ON DETERMINING OPTIMAL DISTANCES BETWEEN EXPLORATION WORKINGS IN CONNECTION WITH DIFFERENT SIZES OF PLOTS AND THE ANISOTROPY OF ORE BODIES

L. M. Petrovsky

(Presented by the scientific seminar of the departments of mineralogy, petrography, and geology and exploration of mineral deposits)

In the original monographic book by B. S. Levonik¹, among other things, the question of determining optimal distances between exploration workings is considered. For this purpose the following formula is proposed for determining the relative error \(\varepsilon\) (in percent):

\[ \varepsilon = 0{,}1\,K_1 \cdot N, \tag{1} \]

where \(0{,}1\) is an empirical coefficient;

\(K_1\) is the coefficient of variability, defined as the ratio of the maximum frequency of the variation series to its mean frequency and taken within the range from \(0{,}1\) to \(5\text{—}10\);

\(N\) is a “number” denoting the distance between exploration workings, m.

The solution of specific problems by formula (1) is proposed by the author of the book for a series of numbers \(N\) from 1 to 1000 (Table 1).

From the table it is evident that, first, the distance between exploration workings determined on the basis of formula (1) is in simple proportional dependence on the values of \(K_1\) and \(\varepsilon\); in this respect formula (1) does not differ, for example, from the empirical formula of V. A. Prokin (1961). The actual character of this dependence, however, as is known, usually proves to be more complex, and establishing it remains a task of primary importance. Second, when compared with the corresponding errors, they prove to be completely unrealistic,

Table 1

Optimal linear distances between exploration workings
for a specified exploration accuracy (errors \(\varepsilon\))

calculated distances for a simple group of deposits (coal and oil-and-gas). The latter is connected both with an unsuccessfully chosen scale of variability and with an unclear relative dimensionality adopted for it, if one takes into account that, according to its definition, the coefficient \(K_1\) in absolute value must always be greater than 1 (as is also the NP index according to V. V. Bogatsky (1963)) and if one attempts to compare the scale chosen by him, for example, with the known scale of variability according to the coefficient of variation \(v\).

The principal shortcoming of formula (1), as of all known formulas analogous to it, is that it does not take into account the dependence of the distance between exploration workings on the size of the exploration targets. The latter may not only be small, medium, or large within each given group of deposits, but also vary differently within each of these categories of targets for different groups of deposits. The importance of taking this factor into account when choosing the optimal density of the exploration network was strongly emphasized at the special session on developing the principal measures for increasing the reliability of the results of geological exploration work (Kogan, 1965).

In connection with the latter, methodologies prove more preferable that determine the necessary and sufficient uniform number of intersections on the basis of probability theory and mathematical statistics, proceeding from which, for an object with a given absolute area, the distance between them is found (Smirnov, 1957; Bogatsky, 1963). In this case it remains only to find a way of correcting such a number of intersections in accordance with the differentiation of areas by their relative sizes.

An expression of one of such methodologies could be another formula from B. S. Levonik’s book, called the basic formula of relative error:

\[ \varepsilon=\pm \frac{100K_1}{N\cdot K_2}\sqrt{\sum [(N_{\mathrm{h}}-n_{\mathrm{h}})P_{\mathrm{h}}\cdot q_{\mathrm{h}}]}, \tag{2} \]

where \(N\) is the “general” population, or the number of all geological-exploration workings;

\(K_1\)—the same as in formula (1);

$K_2$—the exploration coefficient, expressing the density of the exploration network and adopted for category $A$ as equal to 1, for category $B$ as from $1/2$ to $1/4$, and for category $C_1$ as from $1/5$ to $1/10$;

$N_h$—the topological sample, or the number of workings that determine explored reserves of some category—$A$, $B$, and $C_1$;

$n_h$—the random sample, or the number of exploration workings of a given reserve category taken for the sample, with the aim of shortening the process of studying the population $N$; $n_h$ may vary from 1 to $N_h$;

$P_h$—the probability corresponding to each group of the random sample; $P_h=n_h/N_h$;

$q_h$—the non-probability corresponding to each group of the random sample; $q_h=1-P_h$.

Let us emphasize that formula (2) determines the relative error for each given category of reserves, which is confirmed, in particular, by the adjustment of the value $N$ by means of the coefficient $K_2$. However, two comments must be made here. First, the index $K_2$ is incorrectly used in formula (2) (especially as applied to category $A$, for which $K_2=1$). Second, the number of workings thus determined for each category $N_{A,B,C_1}$ presupposes its extension, and consequently also the errors $\varepsilon_{A,B,C_1}$, to an area of one and the same size (to the entire area of the site). In the practice of detailed exploration, however, in deposit areas of different complexity there occur different ratios of reserve categories (of the areas of their distribution), which can be expressed through the ratio index $R_{A,B,C_1}$, characterizing the share of each reserve category within the limits from 0 to 1. Taking these comments into account, formula (2) should have the following form:

\[ \varepsilon_{A,B,C_1} = \frac{ 100K_1\left(K_2^A\cdot R_A+K_2^B\cdot R_B+K_2^{C_1}\cdot R_{C_1}\right) }{ N\cdot K_2^{A,B,C_1}\cdot R_{A,B,C_1} } \sqrt{\sum\left[(N_h-n_h)P_h\cdot q_h\right]} \]

or

\[ \varepsilon_{A,B,C_1} = \frac{100K_1}{N_{A,B,C_1}} \sqrt{\sum\left[(N_h-n_h)P_h\cdot q_h\right]}, \tag{2'} \]

where

\[ N_A:N_B:N_{C_1} = K_2^A\cdot R_A:K_2^B\cdot R_B:K_2^{C_1}\cdot R_{C_1} = {}^1/\varepsilon_A:{}^1/\varepsilon_B:{}^1/\varepsilon_{C_1}. \]

The expression entering formulas (2) and (2′),

\[ \sqrt{\sum\left[(N_h-n_h)P_h\cdot q_h\right]}, \]

characterizes the probability of the accuracy of determining the relative error. Solving formula (2′) with respect to it, we obtain that

\[ \sqrt{\sum\left[(N_h-n_h)P_h\cdot q_h\right]} = \frac{\varepsilon_{A,B,C_1}\cdot N_{A,B,C_1}}{100K_1}. \tag{3} \]

Since, in normally explored deposits (sites),

\[ N_A:N_B:N_{C_1} = {}^1/\varepsilon_A:{}^1/\varepsilon_B:{}^1/\varepsilon_{C_1} = K_2^A\cdot R_A:K_2^B\cdot R_B:K_2^{C_1}\cdot R_{C_1}, \]

then, for given $N$ and $K_1$, the right-hand side of equation (3) will remain constant for any ratios between $\varepsilon_{A,B,C_1}$ and $N_{A,B,C_1}$; in analogous derivative equations of other formulas (Smirnov, 1957; Bogatskii, 1963), this part corresponds in its content to the probability coefficient $t$.

Calculations show that, of all possible values of the expression

\[ \sqrt{\sum\left[(N_h-n_h)P_h\cdot q_h\right]}, \]

the values of it beginning from 2 to 3 and higher deserve practical attention, which, as is known, ensures, at a probability of 95–99.7% and higher, a correct estimate of the specified quantity.

relative errors. For the purposes of sufficiently reliable exploration of deposits explored by drilling and mining-and-drilling systems, the value of 99.7% may practically be taken as 100% probability, which makes it possible to take as a constant value the expression

\[ \sqrt{\sum\left[(N_h-n_h)P_h\cdot q_h\right]} \]

or, for the coefficient \(t\), the number 3.

Taking the latter into account, the probabilistic determination of the sample size of the number of intersections (workings) for detailed exploration of deposits explored by drilling and mining-and-drilling systems should be carried out according to the following formula:

\[ N=\frac{300K(K_A\cdot R_A+K_B\cdot R_B+K_{C_1}\cdot R_{C_1})} {\varepsilon_{A,B,C_1}\cdot K_{A,B,C_1}\cdot R_{A,B,C_1}} \quad \text{or} \quad N_{A,B,C_1}=\frac{300K}{\varepsilon_{A,B,C_1}}, \tag{4} \]

where \(N\) is the number of all elementary exploration blocks on the site,

\[ N=N_A+N_B+N_{C_1}; \]

\(\varepsilon_{A,B,C_1}\) is the permissible relative error in determining the mean value of the parameters;

\(K_{A,B,C_1}\) is the same as \(K_2\) in formula (2), where \(K_B\) and \(K_{C_1}\) may take values from \(1/2\) to \(1/16\) and smaller when \(K_B>K_{C_1}\);

\(K\) is an index of the variability of the least stable parameter for a series of deposits explored by drilling and mining-and-drilling systems.

It seems to us that, at the present stage of study, for a quantitative assessment of the variability of parameters at the deposits under consideration, one should use either the NP index (Bogatskii, 1963) or the coefficient of variation \(v\) (Smirnov, 1957; Petrov, 1963). In the first case, the range of variability of a parameter at these deposits can apparently be represented by stable-maximum NP values within the limits from 1 to 4–5; in the second, from 0 to 100%. Since formula (4) corresponds to \(K\)-NP, in order to use the coefficient \(v\) in it it is necessary to find the function \(F(v)=\text{NP}-K\), provided that the scale of \(v\) values corresponds to the scale of NP values. For this purpose, let us take as a conventional standard of the least variability on the scale of the coefficient \(v\) the value of 10%. Then the desired function may be expressed by the following formula:

\[ K=\left[2\cdot \lg(10+v)-\lg\left(\frac{100-v}{10}\right)\right], \tag{5} \]

where \(\lg\) is the common logarithm.

When \(v\) varies from 1 (practically from 10%) to 100%, the value of \(K\) according to formula (5) will vary within the limits from 1.0 to 4.2.

Formula (4) determines the necessary and sufficient number of uniformly distributed elementary (unit) exploration blocks, and not the number of exploration workings, which may be equal to or greater than the number of elementary blocks depending on whether the block is supported by one central working (according to the principle of the nearest district) or is bounded by four workings (nodal grids).

The choice of one of these grids of unit exploration blocks for the analytical calculation of the number of workings, other conditions being equal, depends on the detail of exploration. In estimating reserves in category \(C_1\), the contour of which is drawn on the basis of near or distant extrapolation, the calculation of the number of \(C_1\) workings should be based on grids constructed according to the principle of the nearest district (\(C_{c1}=N_{c1}\)). In estimating reserves in categories A and B, the contour of whose blocks is usually drawn through workings, the calculation of their number \(C_{A,B}\) should be based on nodal grids and carried out according to the formulas:

Table 2

Optimal design distances between exploration boreholes, depending on the degree of complexity and the size of coal-bearing areas

Note. On areas of complex structure whose exploredness indicators, for example, are 3 times greater than the large indicator of the discontinuity of areas with simple structure, the density of the exploration grid increases accordingly by this ratio.

for a square form of the area of distribution of reserves of each of categories \(A\) and \(B\)

\[ C_{A+B}=\left(\sqrt{N_{A+B}}+1\right)^2; \]

for a rectangular form of this area with a side ratio of \(1:2\)

\[ C_{A+B}=\left(\sqrt{\frac{N_{A+B}}{2}}+1\right)\left(2\sqrt{\frac{N_{A+B}}{2}}+1\right), \quad \text{etc.} \tag{6} \]

A study of nodal grids of elementary exploration blocks has shown that the number of workings and interpolation elements formed by each pair of neighboring workings changes in a definite disproportionate dependence on the absolute number of blocks, which reflects a regular increase or decrease in the specific number of workings and interpolation elements with a corresponding decrease or increase in the number of blocks (Petrovskii, 1965). Since a change in the absolute number of elementary exploration blocks expresses a corresponding proportional change in the relative dimensions of exploration objects, the natural disproportionate dependence noted above may be taken as the basis for determining the optimal number of unit exploration blocks and workings in connection with different relative sizes of areas.

Let us assume that the probable number of unit exploration blocks \(N\), determined by formula (4), meets the optimal requirements for exploration of areas of average size, with some area \(S_{\mathrm{av}}\) constant for a given series of deposits. Then the calculated number of blocks \(N_{\mathrm{sec}}\) for a particular exploration area with area \(S_{\mathrm{sec}}\) would have to be corrected by the ratio \(S_{\mathrm{sec}}/S_{\mathrm{av}}\). In this case, the calculated distance between exploration workings, other conditions being equal, would remain constant for areas of any size within some series of them from \(S_{\mathrm{sec}}^{\min}\) to \(S_{\mathrm{sec}}^{\max}\), just as is the case, for example, in Table 1, and also in the practice of exploration geologists who seek to maintain a constant distance between exploration workings on the basis of the so-called principle of analogy, without taking the size of areas into account (Kogan, 1965).

Another, more numerous group of researchers regards as a primary necessity the keeping constant of the number of intersections (workings) corresponding to the necessary and sufficient sample (Smirnov, 1957; Bogatskii, 1963; Petrov, 1963). In this case the distance between exploration workings on specific areas will not remain constant, but will be determined wholly by the absolute dimensions of the areas.

The actual nature of the change in the density of the exploration grid on areas of different size, as is known, must be accompanied by a corresponding change both in the number of elementary blocks or workings \(N_{\mathrm{sec}}\) and in the distance between them \(l\) (Petrovskii, 1965). A schematic diagram reflecting the essence of the viewpoints of the two groups of analytical methods considered above and of the proposed optimal variant for calculating the density of the exploration grid is shown in Fig. 1. From the latter it is evident that optimization of the basic parameters of the density of the exploration grid is achieved by using the natural or logically arbitrary disproportionate dependence, noted above, of the values of these parameters on the dimensions of the areas. In the diagram it is assumed that, when calculating the density of the exploration grid on large areas, one should proceed from the conditions of distribution over the interval of area increments \(S_{\mathrm{sec}}/S_{\mathrm{av}}\) within the limits from the 1st to the 2nd or from the 2nd to the 3rd sta-

Fig. 1. Principal schematic of the dependence of the calculated parameters of the exploration grid on site sizes according to the analytical methods of V. A. Prokin and B. S. Levonik (a), V. I. Smirnov and V. V. Bogatskii (b), and the proposed optimal variant (c)

...statistical sample of single exploration blocks with a volume of, respectively, \(1.5N\) (point 1) or \(2N\) (point 2).

For small sites, proceeding above all from economic considerations, it is proposed in such calculations to take the statistical sample of elementary blocks within the limits from \(N\) to \(\frac{3}{4}N\). We shall now specify the conditions for applying the optimal variant of determining the density of the exploration grid for different groups of deposits, with the aim of deriving the mathematical form of the correction coefficient \(\varepsilon\), which is a function of \((S_{\mathrm{uch}}/S_{\mathrm{sr}})\).

To characterize the average sizes of sites, a certain area \(S_{\mathrm{sr}}\), constant for each group of deposits, was adopted above; in fact, it should be regarded as the upper limit \(S_{\mathrm{sr}}^{\mathrm{upper}}\) of a certain series of average site sizes with an interval of areas from \(S_{\mathrm{sr}}^{\mathrm{lower}}\) to \(S_{\mathrm{sr}}^{\mathrm{upper}}\). Then, for any series of sites whose \(S_{\mathrm{uch}}^{\max}\) will lie within the limits from \(2S_{\mathrm{sr}}^{\mathrm{upper}}\) to \(3S_{\mathrm{sr}}^{\mathrm{upper}}\), the graphical behavior of the function \(N_{\mathrm{uch}} = f \cdot N\) on individual sites of the series must have the following general character (Fig. 2, a). The change in the value of the coefficient \(f\) on each given segment of this curve, with its constant angle of inclination to the abscissa axis, depends on the specific position of \(S_{\mathrm{uch}}\) in the intervals of the series from \(S_{\mathrm{sr}}^{\mathrm{upper}}\) to \(S_{\mathrm{uch}}^{\max}\) and from \(S_{\mathrm{uch}}^{\min}\) to \(S_{\mathrm{sr}}^{\mathrm{lower}}\), and is described by the following corresponding expressions:

\[ f_{\max}=\left(1+\frac{S_{\mathrm{uch}}-S_{\mathrm{sr}}^{\mathrm{upper}}}{S_{\mathrm{uch}}^{\max}-S_{\mathrm{sr}}^{\mathrm{upper}}}\right) \qquad \text{and} \tag{7} \]

\[ f_{\min}=\left[1-\left(\frac{S_{\mathrm{sr}}^{\mathrm{lower}}-S_{\mathrm{uch}}}{S_{\mathrm{sr}}^{\mathrm{lower}}-S_{\mathrm{uch}}^{\min}}\right)0.25\right]. \tag{8} \]

Fig. 2. Character of the behavior of the general optimal curve of the function \(N_{\mathrm{uch}}=f\cdot N\) (a), with approximate partial scales of variation of the areas of small, medium, and large sites for stratiform ore (c) and coal (b) deposits.

As applied to a series of areas of sites of each given group of deposits, \(f_{\max}\) and \(f_{\min}\) may vary depending on the specific values of \(S_{\mathrm{cp}}^{\mathrm{top}}\), \(S_{\mathrm{cp}}^{\mathrm{lower}}\), \(S_{\mathrm{uch}}^{\max}\), and \(S_{\mathrm{uch}}^{\min}\), as well as on their ratios (broad investigations in this direction are still necessary). For multilayer coal deposits and ore deposits of the first two groups (Smirnov, 1957), for example, such series may be represented in the following form (Fig. 2, b, c).

Thus, the number of unit exploration blocks determined on the basis of all analytical methods must be corrected, taking into account the relative sizes of the sites, by multiplying it by the differentiated coefficient \(f\). Then the formula recommended for this purpose, formula (4), will take the following final form:

\[ N_{A,B,C_{\mathrm{t}}}^{\mathrm{uch}}=\frac{300K}{\varepsilon_{A,B,C_{\mathrm{t}}}}\,f. \tag{9} \]

To determine the optimal distances between exploration workings, it is first necessary to calculate the optimal sizes of unit exploration blocks \(S_{A,B,C}^{\mathrm{bl}}\) according to the formula

\[ S_{A,B,C}^{\mathrm{bl}}=\frac{R_{A,B,C}\cdot S_{\mathrm{uch}}}{N_{A,B,C}^{\mathrm{uch}}}, \tag{10} \]

and then—the dimensions of the short \((b_{A,B,C_{\mathrm{t}}})\) and long \((l_{A,B,C})\) sides of them, taking into account the degree of anisotropy of the parameters and properties of the bodies of useful minerals for detailed exploration of the given site. This latter, very important requirement can be fulfilled on the basis of decomposing the value \(S_{A,B,C}^{\mathrm{bl}}\) into the factors \(b_{A,B,C}\) and \(l_{A,B,C}\), provided that the following relation is observed:

\[ K_{\text{fall}}:K_{\text{strike}}=n:m=B_{A,B,C_1}/b_{A,B,C}:Z_{A,B,C}/l_{A,B,C}, \tag{11} \]

where \(K_{\text{fall}}\) and \(K_{\text{strike}}\) are indices of variability of the least stable parameter of the body of the mineral deposit, respectively along the dip and along the strike (Petrov, 1963);

\(n\) and \(m\) are, respectively, the number of intersections or interpolation elements in the direction of dip and strike of the body;

\(B_{A,B,C_1}\) and \(Z_{A,B,C}\) are the dimensions of the body in its width (dip) and length, within the contour of each reserve category.

Since the quantities \(b_{A,B,C}\) and \(l_{A,B,C}\) are determined respectively by the expressions \(S^{\mathrm{bl}}_{A,B,C_1}/l_{A,B,C_1}\) and \(S^{\mathrm{bl}}_{A,B,C_1}/b_{A,B,C}\), after their arbitrary substitution into equation (11) the problem is practically reduced to determining one of the sides of the elementary block by the following formulas:

\[ b_{A,B,C_1}= \sqrt{\frac{K_{\text{strike}}\cdot S^{\mathrm{bl}}_{A,B,C}\cdot B_{A,B,C_1}} {K_{\text{fall}}\cdot Z_{A,B,C_1}}}, \qquad l_{A,B,C_1}= \sqrt{\frac{K_{\text{fall}}\cdot S^{\mathrm{bl}}_{A,B,C_1}\cdot Z_{A,B,C_1}} {K_{\text{strike}}\cdot B_{A,B,C_1}}}. \tag{12} \]

Obviously, on the basis of formulas (12), after substituting into them expressions (10) and (9), it would be possible to obtain generalized formulas for the direct calculation of optimal distances between boreholes. However, until the actual mathematical form of the relation between the variability index \(K\), the error \(\Sigma\), and the number of elementary blocks \(N\) entering formula (4) has been established, this would prove premature and would also go beyond the limits of the present communication.

As an example, Table 2 gives a calculation of the optimal density of a borehole grid as applied to simple and very complex coal-measure areas; as a prototype of the variability of coal seams in simple areas, the Chernogorsk deposit was adopted, the variability indices of the NP coal seams for which were taken from the work of V. V. Bogatskii (1963).

LITERATURE

1. V. V. Bogatskii. Mathematical Analysis of an Exploration Grid. Gosgeoltekhizdat, 1963.

2. I. D. Kogan. The influence of geological and other factors on the reliability of calculated reserves of mineral deposits. Materials of Meetings and Seminars. Session of the Expert Geological Council, State Geological Committee of the USSR, Moscow, 1965.

3. A. A. Petrov. Application of mathematical statistics to the solution of basic problems in the exploration of deposits. Soviet Geology, 1963, No. 9.

4. L. M. Petrovskii. On determining the rational density of an exploration grid at coal deposits. Proceedings of Higher Educational Institutions. Geology and Exploration, 1965, No. 10.

5. V. A. Prokin. Selection of a rational exploration method for the Sibaevskoe deposit. Collection of Materials on the Geology of Nonferrous, Rare, and Noble Metals. Central Scientific-Research Mining-Exploration Institute, issue 5, Moscow, 1961.

6. V. I. Smirnov. Geological Principles of Prospecting and Exploration of Ore Deposits. Moscow State University Press, 1957.

1. Questions of Economic Geology. Academy of Sciences of the USSR, 1963. ↩