h(R) = R^{(0)}\) at \(R = 0\).

Let us now consider certain specific, simplest hypotheses.

theses encompassing the admissible mechanisms of failure of elastic-viscous rocks.

Hypothesis \(H_n\). The simplest possible scheme of the failure mechanism of viscous and elastic-viscous rocks assumes \(h(R) \approx C = \mathrm{const}\). In this case, evidently, the distribution of \(R^{(M)} - R^{(0)}\) will obey the normal law, since in this case this quantity is the sum of a large number of independent terms determining its asymptotic normality. At the same time, let us emphasize that the equality \(R^{(k)} = R^{(k-1)} + C\), used to justify the normal distribution, and the assumption of independence and of the same order of the random variables \(\eta_k\), mean that the average rate of damage accumulation remains approximately constant as stress levels increase. The latter, obviously, holds if one proceeds from the hypothesis that rock fails at a specified linear deformation of the specimen. This hypothesis is approximately fulfilled, as a rule, under accelerated regimes of mechanical testing of rocks. The propositions noted above are illustrated by Fig. 1, which shows that the normal law approximates rather well the empirical—

Fig. 1. Approximation by the normal law of empirical distributions of specific micropenetration resistances.

... distributions of specific penetration resistances (“instantaneous” resistance), each of which was obtained on an experimental plot measuring \(1.5 \times 1.5\) m for rocks of the Telek phosphate deposit.

Under slow regimes of mechanical testing, the hypothesis of constancy of the mean rates of damage accumulation can approximately be satisfied only for viscous rocks, which are usually characterized by low values of yield limits. In this case, one may also expect the validity of hypothesis \(H_n\). For elastic-viscous rocks, the latter condition, as a rule, does not hold, and in this connection there arises the need to formulate other hypotheses.

Hypothesis \(H_e\). In the present case we assume \(h(R)=R\); then, after substituting this function into (4), we obtain that \(\ln R\) must have a normal distribution. Let us note that the equality \(R^{(k)}=R^{(k-1)}+\eta_k R^{(k-1)}\), used to derive the lognormal distribution (under the assumption of the independence and smallness of \(\eta_k\)), means that the mean rate of damage accumulation in the rock during testing decreases as stresses increase. The decrease in the rate of damage accumulation may be regarded as a reflection of the process of strengthening, which creates obstacles to the emergence of new damage. Damage accumulation appears in this light as a process of weakening (fracturing). In other words, the lognormal distribution arises as a consequence of the superposition of two processes: weakening and strengthening.

Thus, analysis of the models \(H_n\) and \(H_e\) has shown that, depending on the mechanism of destruction of rocks under the action of external loads, it is possible a priori to make certain assumptions about the types of distribution laws of their strength characteristics.

In giving an overall assessment of the various models, let us emphasize that they have been obtained on the basis of schematic (idealized) constructions and therefore, naturally, provide only an approximate description of the variability of the engineering-geological indicators of the properties of various rocks. Nevertheless, their use in assessment leads to more reliable results than a simple review of experimental data devoid of any theoretical interpretation.

REFERENCES

1. L. A. Aronova, M. V. Rats. On the application of the statistical theory of strength to the description of the strength properties of rocks. Materials for the Scientific and Technical Conference on Engineering Geology (11–15 May 1967). Moscow, 1967.

2. V. V. Bolotin. Statistical Methods of Structural Mechanics. Moscow, 1965.

3. F. P. Nifantov, V. E. Olikhovatenko, E. A. Pisarev. Engineering-geological conditions of the Third Barandatka section of the Itatka brown-coal deposit (report on topic 93/63). TPI Archives, Tomsk, 1965.

4. F. P. Nifantov, Yu. A. Usynin, N. S. Rogova, E. A. Pisarev. Engineering-geological conditions of the First section of the Berezovo brown-coal deposit (report on topic 99/63). TPI Archives, Tomsk, 1964.

5. E. A. Pisarev. On the methodology of statistical processing of indicators of engineering-geological properties of rocks. Proceedings of TISI, vol. 12, 1967.

6. M. V. Rats. Heterogeneity of Rocks and Their Physical Properties. Moscow, 1968.

7. K. S. Turitsyn. The basic statistical law of distribution of the physical properties of rocks. In: Geophysical Investigations in Solving Geological Problems in Eastern Siberia. Issue 3, Moscow, 1964.

1. For each type of theoretical distribution there correspond the most powerful statistical criteria for testing hypotheses. ↩