FACTORS INFLUENCING THE FORM OF THE DISTRIBUTION LAW 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)

The practical importance of studying distribution laws in geology can be illustrated by numerous examples from the practice of exploration, lithological-petrographic, geochemical, and other investigations (A. B. Vistelius, D. A. Rodionov, A. V. Kantsel, S. I. Smirnov, and others). In engineering-geological terms, knowledge of the distribution laws of indicators of rock properties also acquires the most urgent importance, especially in the process of determining design indicators, constructing systems of engineering-geological sampling of a rock mass, classifying rocks, and so on. In connection with this it is very important to establish which factors influence the form of the distribution function of indicators of the engineering-geological properties of rocks. The present article is devoted precisely to consideration of these factors. We shall not be mistaken if we name the following.

I. The type of distribution to which a given rock property is subject depends on the scheme of action of the elements of effective heterogeneity upon that property. At the same time, the scheme of action of effective heterogeneity also determines the character and degree of influence of the size of the samples studied on the distribution parameters of the property being studied. This phenomenon is widely known under the name of the scale effect. M. V. Rats showed [4] that rocks are characterized by the following principal actions of heterogeneity, and by the corresponding distributions of properties and scale effects.

1. The scheme A—N—R (additivity—independence—equivalence) is the normal law of distribution of indicators of the corresponding rock properties, which follows from the following assumptions. Suppose that a rock sample consists of \(n\) elements of effective heterogeneity, each of these elements being characterized by the value \(\xi_i\) of the given property \((i = 1, 2, \ldots, n)\). Then the property of the sample as a whole is nothing other than \(M\xi = \frac{1}{n}\sum_{i=1}^{n}\xi_i\), where \(M\xi\) is the mathematical expectation of the random variable \(\xi\). If the action of the heterogeneity elements on \(\xi_i\) is additive, independent, and equivalent, and \(n \to \infty\), then, by virtue of the fulfillment of the conditions of the central limit theorem, one may expect a normal distribution of the given property. According to M. V. Rats (1968), in this case one should expect a scale effect of type II, obeying the rule \(\sqrt{n}\), and the absence of scale effects of types I and III: the mean value and the asymmetry of the distribution of the property do not depend on the size of the samples.

1. Scheme A—B—P (additivity—interdependence—equiprobability). The distribution of the characteristics of the properties is lognormal; scale effects of the second and third kinds: the coefficient of variation changes more slowly than according to the rule \(\sqrt{n}\); as the sample size increases, the skewness decreases. With an unlimited increase in sample size, the A—B—P scheme approaches the A—N—P scheme.

2. Scheme I—N (selectivity—independence); characteristic of rock attributes is the Weibull distribution, a scale effect of the first kind (a decrease in the mean value with increasing sample size), as well as a scale effect of the second kind, the degree of manifestation of which depends on the intensity of manifestation of the scale effect of the first kind.

3. Lower-order heterogeneity substantially complicates the pattern of distribution of the properties of rocks and the manifestation of scale effects; in particular, when the area of influence of the experiment is increased, the variance, decreasing, tends in this case toward a certain constant. Therefore, alongside “pure” distributions and scale effects caused by the effective heterogeneity of rocks, it is appropriate to speak of generalized distributions and scale effects that take into account the influence of lower-order heterogeneity.

II. The form of the distribution function of a given engineering-geological index depends, moreover, on the character of the relations of this index with other characteristics of the rocks. We shall show this by an example, considering here the simplest problem—the problem of the distribution law of a random function of one random argument.

There is a continuous random variable \(\xi_1\) with distribution density \(f(x)\). Another random variable \(\xi_2\) is related to it by the dependence \(\xi_2=\varphi(\xi_1)\). It is required to find the distribution density of the variable \(\xi_2\). In the mathematical sense this problem was solved in the work of E. S. Wentzel [2]. For example, it is not difficult to obtain the very important practical conclusion that a linear random function of a random argument subject to the normal law is also subject to the normal law.

III. The form of the distribution law of particular indices of the engineering-geological properties of rocks depends on the character of the variability of these properties in space. We shall show an investigation of this question by the example of studying the influence of a spatial trend on the form of the corresponding distribution law for the characteristics of some property of rocks \(x\).

Suppose that a certain sample has been taken from a rock mass in the depth interval \([z_1, z_2]\). The sampling density is taken to be constant. In addition, it is known that a given property of the rocks is expressed by some function of the sampling depth, \(a(z)\). It is required to investigate, for such a case, the distribution of the engineering-geological characteristic of the rocks \(x\). For a given fixed sampling depth \(z\), let us specify the distribution of the index \(x\) in the form

\[ f_z(x)=\frac{1}{\sqrt{2\pi}\,\sigma}\exp-\left\{\frac{[x-a(z)]^2}{2\sigma^2}\right\}, \tag{1} \]

and, consequently, the probability that at the fixed depth \(z\) the investigated characteristic of the rock varies within the limits from \(x\) to \(x+dx\) will be

\[ P_x^{(z)}=\frac{1}{\sqrt{2\pi}\,\sigma}\exp-\left\{\frac{[x-a(z)]^2}{2\sigma^2}\right\}\,dx. \]

Let, further, the probability that an observation taken at random from the sample was taken in the interval from \(z\) to \(z+dz\) be equal to

\[ P_z=\psi(z)\,dz. \]

The probability of the joint occurrence of the events (“the sample was taken in the interval from \(z\) to \(z+dz\)” and “the index determined from this sample lies within the limits from \(x\) to \(x+dx\)”) is, by the multiplication theorem of probabilities, written as

\[ P_x^{(z)} p_z = \frac{1}{\sqrt{2\pi}\,\sigma} \exp\left\{-\frac{[x-a(z)]^2}{2\sigma^2}\right\}\psi(z)\,dx\,dz . \]

Let us now compute the probability that an observation taken at random from the sample has a value within the limits from \(x\) to \(x+dx\). Let the sample consist of \(N\) determinations. Renumber them in the order of selection from 1 to \(N\). Then the required probability \(p_x\) is equal to the probability that the observation has a value from \(x\) to \(x+dx\) and at the same time corresponds to a specimen either with number 1, or with number 2, ..., \(j\), or with number \(N\). Denoting the corresponding probabilities by \(p_1, p_2,\ldots, p_N\), by the addition theorem of probabilities we find: \(p_x=p_1+p_2+\ldots+p_N\). Assuming that \(N\) is large, and taking into account that \(p_i=p_x^{(z_i)}\times p_{z_i}\), the summation may be replaced by integration, which gives

\[ p_x = dx \int_{z_1}^{z_2} \frac{1}{\sqrt{2\pi}\,\sigma} \exp\left\{-\frac{[x-a(z)]^2}{2\sigma^2}\right\}\psi(z)\,dz . \]

This means that the probability density of the engineering-geological index \(x\) will be

\[ f(x)= \frac{1}{\sqrt{2\pi}\,\sigma} \int_{z_1}^{z_2} \exp\left\{-\frac{[x-a(z)]^2}{2\sigma^2}\right\}\psi(z)\,dz . \]

Now consider the density \(\psi(z)\). Let the rock specimens, after being selected, be thoroughly mixed, and suppose it is required to calculate the probability that a specimen drawn at random has number \(j\). This probability is equal to

\[ \frac{1}{N}. \]

By assumption, the density of rock sampling is specified as constant. This means that the interval of selection of a specimen does not depend on its number. Denote this interval by \(\Delta_z\); then the entire interval of the section within which all rock specimens have been selected will be represented as \(z_2-z_1=N\cdot\Delta_z\). It follows that the probability of drawing a specimen with number \(j\) can be represented in the form

\[ \frac{1}{N}=\frac{\Delta z}{z_2-z_1}. \]

On the other hand, if a specimen has number \(j\), this means that it was selected within the depth interval from \(z=(j-1)\Delta_z\) to \(z+\Delta_z\). Assuming that the interval \(\Delta_z\) is small, we obtain the probability that a specimen taken at random was selected within the limits from \(z\) to \(z+dz\) as

\[ p_z=\frac{dz}{z_2-z_1}, \]

and, consequently, the probability density of the random variable \(z\)—the depth of selection of a particular rock specimen—is equal to

\[ \psi(z)=\frac{1}{z_2-z_1}. \]

This, as is known, is a uniform distribution (Fig. 1, a). Thus, finally, the probability density of \(x\) has the form

\[ f(x)= \frac{1}{z_2-z_1}\cdot \frac{1}{\sqrt{2\pi}\,\sigma} \int_{z_1}^{z_2} \exp\left\{-\frac{[x-a(z)]^2}{2\sigma^2}\right\}\,dz . \tag{1, a} \]

Figure labels:
— normal law
– – – law of equal probability

Fig. 1. a) curve of a uniform distribution; b) family of theoretical distribution curves under a linear shift along the spatial coordinate; \(z\) is the center of grouping of indicators of the engineering-geological properties of rocks.

Let us also note that, if for fixed \(z\) the probability density of \(x\) is given as \(f_z(x)\), then, with constant sampling density, the following formula will hold:

\[ f(x)=\frac{1}{z_2-z_1}\int_{z_1}^{z_2} f_z(x)\,dz, \tag{2} \]

where \(f_z(x)\) will be called the fixed distribution, and the distribution \(f(x)\) the distribution in the sample.

Let us now calculate \(Mx\), using for this the following formula

\[ Mx=\int_{-\infty}^{\infty} x f(x)\,dx = \frac{1}{z_2-z_1} \int_{-\infty}^{\infty} x\int_{z_1}^{z_2} f_z(x)\,dz\,dx. \]

or, changing the order of integration,

\[ Mx= \frac{1}{z_2-z_1} \int_{z_1}^{z_2} \int_{-\infty}^{\infty} x f_z(x)\,dx\,dz. \]

Let us denote

\[ M_{x_z}=\int_{-\infty}^{\infty} x f_z(x)\,dx . \]

Since \(f_z(x)\) is the probability density of \(x\) at a fixed depth \(z\), \(M_{x_z}\) is the mathematical expectation of the random variable \(x\) under the assumption that \(z\) is fixed. In particular, for the distribution \(f_z(x)\) specified as (1), we have \(M_{x_z}=a(z)\). The mathematical expectation for the distribution in the sample will now be represented in the form

\[ Mx=\frac{1}{z_2-z_1}\int_{z_1}^{z_2} Mx_z\,dz . \]

or

\[ Mx=\frac{1}{z_2-z_1}\int_{z_1}^{z_2} a(z)\,dz . \tag{3} \]

To compute the variance \(\sigma_x^2\), we use the formula

\[ \sigma_x^2=\int_{-\infty}^{\infty} x^2 f(x)\,dx-M^2x . \]

Since the distribution of \(x\) in the sample is specified by density (1), we have

\[ \sigma_x^2=\frac{1}{z_2-z_1}\int_{-\infty}^{\infty} x^2 \int_{z_1}^{z_2} f_z(x)\,dz\,dx-M^2x . \]

Changing the order of integration, we arrive at the expression

\[ \sigma_x^2=\frac{1}{z_2-z_1}\int_{z_1}^{z_2}\int_{-\infty}^{\infty} x^2 f_z(x)\,dz\,dx-M^2x . \]

For a given \(z\), the variance of the random variable \(x\) in the fixed distribution can be written as

\[ \sigma^2=\int_{-\infty}^{\infty} x^2 f_z(x)\,dx-M^2x_z . \]

Therefore

\[ \int_{-\infty}^{\infty} x^2 f_z(x)\,dx=\sigma^2+M^2x_z, \]

and

\[ \sigma_x^2=\frac{1}{z_2-z_1}\int_{z_1}^{z_2}\sigma^2\,dz+\frac{1}{z_2-z_1}\int_{z_1}^{z_2}M^2x_z\,dz-M^2x . \]

Finally, we have

\[ \sigma_x^2=\sigma^2+\frac{1}{z_2-z_1}\int_{z_1}^{z_2}a^2(z)\,dz- \left[\frac{1}{z_2-z_1}\int_{z_1}^{z_2}a(z)\,dz\right]^2 . \tag{4} \]

Thus, the mathematical expectation \(Mx\) in the sample is the averaged mathematical expectation \(Mx_z\) of the fixed distribution, while the variance in the sample exceeds the variance \(\sigma^2\) of the fixed distribution by the amount

\[ \sigma_x^2-\sigma^2= \frac{1}{z_2-z_1}\int_{z_1}^{z_2} a^2(z)\,dz - \left[ \frac{1}{z_2-z_1}\int_{z_1}^{z_2} a(z)\,dz \right]^2, \]

Let us consider the simplest case, when \(a(z)\) is a linear function of depth,

\[ a(z)=\beta z. \]

According to (1, a), we obtain

\[ f(z)= \frac{1}{z_2-z_1}\cdot \frac{1}{\sqrt{2\pi}\,\sigma} \int_{z_1}^{z_2} \exp\left[-\frac{(x-\beta z)^2}{2\sigma^2}\right]\,dz, \]

After performing the transformation

\[ \exp\left[-\frac{(x-\beta z)^2}{2\sigma^2}\right] = \exp\left[-\frac{(z-x/\beta)^2}{2\sigma^2\beta^2}\right] \]

we represent \(f(x)\) in the form

\[ f(x)= \frac{1}{\beta(z_2-z_1)}\cdot \frac{1}{\sqrt{2\pi}\,\sigma/\beta} \int_{z_1}^{z_2} \exp\left[-\frac{(z-x\beta)^2}{2\sigma^2\beta^2}\right]\,dz, \]

after which we shall have

\[ f(x)= \frac{1}{\beta(z_2-z_1)} \left[ \Phi\left(\frac{z_2-x/\beta}{\sigma/\beta}\right) - \Phi\left(\frac{z_1-x/\beta}{\sigma/\beta}\right) \right] = \]

\[ = \frac{1}{\beta z_2-\beta z_1} \left[ \Phi\left(\frac{\beta z_2-x}{\sigma}\right) - \Phi\left(\frac{\beta z_1-x}{\sigma}\right) \right]. \]

To simplify the equality, set \(\beta z_1=-\lambda_1\), \(\beta z_2=\lambda_2\); then

\[ f(x)= \frac{1}{\lambda_1+\lambda_2} \left[ \Phi\left(\frac{\lambda_1+x}{\sigma}\right) - \Phi\left(\frac{x-\lambda_2}{\sigma}\right) \right]. \tag{5} \]

We now compute the distribution function corresponding to (5)

\[ F(x)=p(x\le \xi)= \int_{-\infty}^{x} f(x)\,dx = \]

\[ = \frac{1}{\lambda_1+\lambda_2} \int_{-\infty}^{x} \left[ \Phi\left(\frac{\lambda_1+x}{\sigma}\right) - \Phi\left(\frac{x-\lambda_2}{\sigma}\right) \right]\,dx. \tag{6} \]

The expression obtained includes the integral of the Laplace function. From the rule of integration by parts there follows the equality

\[ \int_{-\infty}^{t} \Phi(t)\,dt = t\Phi(t)+\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{t^2}{2}\right). \tag{7} \]

Applying formula (7) to the right-hand side of equality (6), we have:

\[ F(x)= \frac{1}{\lambda_1+\lambda_2} \left[ (x+\lambda_1)\Phi\left(\frac{\lambda_1+x}{\sigma}\right) + \frac{\sigma}{\sqrt{2\pi}} \exp\left\{-\frac{(x+\lambda_1)^2}{2\sigma^2}\right\} \right] - \]

\[ - \frac{1}{\lambda_1+\lambda_2} \left[ (x-\lambda_2)\Phi\left(\frac{x-\lambda_2}{\sigma}\right) + \frac{\sigma}{\sqrt{2\pi}} \exp\left\{-\frac{(x-\lambda_2)^2}{2\sigma^2}\right\} \right]. \]

For simplicity let us put \(\lambda_1=\lambda_2=\lambda\); then

\[ F(x)=\frac{1}{2\lambda}\left[(x+\lambda)\Phi\left(\frac{x+\lambda}{\sigma}\right) -(x-\lambda)\Phi\left(\frac{x-\lambda}{\sigma}\right)\right] -\frac{\sigma}{2\lambda\sqrt{\pi}} \left[ \exp\left\{-\frac{(x+\lambda)^2}{2\sigma^2}\right\} -\exp\left\{-\frac{(x-\lambda)^2}{2\sigma^2}\right\} \right]. \tag{8} \]

The corresponding family of theoretical distribution curves is given in Fig. 1, б. All curves of this family are symmetric (asymmetry coefficient \(A=0\)) and flat-topped (excess coefficient \(E<0\)). For large values of the ratio \(\lambda/\sigma\), the curves approach the curve of the law of equal probability (uniform distribution within the limits \(\pm\lambda\)). The latter curve, which for this family is limiting (as \(\sigma\to0\)), is shown in Fig. 1, б by a dash-dot line. The other limiting curve of the family (as \(\lambda\to0\)) is the normal-distribution curve.

Let us now calculate, for distribution (5), the mathematical expectation \(Mx\) and the variance \(\sigma_x^2\) of the investigated index of a rock property. According to (3),

\[ Mx=\frac{1}{z_2-z_1}\int_{z_1}^{z_2} a(z)\,dz =\frac{1}{z_2-z_1}\int_{z_1}^{z_2}\beta z\,dz =\frac{1}{2}\beta(z_2+z_1). \]

Since we have already assumed that \(\beta z_1=-\lambda\) and \(\beta z_2=\lambda\), it follows that \(Mx=0\). To calculate \(\sigma_x^2\) we shall use formula (4):

\[ \sigma_x^2=\sigma^2+\frac{1}{z_2-z_1}\int_{z_1}^{z_2}\beta^2 z^2\,dz -\left[ \frac{1}{z_2-z_1}\int_{z_1}^{z_2}\beta z\,dz \right]^2 \]

\[ =\sigma^2+\frac{\beta^2}{z_2-z_1}\left(\frac{1}{3}z_2^3-\frac{1}{3}z_1^3\right) =\sigma^2+\frac{\lambda^2}{3}, \]

or

\[ \sigma_x=\sqrt{\sigma^2+\frac{\lambda^2}{3}}. \]

The approach set forth above may be applied to the study of the distribution laws of particular indices of the engineering-geological properties of rocks also in more complex cases—for example, when there is a nonlinear trend. In this case, obviously, the generality of all the arguments is not lost. The solution of the particular problem considered here has shown that one of the possible cases of deviation of the distribution law of indices of the engineering-geological properties of rocks from the normal law is associated with the presence of a regular linear spatial variability.

IV. The form of the distribution law of particular indices depends on the method by which sampling points are arranged in the rock mass. For example, in geochemistry and in geological exploration there already exist certain theoretical considerations on this matter [1, 3]. The influence of the geometric features of the sampling network on the form of the distribution law of engineering-geological characteristics of rocks has been shown, with numerous examples, in the work of E. N. Kolomenskii (1969); however, as a whole, this question still remains insufficiently investigated, despite the fact that it deserves special attention, since only on the basis of its study can the very process of engineering-geological investigations be regulated—in order to improve the reliability of the results—without any additional expenditure of funds.

V. The form of the distribution law depends on the conditions and accuracy with which investigations of a given engineering-geological property of a rock are carried out. The relevance of studying this question is obvious and is unlikely to meet with objection from anyone. However, it must be stated that, at present, as applied to the engineering-geological properties of rocks, the proposition formulated here in fact remains still unstudied; therefore, it is not possible here to say anything definite in this respect, even in the form of a problem statement.

Thus, the principal factors have been examined above in sufficient detail—factors which, jointly or separately, determine one or another form of the distribution law of particular indicators of the engineering-geological properties of rocks, and which should be taken into account in constructing stochastic models of geological processes and in processing the results of observations.

LITERATURE

1. B. I. Belov. On the geochemical meaning of the laws of distribution of substance concentrations. — In: Mathematical Methods of Geochemical Investigations. Moscow, 1966.

2. E. S. Venttsel. Probability Theory. Moscow, 1964.

3. A. M. Margolin. The influence of the character of the nonuniformity of the network of exploration points on the reliability of the results of oil-field exploration. Transactions of VNII, issue X, VII, 1966.

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