PROBABILISTIC-ANALYTICAL METHOD FOR DETERMINING THE DENSITY OF AN EXPLORATION NETWORK

L. M. PETROVSKII

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

Of all the existing methods for determining the density of an exploration network, as is known, only analytical methods make it possible, on the basis of preliminary exploration data for a deposit, to determine the density required for the detailed exploration stage of individual areas. However, the widely known method of a prescribed error in reserve estimation [6] and the closely related probabilistic-statistical method of V. V. Bogatskii [1] are rarely applied in practice. The main reasons for this are:

1) the independence of the calculated number of intersections from the relative dimensions of the areas of detailed exploration;

2) the incorrect refraction, in the indicated methods, of the well-known premise concerning the primary importance of the number of observations at the exploration target: the calculated minimally guaranteed number of observations for the exploration of reserves of a given category is extended to the entire area of the target, and not to that part of it within the contour of the corresponding reserves.

The absence of officially adopted values of permissible errors in estimating reserves by different categories also restrains the broad application of analytical methods for determining the density of an exploration network. Meanwhile, the need for an objective analytical method for calculating the density of an exploration network—one that differentially takes into account the features of the geological structure of deposits, the character of their individual objects, and the tasks of exploration—in connection with the progressively increasing annual volume of geological exploration work and the necessity of strict planning, is becoming increasingly obvious.

Below, a universal probabilistic-analytical method is proposed for determining the density of a network of boreholes. It is based on determining a sufficient number of elementary interpolation objects (unit exploration blocks), ensuring the calculation ...

of the mean values of the attribute with a prescribed error over area-averaged sections of all deposits explored by drilling and mining-and-drilling systems. This so-called base number of elementary exploration blocks is then adjusted for each given object, depending on the degree of variability of the attribute and the area of the section.

In doing so, it must be borne in mind that the corresponding base number of boreholes may be equal to or greater than the number of elementary exploration blocks, depending on whether the block rests on one central borehole (according to the nearest-neighbor principle) or is outlined by four boreholes (node grids). The choice of one of these exploration grids for the analytical calculation of the number of boreholes, all other conditions being equal, depends on the degree of detail in the exploration of the deposit: for estimating reserves in categories $A$ and $B$, the contour of which is usually drawn along workings, the calculation of boreholes should be based on node grids (according to formula (5)); for estimating reserves in categories $C_{1,2}$, the contour of which is drawn on the basis of close or distant extrapolation, the calculation of boreholes should be based on grids of blocks constructed according to the nearest-neighbor rule.

The base number of blocks or boreholes is determined on the basis of the probabilistic method of investigation [2]. According to the latter, any random sample of uniformly distributed values of an attribute (measurements of thickness or determinations of grade) may be characterized by the occurrence of one of the following incompatible events (possible exploration outcomes):

event Д — the mean value of the attribute is determined with a greater number of negative deviations of the individual values of the attribute from the true mean;

event Е — the same, with a greater number of positive deviations;

event Ж — the same, with an equal number of positive and negative deviations.

The totality of all three incompatible events forms a complete group of events. Consequently, the sum of the probabilities $P$ of these events constitutes the total probability.

$$ P(\text{Д}) + P(\text{Е}) + P(\text{Ж}) = 1. $$

For objects with simple laws of variation of geological parameters, explored by drilling and mining-and-drilling systems, the problem under consideration is practically symmetric with respect to the numbers of positive and negative deviations of individual values from the true mean (in the absence of systematic error, taking into account logging data, in the determinations of individual values). Then $P(\text{Д}) = P(\text{Е})$, or $2P(\text{Д}) + P(\text{Ж}) = 1$, and the probability of the favorable event Ж that interests us is determined from the expression $P(\text{Ж}) = 1 - 2P(\text{Д})$.

The probabilities of the symmetric events Д and Е are calculated from the number of elementary events (members of the sample) and their probability, which is obviously equal to $1/2$. The total number of elementary events for the exploration grid of a section may be represented as the product of $n$ and $m$ interpolation elements between neighboring measurements of the attribute along two mutually perpendicular exploration directions. Then the probability of event Д may be calculated from the expression:

$$ P(\text{Д}) = \left(\frac{1}{2}\right)^n \left(\frac{1}{2}\right)^m Zn\, Zm, $$

where $Zn, Zm$ are the numbers of arrangements or combinations of elementary events along the two principal directions of the exploration grid, calculated by the formula for combinations [3].

The results of the calculation of the probabilities of a favorable outcome of exploration of areas (event Ж) for different numbers of elementary interpolation objects (unit exploration blocks) are given in Table 1. The table shows that estimating the accuracy of the computed mean

Table 1

value with a probability greater than 0.9 (with an error of less than 10%) causes a sharp increase in the number of exploration blocks. It is obvious that, from the standpoint of economic expediency, detailed exploration of geological objects should be limited to a probability of a reliable estimate of the parameters equal to 0.9. With this probability, in exploration practice, the reserves of the corresponding deposits are estimated as category A; the probabilities of estimating reserves of categories B and C₁, as a rule, are taken as equal to 0.8 and 0.6, respectively. Table 1 also shows that, depending on the shape of the area over which the exploration blocks are located, achieving a specified level of probabilistic estimation of the mean value of a parameter requires an unequal number of unit exploration blocks. A comparatively smaller number of them is required for the main part of rectangular and square areas, and the largest number is required for exploring the peripheral strip of mineral bodies at their outcrops beneath cover and for exploring zones adjacent to tectonic faults (Table 2).

Table 2

The base numbers of elementary exploration blocks given in Table 2 ensure calculation of the mean values of parameters with a specified error on geological objects explored by drilling and mining-and-drilling systems. The limiting variability of parameters on these objects usually does not exceed 100% in terms of the coefficient of variation \(V\) and, apparently, may be estimated by the value of the index of nonuniformity \(NP\), according to V. V. Bogatskii [1], within limits up to 8. These conditions are met by the first two groups of ore deposits [6], as well as by all coal deposits. Then the necessary number of elementary exploration blocks \(N^{1}_{A,B,C}\) at each of such deposits with a specific variability of the principal (least persistent) parameter must be calculated by the formulas

\[ N^{1}_{A,B,C}=N_{A,B,C}\,\frac{V}{100} \quad \text{or} \quad N^{1}_{A,B,C}=N_{A,B,C}\,\frac{NP}{8}. \tag{1} \]

The numbers of elementary exploration blocks \(N^{1}_{A,B,C}\) determined by formulas (1) guarantee calculation of the mean values of parameters with a specified error within the outline of each category of explored reserves. Then the density of the exploration grid for exploration of reserves of the corresponding category, determined by the sizes of the area of elementary exploration blocks \(S^{\mathrm{bl}}_{A,B,C}\), must be calculated by the formulas*:

\[ S^{\mathrm{bl}}_{A,B,C} = \frac{100S_{A,B,C}}{V\cdot N_{A,B,C}} \quad \text{or} \quad S^{\mathrm{bl}}_{A,B,C} = \frac{8S_{A,B,C}}{NP\cdot N_{A,B,C}}, \tag{2} \]

where \(S_{A,B,C}\) are the areas of the outlines of reserve exploration by categories \(A\), \(B\), and \(C_1\).

It follows from the formulas that, for a constant base number of elementary exploration blocks (Table 2), the density of the borehole grid for exploration of reserves of a given category depends, other conditions being equal, on the absolute area of the outlines of reserves of different categories. A possible different density of the borehole grid for different ratios of reserve categories (with a constant number of elementary exploration blocks and boreholes in the outline of each reserve category) will determine different reliability or accuracy of reserve delineation. Consequently, achievement of the optimal accuracy of delineating reserves of each category, with a specified number of elementary exploration blocks, must be determined by the optimal ratio of categories. The general requirements for the optimal ratio of reserve categories for different groups of deposits of solid minerals are contained in the Classification of Reserves [4]. Denoting the share of reserves in the outline of the corresponding category from the total reserves of an area of area \(S\) by \(K_{A,B,C}\), formulas (2) can be written in the form

\[ S^{\mathrm{bl}}_{A,B,C} = \frac{100K_{A,B,C}\cdot S}{V\cdot N_{A,B,C}} \quad \text{or} \quad S^{\mathrm{bl}}_{A,B,C} = \frac{8K_{A,B,C}\cdot S}{NP\cdot N_{A,B,C}}. \tag{2a} \]

Theoretical studies of the character of the dependence of the ratios \(S_A\) and \(S_B\) on \(S^{\mathrm{bl}}_A\) and \(S_B\), for different relative numbers \(N_A\) and \(N_B\), can be represented visually on a diagram (Fig. 1). It follows from the diagram that the required ratio \(S_A\) and \(S_B\) corresponds to definite combinations of \(N_{A,B}\) and \(S^{\mathrm{bl}}_{A,B}\).

The selection, entering into formulas (2a), of the necessary and sufficient base number of elementary exploration blocks \(N_{A,B,C}\) for areas of different sizes should be carried out differentially. Experimental studies conducted for areas of coal deposits—

* As more refined indicators of parameter variability are developed, these formulas may accordingly be easily transformed.

Fig. 1. Nature of the dependence of the relative sizes of elementary exploration blocks \((S_A^{\mathrm{bl}}\) and \(S_B^{\mathrm{bl}})\) on prescribed ratios of the areas of the reserve-contour categories A and B \((S_A\) and \(S_B)\) and the number of blocks \(N_{A,B}\). Schematic diagram for differentiating the density of the exploration grid for deposits (areas) of very simple (group I), complex (group II), and very complex (group III) structure

showed [5] that the number of elementary exploration blocks determined by analytical methods for the detailed exploration stage meets the optimal exploration requirements only for areas of average size (about \(15\ \text{km}^2\)). In small areas (\(6\ \text{km}^2\) and less), the necessary number of elementary exploration blocks should amount from the calculated number of blocks \(N^{1}_{A,B,C}\), taken as 100%, to 75%, whereas in large areas (up to \(40\ \text{km}^2\))—to 200%. For exploration targets intermediate in area, the necessary number of elementary exploration blocks may be determined by interpolation between the indicated limits.* From the data presented it is evident that exploration of small areas, other conditions being equal to those of medium and large areas, is of low effectiveness.

Denoting by \(N^{11}_{A,B,C}\) the optimal number of elementary exploration blocks, corrected in accordance with the specific variability of the principal attribute and the relative size of the area, we obtain the final formulas for determining their optimal sizes:

\[ S_A^{\mathrm{bl}}=\frac{K_A \cdot S}{N_A^{11}}, \qquad S_B^{\mathrm{bl}}=\frac{K_B \cdot S}{N_B^{11}}, \qquad S_C^{\mathrm{bl}}=\frac{K_C \cdot S}{N_C^{11}}. \tag{3} \]

For the practical determination of the optimal number of elementary exploration blocks \(N^{11}_{A,B,C}\), it is necessary to construct special nomograms. As an example, such nomograms have been constructed for areas—

* At ore deposits of the first two groups [6], the same limits of the necessary number of elementary exploration blocks may evidently be adopted for the following sizes of exploration targets: small—up to \(1\ \text{km}^2\), medium—\(1\)–\(6\ \text{km}^2\), and large—\(6\)–\(15\ \text{km}^2\).

Fig. 2. Nomograms for determining the optimal number of unit exploration blocks \(N_{A,B,C}\) at coal deposits (sites) of groups I (a), II (b), and III (c): 1 — sites small in area (lower boundary for medium sites and lower boundary for large sites under interpolation); 2 — sites medium in area; 3 — sites large in area.

of coal deposits on the basis of the above grouping of them by the sizes of objects of detailed exploration (Fig. 2). In this case the construction of the nomograms is limited along the ordinate axis by the value \(NP=4\), which in practice covers the maximum variability of the indices of coal quality and seam thickness at all coal deposits. The differentiation of the density of the borehole network in detailed exploration in areas differing in tectonic complexity is achieved by changing the ratios both between \(S_B\) and \(S_A\), and between \(N_B\) and \(N_C\) (Fig. 1). In accordance with this differentiation, for each of the three distinguished groups of coal deposits (areas)* the nomograms were constructed under the following conditions: group I (simple)—\(K_{A,B}=0.3\) and \(N_C:N_B:N_A\) as \(0.06:0.25:1\) (Fig. 2, a); group II (complex)—\(K_A=0.15\); \(K_B=0.4\) and \(N_C:N_B:N_A\) as \(0.12:0.5:1\) (Fig. 2, b); and group III (very complex)—\(K_B=0.0\); \(K_A=0.5\) and \(N_C:N_B:N_A\) as \(0.25:1:1\) (Fig. 2, c). The specified values of \(K_{A,B}\) and the ratios of the base numbers of unit exploration blocks ensure equally reliable accuracy in outlining reserves of the same category for each of the three groups of areas. The procedure for determining the values \(N^{11}_{A,B,C}\) from the nomograms is shown by the example of selecting \(N^{11}_{A}\) for areas with an area of 6, 12, and 24 \(\text{km}^2\) of group I (Fig. 2, a).

As an example, a calculation is given of the optimal density of the borehole network for areas of different tectonic groups of coal deposits and of different sizes (Table 3). To reduce the table and facilitate comparison of the data, the calculation for areas of all groups was carried out for a square borehole network and for one and the same maximum variability of the parameter (\(NP=2\)).

Table 3

Note: the calculated numbers of boreholes do not include additional boreholes and workings necessary for tracing coal seams at their outcrops beneath drifts and individual tectonic faults, and for carrying out special sampling.

* These three groups of coal deposits are distinguished with account taken of the changes and additions introduced into the Classification of Reserves [4], and of the proposals in the new draft “Instructions for the Application of the Classification of Reserves to Deposits of Coal and Oil Shale,” which is under development.

References

1. V. V. Bogatskii. Mathematical Analysis of the Exploration Network. Gosgeoltekhizdat, 1963.
2. E. S. Venttsel. Probability Theory. Nauka, 1964.
3. M. Ya. Vygodskii. Handbook of Elementary Mathematics. Gosizdat, 1959.
4. K. V. Mironov. Prospecting and Exploration of Coal Deposits. Nedra, 1966.
5. L. M. Petrovskii. On determining the rational density of an exploration network at coal deposits. Izvestiya Vuzov. Geology and Exploration, No. 10, 1965.
6. V. I. Smirnov. Geological Foundations of Prospecting and Exploration of Ore Deposits. Moscow State University Press, 1957.