PROCEEDINGS

of the Tomsk Technological Institute

of Emperor Nicholas II.

Vol. 14. 1909. No. 2.

IV.

S. Yu. Doborzhinsky.

SOME GENERAL RULES FOR CONDUCTING EXPLORATION WORK.

With a table of drawings.

1–59.

PREFACE.

The literary data concerning exploration work are quite scant, despite the fact that reports are almost always printed on more or less extensive explorations that have been carried out. In these reports the method of conducting the work is described from the technical and economic points of view; sections of boreholes and prospecting pits, geological sections, as well as geodetic and geological plans of the locality, are presented, together with final conclusions as to the reliability of the deposits.

Unfortunately, most of these works are characterized by a routine approach, which can perhaps be explained by the fact that explorations are usually entrusted to young persons, far from being specialists in the given branch, who therefore are compelled to adhere almost blindly to the individual instructions found in manuals of mining art and in reference books. Special literature—more extensive textbooks on exploration that give general rules for conducting these works, and not merely a description of them from a handicraft point of view—is almost entirely lacking not only in Russian, but also in other languages.

The reason why Western Europeans do not concern themselves with the theory of exploration is understandable: in countries of old culture, all deposits have already been explored in one way or another, chiefly thanks to the working of minerals in these deposits. As for the English and Americans working in new, virgin countries, one would of course expect more from them than they have provided up to the present time. The same may be said of Russian mining engineers.

Up to the present time we have had to make use either of personal experience, or of the instructions contained in the works of S. Voislav: “Exploration of Useful Minerals” and “Investigation of the Ground and Exploration of Useful Minerals by Means of the Hand Auger.” I am far from cri-

nevertheless, I find that they require certain additions.

The aim of the present work of mine consists precisely in a more general and more rigorous formulation of the rules and principles governing exploration works. I cannot suppose that all the questions raised by life have been examined and resolved by me; no! my modest labor is only the first steps on the path toward that for which one must strive!

I consider it necessary to note that, since my work is in some respects a supplement to Wojsław’s work, I have avoided repeating everything that can be found in the above-mentioned works of that author; moreover, I consider it necessary to say that almost all the technical indications that can be found in the proposed work and that do not follow directly from theoretical reasoning have been taken by me from the practice of exploration work—my own or that of other persons—in cases when I had the opportunity to observe the works and their final results. I did not make direct use of the periodical literature on exploration, that is, I borrowed from it neither facts nor views, although I cannot assert that one or another of the ideas expressed by me did not arise under the influence of an article I had read.

The elementary mathematical investigation of exploration data and methods of work, as well as the conclusions drawn on the basis of these investigations, belong to the author of the present work. I cite very few numerical data, chiefly because in most cases there are none at all; for example, numerical data concerning the accuracy of determining the thickness of the drilled rocks as a function of the drilling system are wholly absent; in those cases where the required numerical data can be found in handbooks, I have considered it superfluous to repeat them, in view of the above-mentioned character of the present work, which is only a certain supplement to the existing literature, but in no case a course in exploration work—rather, an aid, a source of material for one.

INTRODUCTION.

The general rules for carrying out exploration work follow directly from the laws of geology; however, in view of the extremely varied geological structure of the districts containing mineral resources, and also in view of the numerous types of ore deposits, the methods of exploration must be applied to each individual case accordingly and modified; so that the director of the work almost always has a broad field for personal initiative, for the application of his own powers of observation, resourcefulness, and scientific knowledge.

The purpose of exploration is to determine the reserve of a mineral in a given locality, field, and also its reliability in the sense of the presumed profitability of the future mining enterprise based on the area being explored. We shall consider only the first question, since the second is connected not only with the art of mining, but also with the economic and financial conditions at the given moment.

Thus, determining the reserve of a given mineral on a given area is reduced to two tasks: to determining 1) the volume of the deposit, bed, vein, etc., and 2) the percentage content in it of the useful mineral.

Determining the volume is nothing other than determining the geometric form of the deposit by means of a sufficient number of measurements. The percentage yield is the determination of the composition from samples taken from various places of the deposit, naturally—as far as possible—correspondingly and systematically distributed within it.

For both purposes serve, as is generally known, boreholes, shafts, exploration mines, adits, and drifts; on the appropriate distribution of these, as well as on their number, depends the more or less accurate final result of the work.

VI

Exploration is always constrained in one respect: by funds; there are almost no explorations with which the owner of a deposit, or in general the person for whose benefit it is being carried out, would in the end remain completely satisfied. If the explorations have yielded a negative result, then, of course, the money has been spent in vain, and therefore it turns out that it was wasted, and, of course, the engineer is to blame; but even in the case of the most brilliant outcome, there usually remains some dissatisfaction with excessively high expenditures, since the reflection appears quite natural that, had development been undertaken even without these expenditures, the result of the enterprise would have been favorable.

In view of this, the question of the economical conduct of work is a matter of primary importance, and therefore it is extremely important that not only the person carrying out the work be able clearly to account to himself for the extent to which each of the works undertaken separately affects the accuracy of the final conclusion, but also be able to justify his actions before another, less knowledgeable person.

Unfortunately, geological data on the basis of which one can directly justify one’s assumptions are by no means always available; this is possible only in the case when a fairly regular deposit is being explored, and in a locality adjacent to one where the same deposits are being worked; but in the case of exploration of deposits whose geological character has not yet been studied exactly, as well as in the cases of pocket-like or stockwork deposits, one very often has to act by feel; for these cases it is not superfluous to make use of the mathematical theory of probability.

Chapter One.

Exploration of Stratified Deposits.

On the exploration of stratified deposits in general.

The exploration of stratified, and consequently chiefly coal-bearing, deposits differs in that the methods for carrying it out are more highly developed, in view, first, of the evident regularity of these deposits, and second, because, owing to the enormous development of the modern coal business, such exploration occurs more often than others. Let us distinguish the following cases: 1) inclined beds of a useful mineral have outcrops at the surface, or are covered only by a thin layer of alluvium, which is not found everywhere within the area being explored; 2) gently dipping beds are covered by a thick layer of newer formations; 3) steeply dipping beds lie beneath a thick layer of barren rocks.

Since the first of these cases has been examined in sufficient detail in Voislav’s work Exploration of Useful Minerals, and the second—partly, this gives me the opportunity not to dwell on many details.

In addition to the subdivision just indicated, exploration may also be subdivided into: 1) exploration of parts of deposits little studied or not studied at all; and 2) exploration of deposits studied in detail, that is, those for which there exist general stratigraphic maps and detailed geological descriptions.

Outcrops at the surface exist.

In this case the explorations have, as it were, been made by nature itself, and they need only be verified by means of exploration trenches, comparing step by step the artificial section obtained with the stratigraphic map—of course, if such a map exists—by measuring the distance of each bed encountered from a reference bed that is quite clearly observed on the ground and plotted on the stratigraphic map—

Thus, such a marker bed is either a bed of the useful mineral, distinguished by its thickness or by the characteristic properties of its floor and roof, or sometimes a bed of barren rock sharply different from the surrounding rocks, and so forth—for example, beds of limestone in the coal formation in southern Russia.

The discrepancy between determinations made from maps and by direct measurement is checked in other exploratory trenches. If the upper layers of the rocks are greatly disturbed, so that there is doubt as to how to determine the dimensions and angle of dip of the bed, and sometimes also in order not to spoil the surface of the land, one resorts to shurfing or to hand drilling, despite the absence of overburden. In this case, as is known, the fundamental principle is the rule that each bed should be penetrated twice by two adjacent exploratory workings: once at the outcrop, and the second time at depth.

On the basis of the data obtained in this way, sections of the deposit are drawn along the exploratory lines, and the horizons of the beds are entered on the situational plan from the sections.

If no general bed map exists, then, before beginning systematic exploration, the outcrops of the beds should be marked on the situational plan, using small excavations in those places where the outcrops are visible but not clear. In such preliminary investigations, traces of old peasant workings are sometimes of considerable use. After the dip and strike have been determined in several places, the selection of exploratory lines is undertaken, and then the course of the work is almost the same as that described above.

Calculation of the Reserve of a Stratified Deposit

Exploration ultimately brings us to the point where we have the horizontal extent of each bed, measured on the bed plan; the inclined length, whose horizontal projection is marked on that plan; and the thickness of the bed. From these data it is very simple to calculate the volume of the bed, and consequently its weight content, by multiplying the volume by the yield per cubic unit¹.

The totality of such calculations gives us the total reserve of the mineral enclosed within the bowels of the earth in the area being explored.

Inaccuracies in the general result of the calculation may arise from—

of three causes: from errors of measurement in the geodetic part of the work, which ultimately affect the length and form of the horizontal crooked line of extension of the seams; and from the inaccuracy of determining the angles of dip of the seam and its thickness, obtained either by direct measurement in prospecting trenches and shafts on the exposed planes of bedding, or else computed from the depths at which the seam is intersected by boreholes; and, finally, from failure to take into account unconformities in stratification. Thus, if the horizontal length of the line of extension is erroneously determined as \(S+\Delta S\) instead of \(S\), the inclined length as \(L+\Delta L\) instead of \(L\), and the thickness as \(d+\Delta d\) instead of \(d\), where \(\Delta S\), \(\Delta L\), and \(\Delta d\) may have either positive or negative values, the error in computing the reserve will be

\[ \Delta O=(S+\Delta S)(L+\Delta L)(d+\Delta d)-SLd. \]

The relative error \(\dfrac{\Delta O}{O}\), if we neglect the comparatively small quantities

\[ \frac{\Delta S\Delta d}{Sd},\quad \frac{\Delta L\Delta d}{Ld},\quad \frac{\Delta L\Delta S\Delta d}{LSd}, \]

will be equal to

\[ \frac{\Delta O}{O}=\frac{\Delta S}{S}+\frac{\Delta L}{L}+\frac{\Delta d}{d}, \tag{1} \]

that is, to the algebraic sum of the errors of the three measurements.

The error \(\dfrac{\Delta S}{S}\) is determined chiefly by the accuracy of the plan, \(\dfrac{\Delta d}{d}\) by the accuracy of borehole or direct measurements. In the present chapter we shall deal only with \(\dfrac{\Delta L}{L}\), since the accuracy of drilling determinations will have to be discussed later, as will errors in determining the strike.

If the angle of dip of the seam is \(\eta\), and \(\Delta \eta\) is the error in its determination, then the magnitude of the error in its inclined length may be determined depending on whether we compute the reserve of the mineral only to some depth \(h\), or, without limiting ourselves by depth, only within a certain horizontal area. The first restriction is rather artificial and is significant only in determining the reserve of a mineral for extraction by mines at a presumed depth.

Thus, if \(h\) is the depth to which it is proposed to carry out pros—

…working, the inclined length will be: \(L=\dfrac{h}{\sin\eta}\); an inaccuracy in determining the dip angle will be reflected in it as follows:

\[ L+\Delta L=\frac{h}{\sin(\eta+\Delta\eta)}, \]

and the error is

\[ \Delta L=h\left[\frac{1}{\sin(\eta+\Delta\eta)}-\frac{1}{\sin\eta}\right]. \]

The relative error is

\[ \frac{\Delta L}{L}=-\frac{\Delta\eta}{\operatorname{tg}\eta+\Delta\eta}, \tag{2} \]

if we take \(\cos\Delta\eta=1\), and \(\sin\Delta\eta=\Delta\eta\).

Formula (2) shows that an error from an incorrect measurement of the dip angle, in the case of steeply dipping seams, has little effect on the determination of the inclined length \(L\), since when \(\eta=90^\circ\), \(\operatorname{tg}\eta=\infty\) and \(\dfrac{\Delta L}{L}=0\); conversely, for gently dipping seams an error of this kind can have a very strong effect on the result of the calculations, since the limit of the expression \(\dfrac{\Delta L}{L}\) as \(\eta=0\) approaches 1.

We shall explain our conclusion by drawings I and II.

In drawing II it is clearly seen that the part of the seam \(CD\), included in the reserve calculation down to the horizon \(XY\), may either decrease to the size \(C'D\), or increase to \(C''D\).

In view of what has been said, in the case of gently dipping seams it is desirable to determine their dip angles accurately, and therefore one should not confine oneself to investigating the outcrops of the seams, but should drill several additional, deeper boreholes, placed at a considerable distance from one another, which, in view of the shallow occurrence of the seams, will not entail large expenditures. It is necessary to note, however, that the error in determining the coal reserve resulting from an inaccurate measurement of the seam dip angle, in the case analyzed above, will in practice have only the consequence that the mine will have to be deepened somewhat more than was assumed, or that its deepening will be stopped at a somewhat higher horizon; neither the one nor the other can have any substantially harmful effect on the course of the enterprise.

If the entire reserve of a given area is calculated according to its horizontal area, then the boundary of the seam must be considered the vertical surface passing through the boundary of the area.

Thus, if the distance of the outcrops of the seam to the boundary of the tract in the direction of the dip of the seam is \(e\), then

\[ L=\frac{e}{\cos \eta},\quad L+\Delta L=\frac{e}{\cos(\eta+\Delta\eta)}, \]

and consequently (Fig. III)

\[ \frac{\Delta L}{L} = \left( \frac{1}{\cos(\eta+\Delta\eta)}-\frac{1}{\cos\eta} \right) : \frac{1}{\cos\eta} = \frac{\Delta\eta}{\cot\eta-\Delta\eta} \tag{3} \]

under the assumption that \(\Delta\eta\) is so small that \(\cos\Delta\eta=1\), and \(\sin\Delta\eta=\Delta\eta\).

Expression (3) shows that an error in \(\eta\), for values of it close to 0, has little effect on the final result. For values \(\eta=90^\circ\),

\[ \frac{\Delta L}{L}=-1, \]

that is, it attains its maximum magnitude, which is, of course, understandable, since in this case a considerable part of the seam may be falsely included in the calculation or incorrectly excluded from it. For mean values of \(\eta\), about \(45^\circ\), the error will approach

\[ \frac{\Delta L}{L}=\frac{\Delta\eta}{1-\Delta\eta}=\Delta\eta, \]

from which we conclude that the accuracy of measuring the angle of dip of seams in the case under consideration has the most substantial significance at very large values of \(\eta\).

It is self-evident that, in the case of several seams, the sum of the errors in determining the reserve of each of them will give us the total error in calculating the entire reserve. As regards the relative error in the calculation of the reserve of a given field, it may be taken as the arithmetic mean error from all the seams, or, more precisely, calculated by

\[ \frac{\Delta o}{o}=\frac{\Sigma\Delta o}{\Sigma o} \]

where \(o, o_1,\ldots\) are the volumes of each seam separately, and \(\Delta o,\ldots\) are the errors in their determination.

Everything said up to this point applies to seams that are regularly inclined, even if interrupted by faults. For folded deposits the determination of the influence of errors on the calculation of the reserve becomes more complicated. Thus, in explorations across the strike, along the heads of seams of folded structure, only the upper ...

elements of the seams, by which the whole section is drawn on the basis of the parallelism of the seams and, consequently, of the constancy of their distances from one another. The sections are drawn by hand; in doing so, the centers obtained by the intersection of perpendiculars restored from adjacent elements of the heads of the seams may serve as a certain aid.

From drawing IV the following is easily seen: for example, the 4th seam is to be drawn in such a way that the head part \(AB\) is determined with its own error \(\Delta\eta_4\); the next part \(BC\), taken parallel to the 3rd seam, with the error of that seam \(\Delta\eta_3\); the part \(CD\)—\(\Delta\eta_2\); \(DE\)—\(\Delta\eta_1\); in general, each succeeding seam is drawn with all the errors of the overlying seams plus its own. In view of this, one should not confine oneself exclusively to explorations along the heads of the seams, but should deepen several more boreholes, if possible in the gently dipping part of the fold, for a more reliable clarification of the disposition of the seams.

To determine the influence of the errors \(\Delta\eta_1, \Delta\eta_2, \ldots\) on the inclined length of some \(n\)-th seam, we compute the errors of the length of each of the corresponding segments of this seam by formula (3). If we denote \(AB=l_4^n\), \(BC=l_3^n\), and so on, then

\[ \Delta l_4^n = l_4^n \cdot \frac{\Delta\eta_4}{\operatorname{cot}\eta_4-\Delta\eta_4}, \qquad \Delta l_3^n = l_3^n \frac{\Delta\eta_3}{\operatorname{cot}\eta_3-\Delta\eta_3}; \]

and, consequently, in general,

\[ \frac{\Delta L}{L} = \frac{1}{L} \sum_1^n l_1^n \frac{\Delta\eta_1}{\operatorname{cot}\eta_1-\Delta\eta_1}. \tag{4} \]

Formula (4) applies to the case of determining the reserve in the field of a given area. In the case of a specified depth, one may use:

\[ \frac{\Delta L}{L} = - \sum_1^n l_1^n \frac{\Delta\eta_1}{\operatorname{tg}\eta_1+\Delta\eta_1}. \]

Sometimes it is necessary to determine the reserve of a mineral in a given field and down to a given depth; and then, in order to clarify the errors of the calculation, one should apply formulas (3) and (2), according to which of the bounding planes intersects the given seam.

Discordance of Stratification

The fundamental principle in compiling sections of stratified deposits is their parallelism; however, strictly speaking, the parallelism of the coal seams chiefly of interest to us is a fiction.

The successive appearance of beds of coal, clay shales, and sandstones is the result of a rhythmic (relatively, of course) uplift and subsidence of the earth’s crust. A coal seam corresponds to the minimum of motion at the moment of the end of uplift and the beginning of subsidence. Between two adjacent coal seams, somewhere midway between them, there must exist formations corresponding to the opposite negative zero moment of motion. These are the principal rhythms, but there may also be secondary ones. Since the amplitudes of such oscillations cannot be equal, for the probability of such a case is very close to zero; and since, during the period of formation of coal seams, there must also have existed a general lowering of the continent, for new strata were being formed upon it; and, finally, since neither this general motion nor the partial oscillations in horizontal extent can be entirely uniform, for the probability of this is again zero, the horizontal plane of the new deposits must always intersect the old ones at some angle. These angles are generally small. The discordance of stratification may compensate itself mutually. Between two separate suites of beds the discordance may be fairly large, amounting to several degrees.

If we neglect the secondary oscillations and consider only the undulations of motion whose traces remain in the form of coal seams, then we find that the lower neutral axis, lying in the barren rock, as well as the upper neutral axes in the coal, are somewhat inclined to the plane of contact of the roof and floor with the seam.

Let \(XY\) be the neutral axis (see drawing V) between two seams; \(D_1\) and \(D_2\) two sections made at a distance \(L\) perpendicular to the axis; \(\Delta\alpha_1, \Delta\alpha_2\) the angles of inclination of the axis to the coal seams; then

\[ d_1 = L \operatorname{tg}\Delta\alpha_1,\quad d_2 = L \operatorname{tg}\Delta\alpha_2; \]

and since \(d_1 + d_2 = D_2 - D_1\), it follows that

\[ D_2 - D_1 = L(\operatorname{tg}\Delta\alpha_1 + \operatorname{tg}\Delta\alpha_2). \]

If, however, we replace the sum of tangents by its expression in terms of the functions \(\sin\) and \(\cos\), we obtain

\[ \frac{D_2 - D_1}{L} = \frac{\sin(\Delta\alpha_1 + \Delta\alpha_2)} {\cos\Delta\alpha_1\cos\Delta\alpha_2}. \]

But \(\cos\Delta\alpha_1\), as well as \(\cos\Delta\alpha_2\), because of the smallness of these angles, may be taken as equal to 1; if, moreover, we denote \(\Delta\alpha_1 + \Delta\alpha_2 = \Delta\alpha\), then finally

\[ \sin \Delta \alpha = \frac{D_{2}-D_{1}}{L}, \tag{5} \]

and in the case of complete pinching out, when \(D_{1}=0\),

\[ \sin \Delta \alpha = \frac{D}{L}. \]

Of course, our formula, derived for barren rock between two coal seams, is also valid for the seams themselves. To give numerical values of \(\Delta \alpha\) is rather difficult: this angle differs for different basins. However, each group, each suite of seams can be characterized by its own fairly constant angle of pinching out. For the Reden group of seams of the Silesian–Dąbrowa basin (Sattelflötze), this angle is about \(5'\) for the shale interbeds between the coal seams of this suite. If it is desirable to verify the calculation of the coal reserve with allowance for the unconformity of stratification, then, in order to obtain numerical data, reference must be made to the workings in the part of the deposit under development.

Since in general the maximum errors from failure to take into account the unconformity of stratification will occur when all the seams thin in one direction (a rather frequent phenomenon), then, having set \(\Delta \alpha\) corresponding to the suite of a given deposit, we draw the deviating seams with the aid of formula 5). It is clear that the deviation of each subsequent seam from its original position increases (Drawing VI). If the direction of pinching out is precisely known, then the above-mentioned correction should be made in the pinching-out of the section itself; but if there are no data for indicating in which direction the pinching out takes place, then in such a case the most probable section will be the section constructed on the basis of the principle of parallelism (Drawing IV).

In any case, however small the errors caused by failure to take into account the unconformity of stratification may be, they should nevertheless be avoided, if possible; or, at any rate, having determined their possible magnitude, one should consciously accept neglecting them.

Gently dipping seams are covered by a thick layer of more recent formations.

Explorations of the type to which we now turn are usually the result of the desire to expand the area occupied by mining operations in districts where extraction began

by the outcrops of the beds directly at the surface; thus general information characterizing the deposit is usually available—namely, the general direction of strike is known, as are the dip and its magnitude, and also what suites may be expected among the beds being worked. Cases of prospecting wholly independent coal basins, discovered beneath more recent formations by means of drilling undertaken either for some industrial purpose or for geological investigations, are not known to me, although there are facts of the discovery of coal seams beneath a thick layer of drift deposits which so far cannot be connected with those already known. In such a case prospecting has to be carried out without the use of any a priori data.

The principal method of carrying out such prospecting is the drilling of deep boreholes that penetrate as many beds as possible. It is self-evident that only a small number of such boreholes can be laid down, owing to their great cost.

On the situation plan of the tract being explored, the horizons of the presumed principal beds and the lines of dislocations are drawn in pencil, as continuations of those already known. The boreholes are placed so that they will test not only our assumptions with regard to the beds, but also the direction of the dislocations; however, boreholes should be laid down at such a distance from the dislocations that they do not fall into them. During drilling, and when the borehole has already been finally completed, an attempt is made to establish the relation of the rocks passed through to those known from the profiles of mines in the part of the basin being worked; for this purpose, not only is a drilling log kept, but, in addition, each borehole is plotted separately. Sometimes, for clarity, the drill cuttings are placed in glass tubes, filling them to scale, so that a kind of natural profile is obtained.

In most cases it is possible to determine the beds passed through by the lithological properties of the beds themselves, as well as of the rocks to which they are subordinate, and likewise by certain characteristic features of the given group or suite in general. In exceptional cases recourse is had to paleontological investigations. After the first boreholes have been completed, new ones are laid down at points determined in part by the progress of the work, in order to clarify unclear stratigraphic relations. Then, from the data obtained, an attempt is made to plot transverse sections (across the strike) of the area being explored, which, however, often proves impossible; in that case additional boreholes are laid down, after which the sections are finally plotted.

The plane of a fault may also intersect none of the drilled boreholes, which are then projected onto it. The faults of which we shall speak play the role of rough drafts; otherwise they might perhaps be called faults of a first approximate accuracy; they are drawn only in pencil. Then, using the said faults, we mark on the plan, in some one plane, the direction of the level lines, which in such a case play the role of a first approximation. With such preliminary processing of the material obtained by exploration, it sometimes becomes clear that it is necessary to lay down yet another borehole; if, however, all the stratigraphic relations are completely clear, then one proceeds to the final drawing of the faults along lines which may not coincide with the original ones, but are carried, as far as possible, perpendicular to the level lines of the beds. In this, the boreholes are projected along lines parallel to the level lines. This procedure, however, gives not entirely accurate results, and therefore auxiliary sections between two neighboring boreholes are often used.

Thus, if \(XY\) is the line of a fault passing between the boreholes \(ABCD\) (see drawing VI), then the auxiliary sections along \(AB\), \(CD\), \(DE,\ldots\) give us the possibility of determining with sufficient accuracy the sequence of bedding at the points of their intersection with the line \(XY\), namely at \(a\), \(b\), \(c\), which serve as the basis for plotting the whole fault. Then we plot the level lines in their final form.

If, during the drawing of transverse sections or level lines and in comparing two neighboring boreholes, it turns out that it is impossible to establish a gradual transition between them with normal or almost normal dip of the beds, then between such two boreholes one must assume the presence of a displacement, the vertical magnitude of which is determined by the position in both boreholes of one and the same bed. The inclinations of the parts broken by displacements should, as far as possible, be determined separately for each part. But since this is sometimes rather difficult, equal inclinations are usually adopted for these parts; the dip is taken as variable only when there are data for such an assumption.

Thus, for example, if in the segment \(AB\) the position of the bed is \(ab\), determined by boreholes I, II; and in the segment \(ED\) it is \(ed\), then in the intermediate segment, for which we have no sufficient direct determinations, we adopt the mean inclination (drawing VII).

The calculation of the reserve is made from the bed map and sections compiled by the indicated methods; in this, the areas delimited by the lines of displacements are considered separately. It is self-evident that the part of the coal directly adjoining the displa—

mi, must be considered unsuitable. Usually, in large dislocations and especially in overthrusts, this part is greater than in small ones, so that we may say the following: the width of the crumpled part near the line of displacement is 10 meters for dislocations up to 25 m in the vertical direction. For larger dislocations this crumpled space increases to 100 meters, and sometimes in overthrusts bands devoid of coal suitable for working are observed as much as 500 m wide.

The inaccuracy in determining the coal reserve in the case we are considering depends, just as in the preceding one, on the inaccuracy in determining the dip and strike of the seams and, moreover, on the insufficient study of the character of the dislocations and their directions. Since nothing strictly definite can be said with regard to dislocations, we shall leave this question aside, drawing the reader’s attention to the fact that sometimes there are cases when, in order to determine the extent of a dislocation, special boreholes have to be drilled.

The relative error in calculating the reserve is given by formula (1), in which, however, the size of the horizons \(S\) depends not only on the accuracy of the plan—that is, on the more or less exact determination of the boundaries of the field accepted for the calculation—but also on the accuracy of the data on the basis of which we have determined their position.

Therefore, without dwelling on the errors that may arise from the inaccuracy of the plan itself—i.e., from geodetic errors—we shall pass directly to the consideration of inaccuracies arising from drilling determinations.

Strike and dip determined from drilling data

Strike and dip may be regarded as a function of three depths determining the position of the seam at the given place, irrespective of whether these depths have been determined directly or obtained by calculation from neighboring points by means of auxiliary sections.

If \(\xi\) is the azimuth of the dip of the seam, \(\eta\) its angle of dip, \(\alpha\) and \(\beta\) the azimuths of the lines connecting the deeper boreholes with the shallower ones, \(l_1\) and \(l_2\) the distances between these boreholes, and the differences of their depths are \(h_1-h_0\) and \(h_2-h_0\), then¹

\[ \operatorname{tg}(\xi-\beta)=\operatorname{cot}(\beta-\alpha)-\frac{h_1-h_0}{h_2-h_0}\cdot\frac{l_2}{l_1}\cdot\frac{1}{\sin(\beta-\alpha)}, \tag{6} \]

\[ \operatorname{tg}\gamma=\frac{h_2-h_0}{l_2}\cdot \frac{1}{\cos(\xi-\beta)} . \tag{7} \]

Since neither the roof nor the floor of a bed is a plane, but rather some wave-like surface, the strike determined from three boreholes, just like the dip, is not, strictly speaking, the strike of the bed in the given space, but only the strike of the plane passing through the points of intersection of the boreholes with the bed. In view of this, theoretically, in order to determine the strike of the bed at a given point, the boreholes should be brought close to that point so that the horizontal distances between them are very small; then formula (6), for very small \(l_1\) and \(l_2\), will give us the direction of the horizontal tangent to the plane of the bed at the given point, and consequently its strictly, mathematically understood strike.

Of course, in practice there is a limit to bringing boreholes closer together, determined by the consideration that, with small differences in depths, owing to the inaccuracy of drilling data, wholly incorrect results of calculation may be obtained; moreover, we require from the boreholes answers to other questions as well, and not only the determination of the strike strictly at a given point. Nevertheless, the reasoning given above shows us that, in the case of folded structure, this condition must be taken into account, and boreholes intended to determine strike and dip must be laid out at such distances from one another that, in the space occupied by them, the roof or floor of the bed could be regarded as a plane.

Thus, in order to understand the strike and dip determined by drilling works with precision, we shall divide the whole field into small areas in which the horizontals of the bed may be considered straight lines (Drawing VIII).

Errors in determining the dip of the bed are reflected in the spacing of the horizontals; their magnitude is given by formulas (3) and (4).

As for the error due to an inaccurate determination of the strike \(\xi-90^\circ\), calling it \(\Delta \xi\), from the triangle \(ABC\) (Drawing IX), in which \(AB\) is the true length \(S\), \(AC\) the erroneously determined \(S+\Delta S\), and, finally, the angle \(\angle BCA=\delta\), we obtain

\[ \frac{AB}{\sin(\delta+\Delta\eta)}=\frac{AC}{\sin(\delta+\Delta\xi)} \]

or, otherwise,

\[ S=(S+\Delta S)\frac{\sin\delta}{\sin(\delta+\Delta\xi)} . \]

If we replace \(\cos \Delta \xi = 1\), and \(\sin \Delta \xi = \Delta \xi\), which is quite permissible in view of the smallness of the angle \(\Delta \xi\), the equality

\[ \frac{\Delta S}{S} = \left[ 1-\frac{\sin \delta}{\sin(\delta+\Delta \xi)} \right] : \frac{\sin \delta}{\sin(\delta+\Delta \xi)} \]

will finally become

\[ \frac{\Delta S}{S}=\cot \delta.\Delta \xi . \tag{8} \]

Thus it is necessary to find the value of \(\Delta \xi\), expressed as a function of the three depths and their distances. Let us turn to eq. (6); since

\[ \operatorname{tg}\Delta \xi = \operatorname{tg}\bigl[(\xi+\Delta \xi-\beta)-(\xi-\beta)\bigr] = \frac{\operatorname{tg}(\xi+\Delta \xi-\beta)-\operatorname{tg}(\xi-\beta)} {1+\operatorname{tg}(\xi+\Delta \xi-\beta)\operatorname{tg}(\xi-\beta)} \]

and, moreover,

\[ \operatorname{tg}(\xi+\Delta \xi-\beta) = \cos(\beta-\alpha) - \frac{h_1+\Delta h_1-h_0-\Delta h_0}{h_2+\Delta h_2-h_0-\Delta h_0} \cdot \frac{l_2}{l_1} \cdot \frac{1}{\sin(\beta-\alpha)}, \]

then, denoting for brevity \(\beta-\alpha=\gamma\), \(h_2-h_0=y\), \(h_1-h_0=x\),

\[ \Delta y=(h_2+\Delta h_2)-(h_0+\Delta h_0)-(h_2-h_0), \]

\[ \Delta x=(h_1+\Delta h_1)-(h_0+\Delta h_0)-(h_1-h_0), \]

for \(\operatorname{tg}\Delta \xi\) we obtain the expression:

\[ \operatorname{tg}\Delta \xi = \frac{ \left( \frac{x+\Delta x}{y+\Delta y}-\frac{x}{y} \right) \frac{l_2}{l_1}\cdot\frac{1}{\sin\gamma} }{ 1+ \left( \cot\gamma-\frac{x+\Delta x}{y+\Delta y}\frac{l_2}{l_1}\cdot\frac{1}{\sin\gamma} \right) \left( \cot\gamma-\frac{x}{y}\frac{l_2}{l_1}\cdot\frac{1}{\sin\gamma} \right) }. \]

And if we discard \(\Delta x\) and \(\Delta y\) in the denominator, as having little influence on the value of the fraction after multiplication, then we find:

\[ \operatorname{tg}\Delta \xi = \frac{ \frac{l_2}{l_1}\,\Delta \frac{x}{y}\,\sin\gamma }{ 1-2\frac{x}{y}\frac{l_2}{l_1}\cos\gamma + \frac{x^2}{y^2}\frac{l_2^2}{l_1^2} } = \Delta \xi, \tag{9} \]

since, in view of the smallness of \(\Delta \xi\), we have the right to take the approximate equality \(\operatorname{tg}\Delta \xi=\Delta \xi\).

If now we substitute into the expression for \(\frac{\Delta S}{S}\) the found value of \(\Delta \xi\) and reintroduce the previous notation, then

\[ \frac{\Delta S}{S} = \frac{ \frac{l_2}{l_1}\, \Delta\left(\frac{h_1-h_0}{h_2-h_0}\right) \sin(\beta-\alpha) }{ 1 - 2\frac{h_1-h_0}{h_2-h_0}\cdot\frac{l_2}{l_1}\cos(\beta-\alpha) + \left(\frac{h_1-h_0}{h_2-h_0}\right)^2 \frac{l_2^2}{l_1^2} } \cot\delta . \tag{10} \]

This expression shows that constant errors, for example those arising from measuring the drill-rod length with an inaccurate tape measure, do not play a significant role in the accuracy of determining the strike of the seam from three boreholes. Indeed, if the error is proportional to the depth, so that \(\Delta h_2=h_2\Delta h,\ \Delta h_1=h_1\Delta h,\ \Delta h_0=h_0\Delta h\), then

\[ \Delta\left(\frac{h_2-h_0}{h_1-h_0}\right) = \Delta h \frac{(h_1-h_0)(h_2-h_0)-(h_2-h_0)(h_1-h_0)} {(h_1-h_0)^2} =0. \]

From this we conclude that the method of measuring the depths of boreholes must, irrespective of other conditions, be strictly the same in all exploration work, and the measurement and record should, as far as possible, be made by the same person.

To determine the influence of the ratio of the quantities entering into equation (10) on the magnitude of the relative error, let us investigate this equation, for brevity denoting

\[ \frac{l_2}{l_1}=n,\quad \frac{h_1-h_0}{h_2-h_0}=v \]

and, finally,

\[ \tg\delta\,\frac{\Delta S}{S}=Z, \]

so that

\[ Z=\frac{n\Delta v\sin\gamma}{1-2nv\cos\gamma+n^2v^2}. \]

Let us differentiate the expression for \(Z\) with respect to \(\gamma\); then

\[ \frac{dZ}{d\gamma} = \frac{ n\Delta v\cos\gamma(1-2nv\cos\gamma+n^2v^2) - n\Delta v\cdot 2nv\sin^2\gamma } {(1-2nv\cos\gamma+n^2v^2)^2}. \]

Since the second derivative is negative, \(Z\) has a max.; therefore, if

\[ \frac{dZ}{d\gamma}=0, \]

then

\[ \cos\gamma(1-2nv\cos\gamma+n^2v^2)-2nv\sin^2\gamma=0; \]

whence we obtain that for max. \(Z\) it is required that

\[ \cos\gamma=\frac{2nv}{1+n^2v^2}; \]

but since it is important for us that \(Z\) be as small as possible, on the basis of the foregoing we conclude that one should strive so that

\[ \cos\gamma \gtrless \frac{2nv}{1+n^2v^2}. \]

Let us now differentiate the expression \(Z\) with respect to \(u\):

\[ \frac{dZ}{du} = \frac{\Delta v\sin\gamma(1-n^2v^2)} {(1-2nv\cos\gamma+n^2v^2)^2}; \]

again, since the second derivative is negative, max. \(Z\) corresponds to

\[ 1-n^2v^2=0. \]

And hence: \(nv=1\), since, by the meaning of the solution of the problem of three boreholes, neither \(n\) nor \(v\) can have negative values.

On the basis of the same consideration as above:

\[ nv \leqq 1. \]

Thus, if in determining the strike we use an equilateral triangle of boreholes, it is desirable that both deeper boreholes should not fall on the same horizontal of the bed; and, conversely, if both deeper boreholes are of approximately the same depth, it is desirable that the lines connecting them with the less deep one should be of different length.

If we take \(nv=1\), then \(\cos \gamma=1\) and \(\beta-\alpha=0\), while \(\dfrac{\Delta S}{S}=\dfrac{0}{0}\), which is quite understandable, since in this case all three boreholes must lie on one straight line, and therefore there must exist the proportion

\[ \frac{h_2-h_1}{h_1-h_0}=\frac{l_2}{l_1}. \]

In general, the investigation of eq. (10) has given us chiefly negative indications: we have learned what arrangement should be avoided. From a practical point of view this is sufficient, since when laying out deep boreholes we are usually so constrained in their number that there is no occasion to pay attention to the secondary advantages of one distribution or another.

Sometimes, when there is a sufficient number of boreholes, they are used in such a way that, grouping them in threes, one determines, for the interval between them, the strike which pertains to the nearest point of the horizontal, and in this way the whole horizontal is traced out. If, in the interval between two points at which the strike is known, there are no direct data for its determination, then the strike is taken as the mean.

Finally, I consider it necessary to note that formula (10) indicates that small errors in determining the depth of the bed by boreholes have little effect on the length of the horizontal.

Let us now pass to determining the influence of errors in measuring the depth of boreholes on the accuracy of the computed angle of dip:

\[ \operatorname{tg}\Delta\eta=\operatorname{tg}\bigl[(\eta+\Delta\eta)-\eta\bigr] =\frac{\operatorname{tg}(\eta+\Delta\eta)-\operatorname{tg}\eta}{1+\operatorname{tg}(\eta+\Delta\eta)\operatorname{tg}\eta}; \]

but from eq. (7)

\[ \operatorname{tg}(\eta+\Delta\eta) =\frac{y+\Delta y}{l_2}\cdot \frac{1}{\cos(\xi+\Delta\xi-\beta)}, \]

and hence

\[ \tg \Delta \eta = l_{2}\, \frac{(y+\Delta y)\cos(\xi-\beta)-y[\cos(\xi-\beta)\cos\Delta \xi-\sin(\xi-\beta)\sin\Delta \xi]} {l_{2}^{2}[\cos(\xi-\beta)\cos\Delta \xi-\sin(\xi-\beta)\sin\Delta \xi]\cos(\xi-\beta)+y^{2}+y\Delta y}. \]

But one may put \(\sin\Delta \xi=\Delta \xi,\ \cos\Delta \xi=1\), and in view of this

\[ \tg \Delta \eta = l_{2}\, \frac{\cos(\xi-\beta)\Delta y-\sin(\xi-\beta)\Delta \xi} {l_{2}^{2}[\cos^{2}(\xi-\beta)-\sin(\xi-\beta)\cos(\xi-\beta)\Delta \xi]+y^{2}+y\Delta y} \]

and finally, discarding \(\Delta \xi\) and \(\Delta \eta\) in the denominator as having little effect on the final result, and also taking \(\tg\Delta\eta=\Delta\eta\), we obtain

\[ \Delta \eta = l_{2}\cos(\xi-\beta)\, \frac{\Delta y-\tg(\xi-\beta)\Delta \xi} {l_{2}^{2}\cos^{2}(\xi-\beta)+y^{2}} \]

or

\[ \Delta \eta = l_{2}\sqrt{1+\tg^{2}(\xi-\beta)}\cdot \frac{\Delta y-\tg(\xi-\beta)\Delta \xi} {y^{2}+l^{2}+l^{2}\tg^{2}(\xi-\beta)}. \tag{11} \]

For the further development of this expression it would be necessary to substitute into it the values for \(\tg(\xi-\beta)\) and \(\Delta \xi\) determined earlier, but then very complicated formulae are obtained, which are ill suited to simplification; therefore formula (11) is more conveniently left in its present form.

Combining formula (11) with formula (3), we obtain the possibility of determining

\[ \frac{\Delta L}{L}. \]

The third element of the reserve of the mineral—the thickness of the seam—is determined by its vertical section by the borehole. Thus, if the thickness of the seam is \(d\), and in the borehole \(d_{0}\) of coal has been traversed, the angle of dip is \(\eta\), then

\[ d=d_{0}\cos\eta. \]

But \(d_{0}\) is determined with some error \(\Delta d_{0}\), and the angle \(\eta\), respectively, is \(\eta+\Delta \eta\); therefore

\[ d+\Delta d=(d_{0}+\Delta d_{0})\cos(\eta+\Delta\eta); \]

whence

\[ \frac{\Delta d}{d} = -1+ \left(1+\frac{\Delta d_{0}}{d_{0}}\right) (\cos\Delta\eta-\tg\eta\sin\Delta\eta). \]

Substituting in place of \(\cos\Delta\eta\) and \(\sin\Delta\eta\) their approximate values \(1\) and \(\Delta\eta\), and discarding \(-\frac{\Delta d_{0}}{d_{1}}\Delta\eta\,\tg\eta\), we obtain

\[ \frac{\Delta d}{d} = \frac{\Delta d_{0}}{d_{0}} - \Delta\eta\,\tg\eta. \tag{12} \]

The magnitude \(\frac{\Delta d_0}{d_0}\) depends not only on the rigor of the measurements and of the supervision of the course of the work, but also on the drilling system. Thus, for example, in diamond drilling, when cores of the penetrated rocks are obtained, the errors of measurement, with some effort and care, can be brought almost to zero; conversely, in drilling with washing, especially in machine work, when the drill advances very rapidly, the error \(\frac{\Delta d_1}{d_0}\) sometimes reaches astonishing proportions. In drilling by the Raky or Rapid systems it happens, through negligence on the part of the contractors, that entire workable seams are missed. In general, for each drilling system the magnitudes of probable and possible errors should be established. As yet, we do not have such numerical data at our disposal.

Thus we have examined the case of determining strike and dip from three boreholes, as well as the error of this method; it has also been indicated that often the dip and strike are taken as the average from adjacent points, in other words, these quantities have to be determined from boreholes that are rather far apart. In general, it must be said that the data for determining the reserve of a mineral in different parts of the area being studied differ by far from having the same density and accuracy. In view of this, it must be admitted that the question of the degree of accuracy of exploration of bedded deposits has been little developed, but at the same time one can see to what extent this question is complex and involved.

The methods I have indicated for calculating errors make it possible to determine the relative errors in calculating the reserve in areas directly explored; as for other areas, we can judge them only on the basis of analogy. Thus, if, proceeding from general geological indications, we suppose that (see Drawing IX) the area \(B\), lying between areas \(A\) and \(C\), is something intermediate in its geological structure between \(A\) and \(C\), and on the basis of such a consideration we calculate the reserve of the mineral in this area, then, of course, the numerical data obtained have the character only of a probable reserve. As for the degree of this probability, it may be very varied, and in most cases it is difficult to find for it a mathematical, numerical value.

For bedded deposits, especially coal deposits, the probability that an area lying between two explored ones will, in its structure, be a transitional link between them comes very close to unity, so that we are justified in making the generalizations of which ...

was stated earlier. Thus, if we have determined the accuracy of the calculation for certain well-explored areas, then, assuming that the method of carrying out exploration is applied everywhere in one and the same way, we have the right, with a considerable probability close to unity, to suppose that the entire coal reserve is calculated with the same degree of accuracy.

Exploration of steeply dipping seams covered by a thick layer of more recent formations

If, for the exploration of inclined seams whose outcrops come close to the ground surface, the principal method is a row of boreholes laid down at small distances from one another across the strike of the deposit, then for gently dipping seams—especially if they are covered—it is necessary to drill a limited number, but deep, boreholes; in the case we are now considering, both are required, that is, many and deep boreholes.

The exploration lines are directed, as usual, across the strike. If the thickness of the recent formations is \(h_0\), and the angle of dip of the seams is \(\eta\), then the horizontal space explored by a borehole that has penetrated to a depth \(h_1\) into the formations containing the useful mineral will be\(^1\)

\[ e = h_1 \cot \eta, \]

and consequently, in order to explore a field of width \(E\), it is necessary to lay down so many boreholes, and of such depth, that the inequality

\[ \cot \eta \sum_{1}^{n} h_1 \geq E; \]

holds; or, if all the seams must be drilled through by the boreholes so that not one of them is missed, it is necessary that every borehole go somewhat deeper than \(h_1\), to some depth \(h_2\). Then the depth of each borehole is

\[ H = h_0 + h_1 + h_2. \]

If \(E\) is divided into \(n\) equal intervals, then the depth of the boreholes is determined by:

\[ H = h_0 + \frac{1}{n} E \tg \eta + h_2 . \tag{13} \]

Thus, we see that, by increasing the depth of individual boreholes,

\(^1\) This problem has been considered by Voislav, but rather narrowly.

we can reduce their number, and, conversely, by increasing the number, reduce the depth. The method of a large number of boreholes is applied, as we have already seen earlier, when the overburden is of insignificant thickness, or absent; but with a thicker cover every borehole is burdened with the cost of drilling through this cover—let us say, with a certain unproductive expenditure \(C\), and, in addition, with the expenditures associated with laying out a new borehole; let us call these expenditures \(c\). Thus, with \(n\) boreholes the aggregate of the indicated expenditures will be \(n(C+c)\), and this leads us to think that the number of boreholes should be limited.

But on the other hand, if we pay attention to the fact that the cost of a borehole increases not proportionally to its depth, but much more rapidly, then we arrive at the opposite conclusion, namely, that one should be satisfied with a certain, fairly limited depth of drilling.

These contradictory conclusions make it possible for us to draw one quite correct conclusion: in explorations of steeply dipping seams covered by a thick layer of more recent deposits, mechanical drilling must be employed, which gives us the possibility of penetrating, at a comparatively more uniform cost of drilling, into deeper horizons.

To decide the question of what number of boreholes and what depth should be used in explorations of the type now under consideration, let us examine in somewhat greater detail the conditions of their cost. Thus, the pricing of drilling work is in general rather varied; most often the price per linear unit of depth increases periodically, for example every 50 meters.

If the base price for the first \(r\) meters is \(a\) rubles, then for the second \(r\) it will be \(a+b\), for the third \(a+2b\), and so on, so that the price of the entire borehole is determined by the sum of an arithmetic progression

\[ ar+(a+b)r+(a+2b)r+\ldots+(a+mb)r+[a+(m+1)b]r_1; \]

in the expression given, \(r_1<r\), since usually the depth of the borehole is not a multiple of \(r\).

Summing, we find \(k\), the cost of a borehole of depth \(H=mr+r_1\):

\[ k=mar+\frac{m(m+1)}{2}br+ar_1+(m+1)br_1, \]

and since \(m=\dfrac{H-r_1}{r}\), then

\[ k=aH+\frac{1}{2}\left(\frac{H-r_1}{r}+1\right)bH+\frac{1}{2}\left(\frac{H-r_1}{r}+1\right)br_1. \tag{14} \]

If the exploration line is divided into \(n\) equal parts, then, depending on whether we drill boreholes at both boundaries of the explored area or only at one, namely at the one toward which the beds dip, the cost of all the work on this exploration line will be \(K=(n+1)k\) or \(nk\), of course on the assumption that the depth of all the boreholes is the same. For brevity denote \(h_1+h_2=h\), and first consider the second case (\(n\) boreholes), as the simpler one. On the basis of formula (13) we have

\[ n=\frac{E \operatorname{tg}\eta}{H-h}, \]

and consequently

\[ K=\frac{E \operatorname{tg}\eta}{H-h}\left[ aH+\frac{1}{2}\left(\frac{H-r_1}{r}+1\right)bH +\frac{1}{2}\left(\frac{H-r_1}{r}+1\right)br_1 \right]. \tag{15} \]

This expression shows that the cost of the work is directly proportional to \(\operatorname{tg}\) of the angle of dip of the beds, and consequently, for angles of dip close to \(90^\circ\), the expenditures on exploration become disproportionately large, so that boreholes in this case are scarcely applicable; in such a case exploration is carried out by single crosscuts from neighboring shafts. This method is rather widespread in the West.

For simplicity of the further reasoning let us take \(\frac{H}{r}\) to be an integer, so that \(r_1=0\), and then

\[ K=\frac{E \operatorname{tg}\eta}{H-h}\left[ aH+\frac{1}{2}\left(\frac{H}{r}+1\right)bH \right]. \]

Let us determine for what \(H\) the cost \(K\) becomes the least; for this

\[ \frac{dK}{dH}= \frac{E \operatorname{tg}\eta}{(H-h)^2} \left[ \frac{bH^2}{2r}-\frac{bhH}{r}-h\left(a+\frac{b}{2}\right) \right]=0. \]

Whence

\[ H^2-2hH-hr\left(2\frac{a}{b}+1\right)=0, \]

\[ H=h\pm\sqrt{h^2+hr\left(2\frac{a}{b}+1\right)}; \]

and since the negative value of \(H\) has no meaning in the present case, in the end

\[ H=(h_1+h_2)\left[ \sqrt{1+\frac{r}{h_1+h_2}\left(2\frac{a}{b}+1\right)}+1 \right]. \]

Since \(\frac{d^2K}{dH^2}>0\), the given expression corresponds to \(\min K\). It shows that, if it were possible to drill to an unlimited depth at equal costs for each sunk unit, so that \(b=0\), then very deep boreholes would be the most rational.

The formula (15) derived by us applies to the case when no borehole is laid down at one boundary; but it may also be used as an approximate formula with the full number of boreholes (Fig. X), in the case when the exploration line is very long and the overburden is not thick; then the number of boreholes \(n\) is large, and we have the right to take \(n\) instead of \(n+1\). For short lines and thick overburden such a simplification is inadmissible. In general, boreholes should be drilled at both boundaries (Fig. X).

If we were to limit ourselves to borehole 4, then all the seams whose outcrops lie on the right-hand side of boundary \(B\) would remain unexplored. Thus, the actual cost of drilling is given by the formula:

\[ K=\left[\frac{E\operatorname{tg}\eta}{H-h}+1\right] \left[ aH+\frac12\left(\frac{H-r_1}{r}+1\right)bH+ \frac12\left(\frac{H-r_1}{r}+1\right)r_1b \right]. \tag{16} \]

Strictly speaking, the cost of the work increases continuously and gradually as the borehole is deepened, since every subsequent meter requires the expenditure of a greater quantity of energy than the preceding one. In practice, calculations of this kind are avoided only because of the complexity of the computations. In order to decide the question in principle with respect to the depth of the boreholes, let us examine Eq. (16), putting \(r_1=0\); then

\[ K=\frac{b}{2r}\left(1+\frac{E\operatorname{tg}\eta}{H-h}\right) \left(r\frac{2a+b\eta}{3}H+\frac{bH^2}{4}\right); \]

and if, for brevity, we denote \(E\operatorname{tg}\eta=N\), and \(r\frac{2a+b}{b}=M\), then

\[ K=\frac{b}{2r}\left(1+\frac{N}{H-h}\right)(MH+H^2); \]

equating the first derivative to zero,

\[ \frac{dK}{dH} = \frac{b}{2r}\cdot \frac{ 2H^3+(N+M-4h)H^2+2(h^2-Nh-Mh)H+Mh^2-NMh }{ (H-h)^2 } =0, \]

we obtain

\[ H^3+\left(\frac{M+N}{2}-2h\right)H^2+ (h^2-Nh-Mh)H+ \frac{Mh^2-NMh}{2}=0. \]

Since the second derivative is positive for \(H>h\), the derived equation corresponds to \(\min K\). Putting

\[ H=x-\frac{1}{3}\left(\frac{M+N}{2}-2h\right), \]

we obtain the equation

\[ x^3+px+q=0, \]

in which

\[ p=-\frac{1}{3}\left(\frac{M+N}{2}-2h\right)^2+(h^2-Mh-Nh), \]

\[ q=\frac{2}{27}\left(\frac{M+N}{2}-2h\right)^3-\frac{1}{3}\left(\frac{M+N}{2}-2h\right)(h^2-Mh-Nh)+\frac{Mh^2-MNh}{2}. \]

Since \(N\) is much greater than \(M\) and \(h\), we have \(p<0\); on the same grounds \(q>0\); and it is also easy to see that \(\frac{1}{2}q+\frac{1}{3}p<0\), and consequently also

\[ \left(\frac{1}{2}q\right)^2+\left(\frac{1}{3}p\right)^3<0. \]

In view of this all the roots of the equation are imaginary, and for solving our cubic equation it remains for us to apply the trigonometric method, which gives

\[ x_1=\sqrt{-\frac{4}{3}p.\sin\varepsilon},\quad x_2=\sqrt{-\frac{4}{3}p.\sin(60-\varepsilon)},\quad x_3=-\sqrt{-\frac{4}{3}p.\sin(60+E)}; \tag{17} \]

for determining \(\varepsilon\) we have

\[ \sin 3\varepsilon=\sqrt{-\frac{72}{4}\frac{q^2}{p^3}}; \tag{18} \]

\(x_3\), being negative, does not correspond to the problem; as for \(x_1\) and \(x_2\), it is easy to see that when \(3E=90^\circ\), or \(E=30^\circ\), \(x_1=x_2=\sqrt{-\frac{p}{3}}\).

This case will occur when \(h\), in comparison with the other quantities—see (16)—is so small that it may be neglected, that is, when the newest deposits are almost absent. Let us suppose, for example, that

\[ \frac{h}{Etg\eta} \]

is so small that we may neglect not only this, but also

\[ \left(1+\frac{M}{N}\right)\frac{h}{N},\quad \text{as well as } M\frac{h^2}{N^2}\text{ and }M\frac{h}{N}; \]

then:

\[ \frac{x_1}{N}=\frac{x_2}{N}=\frac{1}{3}, \]

and consequently

\[ \frac{H}{Etg\eta}=0. \]

This is nothing other than the principle of shallow boreholes, of which we spoke earlier. For greater clarity let us substitute into the expressions for \(p\) and \(q\), instead of \(M\) and \(N\), their values.

\[ \left. \begin{aligned} p={}&-\frac{1}{3}\left(r\frac{2a+b}{2b}+\frac{E\tg\eta}{2}-2h\right)^{2} +\left(h^{2}-r\frac{2a+b}{b}h-E\tg\eta\cdot h\right),\\[4pt] q={}&\frac{2}{27}\left(r\frac{2a+b}{2b}+\frac{E\tg\eta}{2}-2h\right)^{3} +\left(r\frac{2a+b}{2b}h^{2}-r\frac{2a+b}{2b}E\tg\eta\cdot h\right)\\ &-\frac{1}{3}\left(r\frac{2a+b}{2b}+\frac{E\tg\eta}{2}-2h\right) \left(h^{2}-r\frac{2a+b}{b}h-E\tg\eta\cdot h\right),\\[4pt] H={}&x-\frac{1}{3}\left(r\frac{2a+a}{2b}+\frac{E\tg\eta}{2}-2h\right), \end{aligned} \right\} \tag{19} \]

Of the two values \(H_1\) and \(H_2\), \(H_1 < H_2\), for the max. \(\varepsilon\), as we have seen, is \(30^\circ\); \(H_1\) corresponds to a larger number of boreholes, \(H_2\) to a smaller number, but to deeper ones. The choice of one method or the other depends on secondary technical conditions, or else may be made on the basis of the magnitude \(K\), obtained if in formula (15) we substitute \(H_1\) or \(H_2\).

For completeness of exposition, let us consider several more special cases relating to this type of exploration.

Thus, with improved methods of mechanical drilling, we may, within rather broad limits, regard the price of a running unit as constant; therefore the cost of a borehole will be expressed as

\[ k=C+BH, \]

where \(A\) is the one-time expenditure, and \(B\) is the price of a running unit; then

\[ K=\left(\frac{E\tg\eta}{H-h}+1\right)(C+BH), \]

\[ \frac{dK}{dH} = \frac{-E\tg\eta\cdot\frac{C}{B}-E\tg\eta\cdot h+(H-h)^2}{(H-h)^2} =0. \]

Hence

\[ H=\sqrt{E\tg\eta\left(\frac{C}{B}+h\right)}+h \]

is the value of \(H\) corresponding to \(\min. K\).

If the depth \(H\) is approximately known, then very often the one-time expenses are distributed over the unit of drilling depth, and then

\[ k=BH. \]

\(\operatorname{Min.}K\) will occur at

\[ H=\sqrt{E\tg\eta h}+h. \]

The cost of drilling has not entered into this expression at all. From this we conclude that, in the case where the contractor agrees to a uniform and not excessive rate, deep boreholes are advantageous for the contracting party. As for the methods of exploration of steeply dipping seams in general, as well as the methods of calculating the reserves of the mineral, all that can be said of them is that they are identical with those described earlier, so there is no purpose in repeating that description.

Chapter Two.

Exploration of Vein and Pocket Deposits.

Exploration of vein deposits by means of boreholes.

Vein deposits are very rarely explored by means of boreholes, in view of the fact that, in order to determine the reserve of the mineral in such deposits, it is necessary to establish precisely the content of the mineral per unit volume of the vein rock, while drilling operations in this respect almost never achieve the goal. True, by means of diamond drilling one can obtain cores taken from the vein rock and thus trace its content along a vertical line at a given point; but this method is applicable, first, only to crystalline and, in general, hard rocks, and second, it can yield results that are favorable in an economic sense, if the vein deposit is being explored in detail, only when the vein is explored at its outcrops, that is, when the boreholes are not very deep.

In the case of deep drilling, its cost is incommensurably high in relation to the result obtained. But if the veins come to the surface or are covered by a thin layer of alluvium, then exploration is conveniently carried out by pitting and by a system of crosscuts with exploratory drifts along the strike and dip (raise).

Nevertheless, there are cases when drilling operations are the only possible means in the exploration of a vein deposit. For example, if it is necessary to find the continuation of a vein or a group of veins separated from the part being worked by a series of faults, and searches by crosscuts prove fruitless, if on the level of the workings the crosscuts encounter newer deposits.

It goes without saying that from such explorations one cannot demand precise data on the percentage composition of the vein: they can give only general indications of its character, so that the final result

to give reconnaissance—this is a more or less clear picture of the structure of the deposit and a very rough idea of its reserves. In general, the system for conducting exploratory work is, of course, the same as in the exploration of a steeply dipping bedded deposit; only, in view of the absence of parallelism between the veins and the variable angle of dip both in individual veins and in their suites, when drawing up a plan of exploration the principles by which we were guided for coal deposits lose their force entirely. Thus, if the dip angle for a given suite of veins varies from $\eta_1$ to $\eta_2$, then it is clear that, in order not to miss a single vein in the area being explored, the number of boreholes and their depth must be calculated in the same way as in a bedded deposit for the maximum dip angle $\eta_2$. In this case, of course, the heads of the veins will be drilled through by at least two, and sometimes three, boreholes.

Moreover, since the dip of the veins may assume all possible directions, it is difficult, with a single exploratory line, to obtain a complete picture of the cross-section in the direction of that line. In view of this, in cases where $\eta_1$ and $\eta_2$ are not sufficiently known, and likewise where their direction is unknown, the exploratory lines should be grouped in pairs, at small distances from one another.

With any more or less considerable thickness of the more recent deposits covering the veins, systematic exploration of them by boreholes becomes excessively costly—and therefore it is usually preferred, after having carried out only general exploration that has revealed the presence of veins of the required qualities, not to explore them in detail by drilling, but to proceed directly to the sinking of shafts and to development, which, of course, are preceded by local explorations with the aid of mining workings. Certainly, such a method is associated with great risk: the deposit may prove unsuitable and the expenditures will be wholly unproductive. But since drilling operations would also cost enormous sums, while their utilitarian value for development, even in the event of a favorable outcome of the exploration, is nil, it is preferred to lay down shafts directly. If diamond drilling in hard rocks gives results which, owing to its considerable cost, are not wholly satisfactory for determining the composition of the vein, then all the more does percussion drilling in rocks of medium hardness and in soft rocks lead, in general, to extremely poor results.

In some cases it is not applicable at all, for example, in the exploration of stockwork deposits of galena in dolomites. In general it may be said that, if the mineral being explored differs little in its external appearance from the surrounding rocks, or is finely disseminated in rocks that differ little from the surrounding barren rocks, then drilling

work cannot provide not only precise indications of the properties and reliability of the said deposits, but often, with their aid, it is difficult to determine the presence of the useful mineral. For these cases, prospecting pits or shafts with exploratory crosscuts are recommended, of which we shall speak further below.

If, in the case under consideration, despite everything, it becomes necessary to resort to drilling, then the drill samples must be subjected to careful chemical and microscopic analysis, with unremitting observation of the progress of the work, so that even the most insignificant intercalation of rocks does not escape our attention and is determined with scientific accuracy.

Exploration of nest deposits and stocks.

The remark made in the preceding chapter concerning the difficulty of applying drilling to the exploration of certain ore deposits applies, of course, also to nest deposits. But since prospecting by pits is applicable only in the case where the ore nests lie near the surface, it may be said that, in general, exploration of deeply lying nest deposits is easily feasible only when the rock containing them differs rather sharply from the surrounding rock.

A distinctive feature of nest deposits, as compared with bedded and vein deposits, consists in the fact that the nests are scattered throughout the whole space of the deposit and occupy only part of it, and that the discovery of the useful mineral at any one point gives no right to suppose its presence at another, neighboring one. In accordance with the above, exploration of deposits of this type is, to a certain degree, a game in which chance plays a prominent role.

The aim of such explorations is, first, to find the nests; then—to determine their form and volume, and finally—the percentage content of the useful mineral.

Let us suppose that the character of the deposit is known, and therefore the geological horizon of occurrence of the nests is also known, which determines to a certain degree the depth of boreholes or pits. If, in the given space, whose area is \(F\), there is a nest occupying, in horizontal projection, an area \(f\), then the probability that a borehole laid at random in the area being explored will strike the nest is equal to

\[ p_1 = \frac{f}{F}; \tag{20} \]

with \(n_1+m\) boreholes, the probability that \(m\) finds will be made is determined by the equation

\[ p=\frac{1.2.3\ldots(n_1+m)}{1.2.3\ldots n_1\,1.2.3\ldots m} \left(\frac{f}{F}\right)^m \left(1-\frac{f}{F}\right)^{n_1}. \tag{21} \]

Since we wish to determine the probability of a single find, putting \(m=1\), and \(n_1+1=n\), we obtain

\[ p=n\frac{f}{F}\left(1-\frac{f}{F}\right)^{n-1}. \tag{22} \]

This expression is easily reduced to the following form:

\[ 1-p=\left(1-n\frac{f}{F}\right)^n, \]

namely: if from both sides of the original equation one subtracts unity and carries out the expansion of \(\left(1-\frac{f}{F}\right)^{n-1}\), it turns out that the right-hand side is nothing other than the expansion of the binomial \(\left(1-n\frac{f}{F}\right)^n\).

We pose the question: what must \(n\) be in order that the probability of finding the nest should become certainty, \(p=1\); then, of course,

\[ \left(1-n\frac{f}{F}\right)^n=0, \]

and consequently

\[ n\frac{f}{F}=1. \tag{23} \]

The geometrical meaning of this expression is such that, if the entire field is uniformly covered by boreholes,—for only in this case is the probability of finding ore in each of them the same, and each drilling may be regarded as an independent trial; and if the area around each borehole is equal to \(f=\frac{F}{n}\),—then there is no possibility of failing to strike the nest.

Let us explain what has been said by the following further reasoning. Each borehole explores a certain area: \(\Delta F_1, \Delta F_2,\ldots\); these areas may be unequal. After \((n-1)\) drillings, the explored part of the field will be

\[ \sum_1^{n-1}\Delta F, \]

and consequently, if these \((n-1)\) drillings have given an unfavorable result, then the probability that, with \(n\) drillings, a nest \(f\) situated within the field under exploration will be found is expressed as:

\[ p=\frac{f}{F-\sum_{1}^{n-1}\Delta F}; \]

with a sufficiently large value of \(\sum \Delta F\), the ratio equal to \(p\) approaches unity, so that for some

\[ \sum_{1}^{n-1}\Delta L: \]

\[ f=F-\sum_{1}^{n-1}\Delta F=\sum_{1}^{n}\Delta F-\sum_{1}^{n-1}\Delta F, \]

and the finding of the nest becomes obligatory. Of course, there is no need to suppose that only the last, \(n\)-th, borehole will give a favorable result; the order of drilling here plays no role: we have shown only that, if the whole field has been explored by \(n\) boreholes sufficiently fully, then one of these boreholes must reveal the nest.

Since in general there are no rational grounds for assuming that one borehole explores a portion greater or smaller than another, except that near the boundaries of the field being explored there may be portions smaller than the normal ones, and since we assumed \(\Delta F_1 \gtreqless \Delta F_2,\ldots\) only for the sake of generalization, we shall now take

\[ \Delta F_1=\Delta F_2\ldots=\Delta F_n=\frac{F}{n}, \]

and consequently

\[ f=\Delta F=\frac{F}{n} \]

—the expression derived above.

In our reasoning we assume that the nest sought is in fact located on the area being explored. This assumption was made for simplicity of reasoning, but it is not necessary, since in reality we usually know only that the nest is probably located in the field being explored, and at the same time we are not in a position to express this probability by a numerical value. If we call this indefinite probability \(p_0\), then the probability of finding the nest with one borehole will be

\[ p_1=\frac{f}{F}p_0, \tag{24} \]

and the probability of finding it with \(n\) boreholes

\[ p=n\frac{f}{F}p_0\left(1-\frac{f}{F}p_0\right)^{n-1}. \tag{25} \]

and we must strive so that the result of exploration be equal to the probability of the presence of a deposit; that is, in other words, so that deposit \(f\) be found if it is located on the area being explored. Formula (25), after the same transformation as we described for (22), will take the form

\[ p_0-p=\left(p_0-n\frac{f}{F}p_0\right)^n; \]

therefore, for \(p=p_0\) one obtains

\[ 1-n\frac{f}{F}=0. \]

In our further reasoning, for simplicity, we shall in most cases take \(p_0=1\). If, in general, the area actually occupied by deposits is denoted by \(\Sigma f\), then the probability of finding a deposit with one borehole will be

\[ p=\frac{\Sigma f}{F}. \tag{26} \]

If \(\Sigma f=F\), then obviously every borehole will strike ore.

Since the form of a deposit is not defined by anything, in most cases we have to assume that its horizontal projection is a circle (Fig. XI a), on the grounds that there are no, or at least we have no, rational grounds to suppose that the horizontal dimensions of the deposit have developed in some direction more than in another; thus

\[ f=\pi\frac{d^2}{4}. \]

On the other hand, any three boreholes form a triangle, so that the area \(F\), covered by the network of drilling works, naturally breaks up into a certain number of triangles; by connecting the boreholes four at a time, we can divide the field into rectangles, squares, rhombi, etc. The square layout is most often applied. In a square grid each borehole corresponds to the square at whose center it is located. In a layout in equilateral triangles, each borehole corresponds to a certain regular hexagon.

If the outer boreholes are laid down at the boundaries of the area, so that we may take

\[ na^2=F, \]

where \(a\) is the mutual distance between boreholes, then, connecting this expression with (23), we find that, in order not to miss a single deposit, the equality must hold:

\[ n\pi \frac{d^2}{4} = n a^2, \]

whence

\[ a = \frac{d}{2}\sqrt{\pi} = 0.88d. \tag{27} \]

Voislav gives the following rule for this case: drill boreholes at a distance from one another no greater than the diameter of the very smallest nest suitable for development. It is easy to see, however, that with \(a=d\) a nest may be missed, since the boreholes may fall at the vertices of a square described about the circle—the nest—without touching it.

A completely uniform subdivision of the field by boreholes is achieved if all neighboring boreholes are at equal distances from one another, that is, if the field is divided into equilateral triangles. If \(l\) is the distance between two boreholes, then the area of a regular hexagon at the center of which there is a borehole (Fig. XII) is equal to \(\frac{l^2\sqrt{3}}{2}\), and in order that not a single nest be missed by exploration, the following equality must hold:

\[ n\pi \frac{d^2}{4} = n \frac{l^2\sqrt{3}}{2}, \]

whence

\[ l = d \sqrt{\frac{\pi}{2\sqrt{3}}} = 0.95d. \tag{28} \]

Thus, a quite rational rule is: boreholes should be placed in a checkerboard order, along mutually parallel exploration lines. The distance of the boreholes from one another is \(l\), the distance between exploration lines is \(\frac{l\sqrt{3}}{2}=0.86l\); the length \(l\) is determined by the character of the deposit and by the requirements imposed upon the exploration.

Since the shape of the area being explored is usually incommensurable with the area of the squares or hexagons into which we divide it, then, of course, the equalities on the basis of which we have drawn our conclusions have only an approximate value, quite sufficient for practical purposes. We can say only that what we have explored with complete accuracy is not the area \(F\), but some other area \(F_1\), and moreover \(F_1 \gtrless F\), depending on the arrangement of the outermost boreholes. Of course, in preparing an exploration project one must strive to make the difference between \(F\) and \(F_1\) as small as possible.

If the area is divided into squares, then

\[ p_1 = \sum \frac{f}{a^2}, \]

in the case of regular hexagons,

\[ p_2=\frac{2}{\sqrt{3}}\sum \frac{f}{l^2}; \]

with an equal number of boreholes and \(a=l\),

\[ \frac{p_2}{p_1}=\frac{2}{\sqrt{3}}=1.15, \]

and consequently the recommended method is more advantageous than division into squares, since it is, to a certain degree, a natural method of the most uniform and complete subdivision of an area, and chiefly because a regular hexagon more closely approaches a circle—the nest—than does a square.

For given dimensions of the borehole network, all nests whose dimensions correspond to eqs. (27), (28), or are larger than them, will necessarily be found. The probability that, besides these, further nests will be found whose diameter

\[ d<\frac{a}{0.88}\quad \text{or}\quad d<\frac{l}{0.95} \]

is given by eq. (22).

It is easy to see that, as the diameter of the nests decreases, the probability of finding them decreases very rapidly. Suppose we have a nest of diameter \(d_1\) and \(d_2\); the probability of finding them is:

\[ p_1=\frac{\pi N}{4}\frac{d_1^2}{L^2};\qquad p_2=\frac{\pi N}{4}\frac{d_2^2}{L^2}, \]

where \(N\) depends on the method of subdivision: for a square subdivision, \(1\); for a hexagonal subdivision,

\[ \frac{2}{\sqrt{3}}; \]

hence

\[ \frac{p_1}{p_2}=\frac{d_1^2}{d_2^2}. \]

Thus, the probabilities of finding nests are directly proportional to the squares of their diameters. The probable result of exploration in general may be calculated on the basis of formula (21), which expresses the probability that, in \(n+m\) independent trials, event \(E\) will occur exactly \(m\) times, if for each trial separately this probability is equal to \(p_1\). For our case, this event is the finding of a nest. Thus, the probability of finding \(m\) nests, more exactly \(m\) boreholes, i.e. the probability of detecting ore, is given by the formula indicated above. If there are many boreholes, then, applying Stirling’s formula for simplification, we obtain

$$ p=\frac{(n+m)^{(n+m)+\frac12}}{n^{n+\frac12}.m^{m+\frac12}}\frac{p^m q^n}{\sqrt{2\pi}}, \tag{29} $$

$$ p=\frac{f}{F}, \qquad q=1-\frac{f}{F}. $$

The most probable number \(m=\mu\), that is, the one for which \(p\) attains its greatest value, is determined by the inequalities

$$ (m+n)p+p>\mu>(m+n)p-q, $$

where \(\mu\) is an integer.

Formulae (21), (29) may be used when the character of the deposit, thanks to neighboring workings, is fully known, so that the area of the nests per unit of the earth’s surface is approximately known. In this case they enable us to choose the number \(n+m\) of boreholes and their mutual distance so that the most probable number of finds \(\mu\), as well as the probability that precisely this number of nests will be determined, are sufficient for us. However, we very rarely possess such data; in the majority of cases we know the dimensions of the nests only in part and, moreover, only “from—to,” in general rather vaguely. It is also difficult to predict what horizontal transverse dimension of a nest determines its suitability or unsuitability for working, since here not only the horizontal dimensions but also the vertical ones, and also the percentage composition, play a role.

Thus, if we have only data on the dimensions of the nests, then the probability of discovering a nest of diameter \(d\) on the area being explored \(F\) is expressed—(24), if \(p_0\) is the probability of the existence of a nest in this area, by

$$ p=\frac{\pi d^2}{4F}p_0; $$

\(p\) in most cases is small, since the ratio \(\dfrac{\pi d^2}{4F}\) is small, and so is \(p_0\); in view of this we would have to carry out a great mass of exploratory drilling or pitting in order to bring \(p=p_0\), and \(n\dfrac{\pi}{4}\dfrac{d^2}{F}=1\). In this case it is usually preferred first to explore carefully some part \(DF\) of the entire field in one or several places and, in accordance with the data obtained, to continue exploring the area further or not to explore it.

Assuming that the conditions of occurrence of the nests do not vary over the whole space being explored, we can make further exploration conve—

be a known problem in probability theory concerning the further carrying out of experiments. For this case we have:

Let \(E\) be an event whose probability,

\[ p=\frac{\pi d^{2}}{4F}\,p_{0}, \]

is entirely unknown to us \((p_{0})\). Suppose that the experiment to which the phenomenon \(E\) is proper has been reproduced \(s\) times, and that the event (the finding of ore) occurred \(m\) times. Let us then imagine that the experiment (drilling, sinking of prospecting pits), to which the event \(E\) is proper, is to be reproduced in the future another \(\sigma\) times. The probability that in these \(\sigma\) experiments the event \(E\) will occur \(\mu\) times will be: ^1)

\[ p=\frac{\sigma!(1+s)!(m+\mu)!(s+\sigma-m-\mu)!}{\mu!(\sigma-\mu)!m!(s-m)!(s+\sigma+1)!}. \tag{30} \]

It goes without saying that the method and the layout of the exploration works must be the same in both cases.

Sometimes explorations are conducted in such a way that the entire field under exploration is covered by a sparse network of boreholes, with the expectation, in the event of a sufficiently favorable result, of drilling intermediate boreholes. To determine the probability of finding new nests one may use formula (30). If there are many boreholes, then, to simplify this formula, one may use Stirling’s formula

\[ 1.2.3\ldots n=\sqrt{2\pi}\,e^{-n}n^{\,n+\frac12+Q},\quad \text{where } Q<\frac{1}{12n}. \]

If we discard \(Q\) and substitute the indicated equality in place of \(n!\), \(m!\), \(\mu!\), etc., then after simple reductions we find that

\[ p=\sqrt{ \frac{ \dfrac{1}{2\pi}\, \sigma^{\sigma+\frac12} (s+1)^{s+1+\frac12} (m+\mu)^{m+\mu+\frac12} (s+\sigma-m-\mu)^{s+\sigma-m-\mu+\frac12} }{ \mu^{\mu+\frac12} (\sigma-\mu)^{\sigma-\mu+\frac12} m^{m+\frac12} (s-m)^{s-m+\frac12} (s+\sigma+1)^{s+\mu+\frac32} } }. \tag{31} \]

This formula is also still too complicated for ordinary practical calculations; but since the further development of exploration in areas that have proved sufficiently promising consists in placing new boreholes midway between the existing ones, the total number of boreholes or prospecting pits is increased fourfold, and \(\sigma=3s\). We pose the question: what is the probability that in this case \(\mu=3m\), that is, that the number of boreholes with ore will increase proportionally?

^1) Nekrasov, Theory of Probabilities, p. 101.

\[ p = \frac{ 4s^{3s+\frac{1}{2}(s-1)^3+\frac{3}{2}} }{ \sqrt{6\pi}\sqrt{m(s-m)}\,(4s+1)^{4s+\frac{3}{2}} }. \tag{32} \]

In concluding the chapter on the search for nests by means of drilling or test-pitting, I consider it necessary to note that, if the nests have an elongated form and are oriented in some definite direction, then the dimensions of the grid of exploratory workings must be coordinated with this circumstance; that is, the area \(F\) must be divided into such figures as would most closely correspond to the horizontal projection of the nests. If, for example, the nests have an oval shape, and the ratio of their axes may be expressed as \(a:b\), where \(a\) is the short axis and \(b\) the long axis, then in such a case the exploratory boreholes should be arranged so that they form rectangles \(ab\).

Determination of the Ore Reserve in a Nest Deposit

Here we shall concern ourselves only with the method of determining the volume of the nests under investigation, leaving the factor of percentage composition to the following chapter.

If in some part of the field under exploration the work has been completed, then one proceeds to a more detailed determination of the dimensions and forms of the nests or stockworks whose presence has been discovered. For this purpose it is necessary, between the boreholes that have shown the presence of ore, to drill new ones or to lay out test pits. In exceptional cases, when the grid is very dense in comparison with the size of the nests, one may be satisfied with the data already available; however, no general rule that would fully resolve the question indicated can be given. Simultaneously with the drilling of additional boreholes, vertical sections of the nests should be drawn in several directions, for only the combination of the plan and the sections gives us the possibility of obtaining a clear conception of the form of the nest; this remark applies above all to stockworks.

Transverse sections give us the possibility of determining the law of wedging-out of the given type of nests, and thereby of determining more successfully their transverse dimensions, which exploratory workings cannot indicate with complete accuracy. Until the general form of the nests is not yet fully known, we have to draw the boundaries of a nest between two boreholes, of which one has shown the presence of ore and the other its absence, at the midpoint between them, proceeding from the consideration that equal distances from both boreholes are equally probable (drawing XIV).

Of course, the “law of wedging out” is determined chiefly by the geological character of the nests and by their genesis; therefore, when they have in general already been studied, the task is made easier. In the contrary case, this law becomes clear after a number of sections have been drawn, and then we can introduce the desired corrections on the plan. Transverse sections of stockworks and similar deposits, as well as their horizontal projections, should be drawn at a fairly large scale, such as 1:1000, and, for more valuable minerals, at 1:500. In the case of bed-like nests of deposits, a scale of 1:1000, or even 1:5000, is quite sufficient for horizontal dimensions; the vertical dimensions, however, in this case are often enlarged tenfold.

The volume of stockworks and, in general, of small but thick nests should be calculated as follows: the horizontal projection of the nest is divided, by placing over it a correspondingly ruled sheet or simply by a pencil grid, into separate small squares; corresponding to these squares and to the triangles at the boundaries of the nest, the heights of elementary prisms are determined by means of sections, into which the entire stockwork is mentally divided. The sum of the volumes of these prisms is, understandably, equal to the volume of the stockwork.

The method of calculation I propose, in which the principal role is played by dimensions taken from the drawing, and not by data obtained directly from boreholes or shafts—that is, the results of the individual processing of the data obtained by exploration—may at first glance seem inappropriate; but, on the contrary, all other, more mechanical calculations give crude, incorrect, not to say simply false, results, since it is impossible to formulate them in such a way that they conform to the form of the stockwork, a deposit that ordinarily cannot be brought under the types of geometrically regular bodies.

Flat, lens-shaped deposits—nests—are calculated by their mean thickness, which is obtained as the arithmetic mean from the determinations made by exploration workings; moreover, the parts lying at the boundaries of the nest, with a width adopted in accordance with the law of wedging out, and unsuitable for working because of their small thickness, are discarded.

Sometimes it is preferred to calculate the volumes of separate areas, assuming that each borehole or shaft determines the thickness of the deposit, by squares—in the case of a square network—or by regular hexagons—in the case of division into regular triangles. It is easy to see that, in that part of the field consisting of the indicated regular figures, either system of calculation gives entirely identical results:

\[ V = na^2 \sum_{1}^{n}\frac{h_1 + h_2 + \cdots + h_n}{n} = \sum_{1}^{n} a^2 h_i . \]

The difference, which gives some excess of accuracy in the calculation by sections, is based on the fact that, in the first case, boreholes at the boundaries are included in the count on the same grounds as those lying in the middle of the deposit, although they do not correspond to full squares. Thus, by the first method—

\[ V_1 = \frac{1}{n} f \sum_{1}^{n} h_i \]

and by the second—

\[ V_2 = a^2 \sum_{1}^{n} h_i + \sum_{n_1}^{\,n-n_1} Da^2 h_i . \]

In general the difference obtained is rather small, masked by other inaccuracies of exploratory determinations. The inaccuracy in computing the volume of a sheet-like deposit depends chiefly on the inaccuracy in determining its horizontal and vertical dimensions. The individual error in the horizontal dimensions of the nest, \(\Delta f\), arising when the boundaries of the nest are drawn, depends on the personality of the director of the work, on the state of the information concerning the character of the nests and, finally, on the dimensions of the exploration itself; in general \(\Delta f\) represents something rather vague. If \(\Delta h\) is the error of the exploratory determinations, then, assuming that the calculation was made by the method of the arithmetic-mean thickness, we obtain:

\[ \Delta V = (f+\Delta f)\sum_{1}^{n}\frac{h_i+\Delta h_i}{n} - f\sum_{1}^{n}\frac{h_i}{n} = f\sum_{1}^{n}\frac{\Delta h_i}{n} + \Delta f\sum_{1}^{n}\frac{h_i}{n} + \Delta f\sum_{1}^{n}\frac{\Delta h_i}{n}; \]

finally the relative error, if we neglect

\[ \frac{\Delta f \sum \Delta h_i}{f \sum h_i}, \]

will be

\[ \frac{\Delta V}{V} = \frac{\Delta f}{f} + \frac{\displaystyle \sum_{1}^{n}\Delta h_i}{\displaystyle \sum_{1}^{n}h_i}. \tag{33} \]

With regard to \(\Delta h\), everything said earlier concerning the relative accuracy of borehole determinations remains in force.

As for errors in determining the volume of stockworks, one can only say that, in view of the individual treatment of the immediate data, both the horizontal and the vertical dimensions adopted for the calculation bear the same individual imprint—

work supervisor’s notes. The error \(\frac{\Delta V}{V}\) may be expressed by the same formula (33), with the sole difference that \(\Sigma \Delta h\) is not the error of the measurements, but only a certain consequence of these errors and of the graphical processing of the drilling data. Formula (33) can have no practical application in this case.

Reliability of Deposits in the Sense of Percentage Composition

In this chapter we shall consider veins, stocks and nests, stockworks, and, equally, those beds which, despite their clearly bedded character, approach the type of veins, namely in the case when the bed consists of barren rock, within the mass of which the useful mineral is irregularly scattered.

The general principle of the uniform distribution of samples taken for determining the composition of a deposit may be justified by the following reasoning:

If we have a heterogeneous body of volume \(V\), containing on average \(G\) weight units of useful mineral per unit volume, and we have taken a sample of volume \(v\), then the probability that the content \(g\) per unit volume of the sample is equal to \(G\), if the sample is taken quite at random from somewhere, will be expressed as:

\[ p_1=\frac{v}{V}i; \]

a series of such samples gives us

\[ p_1=\frac{\sum_1^n v}{V}i, \]

where \(i\) is the probability that the volumetric ratios correspond to the weight ratios; \(i<1\) in composition is the probability that the equality exists:

\[ g=\frac{\sum_1^n gv_1}{\sum_1^n v}=G. \]

Let us now imagine that the body under consideration is divided into \(n\) equal-sized parts \(DV\), so that

\[ \frac{V}{DV}=n \]

and that from each such part we take a sample \(v^{(1)}, v^{(2)}, v^{(3)}\ldots\)

The probability that the composition of each such sample \(g^{(1)}, g^{(2)},\ldots\) corresponds to the composition \(G^{(1)}, G^{(2)},\ldots\) of the parts \(DV\), that is, that the equalities \(g^{(1)}=G^{(1)},\ g^{(2)}=G^{(2)},\ldots\) exist, will be expressed respectively

\[ p^{(1)}=\frac{v^{(1)}}{DV} i_1,\quad p^{(2)}=\frac{v^{(2)}}{DV} i_1,\ldots \]

Summing the probabilities, we find:

\[ p_0^{(1)}=\frac{\displaystyle\sum_{1}^{n} v^{(1)}}{DV} i_1, \]

where \(i_1>i\) is the probability that one of the indicated equalities exists; while the probability of the existence of the equality

\[ \frac{1}{n}\sum_{1}^{n} g^{(1)}=\frac{1}{n}\sum_{1}^{n} G^1=g_0=G, \]

the arithmetic mean of our equalities, is

\[ p^{(1)}>p_0^{(1)}. \]

Let us now assume, as we have the right to do, that all samples, both in the first and in the second case, are equal in volume, that is, that

\[ v_1=v_2=v_3=\ldots=v^{(1)}=v^{(2)}=v^{(3)}\ldots \]

Both the first and the second expression for probability determine for us the probability that the arithmetic mean of the composition of the samples is equal to the composition of the entire body under investigation.

The ratio of these probabilities is:

\[ \frac{p^{(1)}}{p_1}>\frac{V}{DV}\cdot\frac{i_1}{i}\quad \text{or}\quad \frac{p^{(1)}}{p_1}>n\omega. \]

The conclusion, at first glance rather surprising, becomes understandable if we pay attention to the fact that, with samples taken without any order, at random, the samples may be extremely diverse, but they may also differ very little from one another, so that, as the number of samples increases, the probability of determining from them the exact composition of the whole body increases only insignificantly, in accordance merely with the ratio of the volumes taken for the samples.

In chemical laboratories, the substance intended for analysis is ground and mixed; then the weighed portion taken for determination corresponds quite exactly to the composition of the full sample, that is, it consists of small particles belonging to the various parts of the sample. On a larger scale, but, of course, more roughly, the same is achieved by the uniform distribution of samples in stockworks, veins, and the like.

The expression for the probability, if we substitute \(DV=\dfrac{V}{n}\), may also be written as follows:

\[ p'=n^2\frac{v}{V}i+Q, \tag{34} \]

where \(n\) is the number of samples, and \(Q=P'-P'_0\). Since the ratio \(\dfrac{v}{V}\) is in general a small quantity, approximately \(\dfrac{1}{100000m}\) or even \(\dfrac{1}{1000000m}\), where \(m\) is an integer, then even with a fairly large number of samples the probability of a fully correct determination of the composition of a vein, a nest, is not large and can be brought up to 1 only in exceptional cases. If we neglect \(Q\) and put \(P'=1\), then

\[ V=n^2vi. \tag{35} \]

This expression shows that the number of samples is of decisive importance for the reliability of determining the percentage composition of a deposit.

In exploration of stockwork deposits by means of shafts, the principle of uniform distribution of samples is achieved by itself in view of the correct distribution of the shafts. The same may be said of the exploration of large bedded deposits by shafts. The matter is otherwise in exploration by drilling: drill samples in general often do not provide exact material for judging the percentage composition of the drilled rocks, although they do make it possible to judge, with a certain degree of accuracy, the character of the deposit.

If we denote by \(p'_0\) the probability that a sample taken from a borehole corresponds exactly to the drilled core, then the probability that the result of the laboratory investigation corresponds to the composition of the deposit in the area whose center is the borehole will be expressed as:

\[ p^1=\frac{\pi r^2 h_0}{DV}\,ip^1_0, \tag{36} \]

which constitutes the probability of the equality

\[ g^1=G^1. \]

With \(n\) boreholes, reasoning as above, we obtain

\[ P_0=n^2\frac{\pi r^2}{F}\,ip^1_0+Q_0, \tag{37} \]

and this is the probability of the equality

\[ \frac{1}{n}\sum_1^n g'=\frac{1}{n}\sum_1^n G', \]

under the assumption that \(h\) is constant. It is easy to see that, in view of the negligible magnitude of \(\frac{\pi d^{2}}{4}\) in comparison with the area of the deposit and, moreover, since \(p'_0<1\), and often can be expressed by a small fraction, \(P_0\) is a small quantity even with a considerable number of boreholes. Thus, drill holes usually do not give us the possibility of determining the composition of the deposit.

By comparing the individual results of laboratory analyses, it is sometimes possible to divide them into such groups that, approximately,

\[ g^{1}=g^{2}=g^{3}=\ldots=g^{n};\quad g^{n+1}=g^{n+2}=\ldots=g^{m};\quad g^{m+1}=g^{m+2}=\ldots=g^{l}. \]

If approximately equal results of analyses correspond to certain parts of the deposit, then from this we conclude that these parts are, in general, homogeneous, since \(p'_0\) is assumed to be the same for every borehole. If, on the basis of such a consideration, we mentally divide (on the plan) the entire deposit into homogeneous areas and sink in each area a pit or an exploratory shaft, then the samples obtained by these explorations will give us the composition of each area (under the assumption of its homogeneity), since every part of a homogeneous body is identical in composition with the whole. If our areas are \(\Delta F_1,\Delta F_2,\Delta F_3,\ldots\), and the corresponding thicknesses of the ore bed are \(h_1,h_2,\ldots\), then the percentage composition of the deposit will be:

\[ G=\sum_{1}^{n}\Delta F_j h_j G_j:\sum_{1}^{n}\Delta F_j h_j. \]

The probability of this equality, in general, cannot be expressed by a numerical quantity: it is the smaller, the more areas \(\Delta F\) there are. This method can be adopted only when, in addition to the indications from boreholes, there are also general geological indications, or indications obtained by working, that the character and composition of the vein—bed are sufficiently constant. If these indications are absent, then the deposit should be divided into equal and, moreover, small areas, while nevertheless taking into account the indications of the boreholes regarding the character and composition of the ore, since thanks to them some increase in the accuracy of the determination is obtained; and in each such area a pit should be sunk. The arithmetic-mean sample will give us the sought percentage composition if \(h\) is everywhere the same; otherwise,

\[ G=\sum \Delta V_j G_j:V. \]

As regards the accuracy of determinations of the composition of the deposit, we must make the following remark: all our reasoning is based on some probability or other that the given determination corresponds to reality; conversely, \(Q=1-P\) will give us the probability that it is false, po–

therefore it seems pointless to go into the details of the measurements and calculations, since all that has been said and all the conclusions sufficiently indicate the nature of these determinations. In cases where the probability has been brought almost to 1, and thereby \(Q = 0\), the significance of errors in measurements and calculations increases; but since such cases occur extremely rarely, we need not dwell on them.

Exploration of Vein Deposits by Means of Mine Workings

The method of exploration just described, which at the same time has the character of preparatory work, is applied not only to veins, but also to ore beds and deposits in which the useful mineral occurs in an unevenly scattered form. The method consists in driving a system of horizontal drifts along the strike and inclined ones along the dip (rise), by which the entire field under exploration is divided into rectangles. Measurements of the thickness of the vein, made in these drifts, as well as samples taken from their various places, enable us to determine with sufficient accuracy the total reserve of the mineral.

a) Calculation of volume.

The volume of a part of the vein in the space under exploration may be calculated in various ways. The simplest method is

\[ V = LEh, \tag{40} \]

where \(L\) is the horizontal length measured along the drifts, \(E\) is the inclined length of the field, and \(h\) is the arithmetical mean of all measurements of thickness.

Calculating the individual rectangles \(ABCD\) (Fig. XVI) and summing them, we find

\[ V = \sum_{1}^{s} le \frac{h_0 + h'_0}{2}, \]

where \(h_0\) is the arithmetical mean of the measurements along the horizontal drift intersecting the area \(ABCD\) through its middle, and \(h'_0\) is that of the measurements along the inclined drift. If the distances between the drifts are equal, then (\(s\) being the number of areas)

\[ V = le \sum_{1}^{s} \frac{h_0 + h'_0}{2}. \]

This expression is somewhat more rational than the preceding one. To obtain it we assume that the thickness of the vein is determined by measurements along the horizontal drift upward and downward at distances \(\frac{e}{2}\), and from the inclined drift at distances \(\frac{l}{2}\). The second calculation is based on

assumption that between two points at which the thickness of the vein is different, there exists a gradual and uniform transition. Of course, such an assumption is not correct: the law of variation of thickness is another matter, but it is not known to us at all, and therefore such an assumption is natural. However, reasoning more strictly, on the basis of the indicated assumption we shall arrive at somewhat different conclusions. Let us imagine that at the points \(a\) and \(b\) the thickness of the vein has been measured; the question is, what thicknesses should be adopted at the intermediate points of the strip \(ab\)?

According to the assumption of a uniform transition from point \(a\) to \(b\), on the basis of the proportionality of the sides (Fig. XVII) of similar triangles:

\[ h_y=(h_{l2}-h_{l1})\frac{y}{e}+h_{l1}=h_{l2}\frac{y}{e}+h_{l1}\frac{e-y}{e}. \tag{38} \]

Let this expression refer to the intermediate points between two horizontal strokes. The thickness of the vein between two inclined strokes is determined by:

\[ h_x=(h_{e2}-h_{e1})\frac{x}{l_1}+h_{e1}=h_{e2}\frac{x}{l_1}+h_{e1}\frac{l-x}{l}, \tag{39} \]

where \(h'_{l1}, h''_{l1}, \ldots, h'_{l2}, \ldots, h_{e1}, h'_{e1}, \ldots, h'_{e3}, \ldots\)—are the heights of the horizontal and vertical strokes at the points \(a, a_1, a_2, \ldots, b, b_1, \ldots, c, \ldots, d\).

The height of the prism obtained from the intersection of both strips, whose widths are \(\Delta x\) and \(\Delta y\) (inclined \(\Delta x\), horizontal \(\Delta y\)),

\[ h_{xy}=\frac{1}{2}(h_y+h_x). \]

We are compelled to take the arithmetic mean of the heights determined by both formulas, since we do not know which of them should be preferred, and in general we do not know how to compose the dependence

\[ h_{xy}=f(h_x,h_y); \]

we know only one thing, namely that

\[ h_x \gtrless h_{xy} \gtrless h_y, \]

depending on whether \(h_x>h_y\) or, conversely, \(h_x<h_y\), that is, that \(h_{xy}\) lies between \(h_x\) and \(h_y\).

Since the prism of which we have mentioned has a very small base, we have the right to assume that the roof of the vein is parallel to the floor at the point \(xy\), and that, in view of this, the height of this prism is everywhere the same—

the slope is the same and equal to \(h_{xy}\). On the basis of what has been said, the elementary volume of the prism in question will be expressed as

\[ \Delta^{2}v_{xy}=\Delta x\Delta y\,h_{xy} =\Delta x\Delta y\,\frac{h_x+h_y}{2}; \]

and consequently the volume corresponding to the rectangle \(EFGH\) is

\[ V_1=\frac12\sum\sum h_x\Delta x\Delta y +\frac12\sum\sum h_y\Delta x\Delta y, \]

where the summation is to be carried out over all elementary prisms within the limits from 0 to \(l\) and from 0 to \(e\). If in the expression for \(V_1\) we substitute the previously determined values of \(h_x\) and \(h_y\) and carry out the operations, remembering that \(h_e\) does not depend on \(x\), and \(h_l\) on \(y\), and also that

\[ \sum_1^n x\Delta x=\frac{l^2}{2}\left(1+\frac1n\right),\qquad \sum_1^m y\Delta y=\frac{e^2}{2}\left(1+\frac1m\right), \]

since \(\Delta x=\dfrac{l}{n}\) and \(\Delta y=\dfrac{e}{m}\), then finally:

\[ V_1=\frac14 le\left[ \frac1m\left\{\left(1+\frac1n\right)\sum_1^m h_{e_2} +\left(1-\frac1n\right)\sum_1^m h_{e_1}\right\} +\frac1n\left\{\left(1+\frac1m\right)\sum_1^n h_{l_2} +\left(1-\frac1m\right)\sum_1^n h_{l_1}\right\} \right]. \]

But \(\sum_1^m h_{e_2}\Delta y\) and \(\sum_1^m h_{e_1}\Delta y\) represent nothing other than the areas of the longitudinal sections along the inclined shafts; let us call them \(F_{e_2}\) and \(F_{e_1}\), and likewise \(F_{l_2}\) and \(F_{l_1}\) the sections of the horizontal workings; in view of this

\[ \sum_1^m h_{e_2}\Delta y=\frac{e}{m}\sum_1^m h_{e_2}=F_{e_2},\qquad \sum_1^n h_{l_2}\Delta x=\frac{l}{m}\sum_1^n h_{l_2}=F_{l_2}, \]

and so on, and our expression will take the form

\[ V_1=\frac14 l\left[\left(1+\frac1n\right)F_{e_2} +\left(1-\frac1n\right)F_{e_1}\right] +\frac14 e\left[\left(1+\frac1m\right)F_{l_2} +\left(1-\frac1m\right)F_{l_1}\right]; \]

and since the quantities \(\dfrac{l}{4n}(F_{e_2}-F_{e_1})\) and \(\dfrac{e}{4m}(F_{l_2}-F_{l_1})\) are comparatively small, then, discarding them, we obtain

\[ V_1=\frac{l}{2}\left(\frac{F_{e_2}+F_{e_1}}{2}\right) +\frac{e}{2}\left(\frac{F_{l_2}+F_{l_1}}{2}\right). \]

Exploration of Vein and Stockwork Deposits

But

\[ l \cdot \frac{F_{e_2}+F_{e_1}}{2}=V_e, \]

that is, the volume calculated by the arithmetic mean of \(F_{e_1}\) and \(F_{e_2}\), and likewise

\[ e \cdot \frac{F_{l_2}+F_{l_1}}{2}=V_l, \]

therefore

\[ V_1=\frac{V_e+V_l}{2} \]

or, further,

\[ V_1=\frac{1}{4}\left(V_{e_1}+V_{e_2}+V_{l_1}+V_{l_2}\right), \tag{41} \]

that is, the arithmetic mean of the volumes computed as though the vein in the given rectangle were formed: 1) by moving the profile of the lower horizontal drift upward along the dip by a distance \(e\), 2) the upper one downward by the same distance, 3) the left inclined drift to the right by a distance \(l\), and the right one to the left by the same distance.

For the calculations one may use the formula

\[ V_1=\frac{l_e}{4}\left[\frac{1}{m}\left(\sum_{1}^{m}h_{e_1}+\sum_{1}^{m}h_{e_2}\right)+\frac{1}{n}\left(\sum_{1}^{n}h_{l_1}+\sum_{1}^{n}h_{l_2}\right)\right], \tag{42} \]

where \(\frac{1}{m}\sum_{1}^{m}h_{e_1}\), \(\frac{1}{n}\sum_{1}^{n}h_{l_1}\), … are the arithmetic mean heights along the four drifts bounding the rectangle. The sum of such volumes will give us the volume of the vein in the explored field. In the expression \(V_1\) there enters the arithmetic mean of four arithmetic means, so that we may write

\[ V_1=l_e\,\frac{h_1+h_2+h_3+h_4}{4}=l_eh_0. \]

If the field is divided into equal rectangles, then

\[ V=l_e\sum_{1}^{s}h_j, \]

where \(s\) denotes the number of these rectangles. Let

\[ \frac{1}{s}\sum_{1}^{s}h_j=h \]

be the arithmetic mean of the mean thicknesses over all the rectangles, and since \(sle=LE\), then, finally,

\[ V=LEh, \]

where, however, \(h\) has a somewhat different meaning than in eq. (40).

The calculation presented is based on the assumption of a gradual and uniform change in the thickness of the vein, and therefore it is valid only to the extent that our assumption agrees with reality. Strictly speaking, we have calculated only the volume occupied by the drifts, and from this volume we judge the adjoining volume. Two horizontal and two inclined drifts adjoin the square \(EFHG\). The volume of the rock extracted from them is calculated by the formula

\[ u=b\frac{e}{n}\left(\sum_1^n h_{e_1}+\sum_1^n h_{e_2}\right) +a\frac{l}{m}\left(\sum_1^m h_{l_1}+\sum_1^m h_{l_2}\right)+\gamma, \]

since we make measurements at intervals of \(\frac{e}{n}\) and \(\frac{l}{n}\) meters; \(\gamma\) is the volume of the four prisms at the corners,

\[ \gamma=ab\left(h_{l_1e_1}+h_{l_2e_2}+h_{l_1e_1}+h_{l_2e_2}\right). \]

The ratio \(\frac{n+\gamma}{v_1}\) may serve as a certain measure of the accuracy of the calculation, and also as its probability in the case when the law of variation of the vein thickness is not fully known to us. Since, strictly speaking, the volume \(V\) is not known to us, instead of the ratio \(\frac{n+\gamma}{v_1}\) it is rational to adopt the ratio of the areas of the rectangle to the areas of the drifts. Thus,

\[ p=\frac{2be+2al}{le}\,\omega+\frac{4ab}{le}\omega \, ^1). \]

If, for simplicity, we denote the ratios \(\frac{a}{e}\) and \(\frac{b}{l}\) respectively by \(\alpha\) and \(\beta\), then the expression for the probability takes the form

\[ p=2(\alpha+\beta)\omega+4\alpha\beta\omega. \tag{43} \]

Of course, within the limits of practical possibility, one should try to make \(p\) as close as possible to 1, which, however, cannot in fact be achieved. Thus, if the field is divided into squares and therefore \(a=b\), then \(\alpha=\beta\); with \(p=1\) and \(\omega=1\), our equation takes the form

\[ 4\alpha^2+4\alpha-1=0, \]

whence

\[ \alpha=\frac{a}{l}=-\frac{1}{2}\pm\frac{1}{2}\sqrt{2}; \]

and since the negative value of \(\alpha\) has no meaning in the given case, then

\(^1\) Here \(\omega\) is the probability that the ratio of the areas corresponds to the ratio of the volumes.

\[ \frac{a}{l}=\frac{1}{2}\left(\sqrt{2}-1\right)=0.205, \]

and approximately \(l=5a\), which, of course, would cost excessively much. Strictly speaking, there is no need to increase \(p\) at once to the limit 1, for \(p\) is, so to say, only a certain minimum probability, which in reality is much greater. Thus, if the assumption on the basis of which we made the calculation regarding the uniform variation of thickness is unconditionally valid, then the probability that the calculation is correct is equal to 1; conversely, if uniformity is absent, then, supposing that the structure of the vein remains the same as in the adits, we express the probability \(p\) by equation (43). In reality, the probability of our calculation is neither \(p\) nor 1, but some quantity lying between them, which, before exploration—especially if the deposit in general has been little studied—we are not in a position to indicate. Therefore, when designing exploration work, one should not be troubled by the small numerical value of \(p\) according to equation (43).

If exploration has been carried out, then, with regard to the correctness of our assumption, we can obtain indications as to uniformity on the basis of the following reasoning. Let \(M\) denote the number of horizontal adits, and \(N\) the number of inclined adits; then along a certain horizontal line passing through the points \(h l_1,\ h'l_2,\ h'l_3,\ldots,\ h'l_M\), under the condition of uniformity,

\[ \frac{2h'l_1}{h'+h'l_3}=1,\quad \frac{2h}{h'l_2+h'l_4}=1,\ldots; \]

if, however, \(h'l_2\) is not strictly the arithmetical mean of \(h'l_1\) and \(h'l_3\), then uniformity is absent. Let us form the equalities:

\[ \left. \begin{aligned} T'_{(l_1h)}&= \frac{2(M-2)} {\dfrac{h'_{l(1)}+h'_{l(3)}}{h'_{l(2)}}+ \dfrac{h'_{l(2)}+h'_{l(4)}}{h'_{l(3)}}+\cdots+ \dfrac{h'_{l(M-2)}+h'_{l(M)}}{h'_{l(M-1)}}} \\[1.2em] T''_{(l_1h)}&= \frac{2(M-2)} {\dfrac{h''_{l(1)}+h''_{l(3)}}{h''_{l(2)}}+ \dfrac{h_{l2}+h''_{l4}}{h''_{l3}}+\cdots+ \dfrac{h''_{l(M-2)}+h''_{lM}}{h''_{l(M-1)}}} \\[1.2em] &\qquad\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \\[0.6em] T'_{(e_1h)}&= \frac{2(N-2)} {\dfrac{h'_{e1}+h'_{e(3)}}{h'_{e(2)}}+ \dfrac{h'_{e(2)}+h'_{e(4)}}{h'_{e(3)}}+\cdots+ \dfrac{h'_{e(N-2)}+h'_{e(N)}}{h'_{e(N-1)}}} \\[0.8em] &\qquad\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \end{aligned} \right\} \tag{44} \]

The number of equalities \(T_{(l,h)}\) corresponds to the number of measurements of heights along the horizontal drift, \(T_{(e,h)}\)—along the inclined one. The closer the uniformity is to strictly mathematical uniformity, the closer every \(T_{(l,h)} - T_{(e_1,h)}\) is to unity. If the arithmetic means of these determinations,

\[ T_{(l,h)}=\frac{1}{s}\sum_1^s T_{(l,h)}, \qquad T_{(e,h)}=\frac{1}{r}\sum_1^r T_{(e_2,h)}, \]

which, just like every \(T<1\), give quantities that differ rather greatly from 1, then new intermediate drifts should be driven and the volume computed anew.

b) Content of the useful mineral.

Simultaneously with the driving of drifts and the measurements of thickness, samples of the vein rock are taken in order to determine, by laboratory means, the mineral content in them; independently of this, the extracted ore may go to concentrating plants for a bulk sample.

Let us assume, first, that the samples from a drift are taken correctly, that is, that every sample really corresponds to a certain part of the drift, and consequently also of the vein—this is feasible if the extracted rock, say over \(\Delta x\) or \(\Delta y\) meters, is piled in a stack, and from it a reduced sample is taken. Thus the probability \(p_0\) that \(\Delta u\), taken for the sample, corresponds to \(\Delta V\), is equal to 1. In addition, let us suppose that the thickness of the vein at the place where the sample is taken is known, since it has been measured. In order to be able to extend the results of the samples to the whole vein, we must also make the same hypothetical assumption as in determining the volume, that is, that the composition of the vein changes gradually and uniformly. In view of the foregoing, if at points \(a\) and \(b\) of the horizontal drift the samples gave determinations \(g_{l_2}\) and \(g_{e_2}\), and along the inclined ones \(g_{e_2}\) and \(g_{e_1}\), then, reasoning in exactly the same manner as before, we shall find:

\[ \begin{aligned} g_x&=g_{e_2}\frac{x}{l}+g_{e_1}\frac{l-x}{l},\\ g_y&=g_{l_2}\frac{y}{e}+g_{l_1}\frac{e-y}{e}. \end{aligned} \tag{45} \]

The content of the elementary prism \(\Delta x\Delta y\), situated at a distance \(x\) from drift \(e\), will be determined by:

\[ \Delta^2 G_{ex}=(g_x h_x)\Delta x\Delta y; \]

and the same, at a distance \(y\) from drift \(l\),

\[ \Delta^2 G_{ly}=(g_y h_y)\Delta x\Delta y; \]

and, finally, of the prism formed by the intersection of two imaginary strips, of horizontal width \(\Delta x\) and inclined width \(\Delta y\), as the arithmetical mean of these determinations:

\[ \Delta^{2}G_{xy}=\frac{1}{2}\left(\Delta^{2}G_{ex}+\Delta^{2}G_{ly}\right) =\frac{1}{2}\left(g_x h_x+g_y h_y\right)\Delta x\Delta y. \]

Summing \(\Delta^{2}G_{xy}\) from 0 to \(l\) in \(x\) and from 0 to \(e\) in \(y\), we finally arrive at the expression

\[ G_{le}=\frac{1}{2}\left(G_l+G_e\right), \]

showing that the reserve corresponding to the given rectangle is equal to the arithmetical mean of the reserves computed on the basis of measurements and samples along the two adjacent horizontal adits and along the two adjacent inclined ones that bound the given rectangle.

Let us now compute \(G_l\) and \(G_e\) separately for each of them:

\[ \begin{aligned} G_l &=\sum\sum \Delta^{2}G_{ly} =\sum\sum\left(g_{l_2}\frac{y}{e}+g_{l_1}\frac{e-y}{e}\right) \left(h_{l_2}\frac{y}{e}+h_{l_1}\frac{e-y}{e}\right)\Delta x\Delta y \\ &=\sum_{1}^{n} g_{l_2}h_{l_2}\Delta x\cdot \sum_{1}^{m}\frac{y^2}{e^2}\Delta y +\sum_{1}^{n}\left(g_{l_1}h_{l_2}+g_{l_2}h_{l_1}\right)\Delta x \sum_{1}^{m}\frac{y(e-y)}{e^2}\Delta y \\ &\quad+\sum_{1}^{n}g_{l_1}h_{l_1}\Delta x \sum_{1}^{m}\frac{(e-y)^2}{e^2}\Delta y . \end{aligned} \]

But since

\[ \sum_{1}^{m}\frac{y^2}{e^2}\Delta y =\frac{1}{3}\left(1+\frac{1}{m}\right)\left(1+\frac{1}{2m}\right)e =\frac{1}{3}\left(1+\frac{3}{2m}+\frac{1}{2m^2}\right)e, \]

\[ \sum_{1}^{m}\frac{y(e-y)}{e^2}\Delta y =\frac{1}{2}\left(1+\frac{1}{m}\right)e -\frac{1}{3}\left(1+\frac{1}{m}\right)\left(1+\frac{1}{2m}\right)e =\frac{1}{6}\left(1-\frac{1}{m^2}\right)e, \]

\[ \sum_{1}^{m}\frac{(e-y)^2}{e^2}\Delta y =e-\left(1+\frac{1}{m}\right)e +\frac{1}{3}\left(1+\frac{1}{m}\right)\left(1+\frac{1}{2m}\right)e =\frac{1}{3}\left(1-\frac{3}{2m}+\frac{1}{2m^2}\right)e, \]

and, moreover, since the degree of accuracy of the calculation being made allows us to discard \(\frac{1}{m^2}\), as well as \(+\frac{3}{2m}\) and \(-\frac{3}{2m}\), for the multipliers at these quantities differ little from one another, then

\[ G_l=\frac{2e}{3}\left[\sum_1^n \frac{g_{l_2}+g_{l_1}}{2}\cdot \frac{h_{l_2}+h_{l_1}}{2}\Delta x +\frac12\left(\sum_1^n g_{l_2}h_{l_2}\Delta x+\sum_1^n g_{l_1}h_{l_1}\Delta x\right)\right] \]

or

\[ G_l=\frac{2}{3}\left[G\left(\frac{l_1+l_2}{2}\right)+\frac12(G_{l_1}+G_{l_2})\right]. \]

Proceeding in an entirely analogous manner, one can find for \(G_e\):

\[ G_e=\frac{2l}{3}\left[\sum_1^m \frac{g_{e_2}+g_{e_1}}{2}\cdot \frac{h_{e_2}+h_{e_1}}{2}\Delta y +\frac12\left(\sum_1^m g_{e_2}h_{e_2}\Delta y+\sum_1^m g_{e_1}h_{e_1}\Delta y\right)\right] \]

or

\[ G_e=\frac{2}{3}\left[G\left(\frac{e_1+e_2}{2}\right)+\frac12(G_{e_1}+G_{e_2})\right]. \]

Thus, the geometric expression of the reserve corresponding to one square will be:

\[ G=\frac13\left[ G\left(\frac{l_1+l_2}{2}\right) +G\left(\frac{e_1+e_2}{2}\right) +\frac14(G_{l_1}+G_{l_2}+G_{e_1}+G_{e_2}) \right], \tag{46} \]

and the corresponding arithmetical expression is

\[ \begin{aligned} G=\frac{el}{3}\Biggl[ &\frac1n\sum_1^n \frac{g_{l_2}+g_{l_1}}{2}\cdot \frac{h_{l_2}+h_{l_1}}{2} +\frac14\sum_1^n (g_{l_2}h_{l_2}+g_{l_1}h_{l_1})\\ &+\frac1m\sum_1^m \frac{g_{e_2}+g_{e_1}}{2}\cdot \frac{h_{e_2}+h_{e_1}}{2} +\frac14\sum_1^m (g_{e_2}h_{e_2}+g_{e_1}h_{e_1}) \Biggr], \end{aligned} \tag{47} \]

where \(G_{l_2},\ldots,G_{e_2}\) are the content (reserve) calculated from the data in one of the adits with their extension over the whole rectangle; \(G\left(\frac{l_1+l_2}{2}\right)\) and \(G\left(\frac{e_1+e_2}{2}\right)\) are reserves calculated from the arithmetic-mean data of two neighboring parallel adits.

If we make the assumption that it is not the percentage content, but the reserve of a square unit of the inclined plane of the vein that changes uniformly and gradually, so that from \(g_{e_1}h_{e_1}\) to \(g_{e_2}h_{e_2}\) there is a gradual and uniform transition, then

\[ g_{e_x}h_{e_x}=g_{e_2}h_{e_2}\frac{x}{l}+g_{e_1}h_{e_1}\frac{l-x}{l}, \]

\[ g_{l_y}h_{l_y}=g_{l_2}h_{l_2}\frac{y}{e}+g_{l_1}h_{l_1}\frac{e-y}{e}. \]

The reserve corresponding to the rectangle bounded by four adits,

\[ G=\frac{1}{2}\sum\sum\left(g_{e_x}h_{e_x}+g_{l_y}h_{l_y}\right)\Delta x\Delta y . \]

Whence, analogously:

\[ G=\frac{el}{4}\left[ \frac{1}{m}\left(\sum_{1}^{m} g_{e_2}h_{e_2}+\sum_{1}^{m} g_{e_1}h_{e_1}\right) +\frac{1}{n}\left(\sum_{1}^{n} g_{l_2}h_{l_2}+\sum_{1}^{n} g_{l_1}h_{l_1}\right) \right] \tag{48} \]

or

\[ G=\frac{1}{4}\left(G_{l_2}+G_{l_1}+G_{e_2}+G_{e_1}\right). \tag{49} \]

In this expression \(G_{l_2}, \ldots, G_{e_2}\) have the same meanings as in (46). If we compare expressions (46) and (49), we shall see that the expression derived here constitutes one of the three terms of which (46) consists. In general, this latter computation is rougher than the former.

As for the a priori probability of our calculations, of course its numerical value cannot be expressed by anything other than the ratio of the areas of the strictly explored space, that is, of the workings, to the explored part of the vein; thus formula (43) is applicable for this case, only \(\omega\) denotes in it the probability that the ratio of the areas corresponds to the ratio of the mineral content. As for the a posteriori probability, that is, after the explorations have been carried out, its determination reduces to determining the accuracy of the assumption placed at the basis of the calculations, and therefore, for case 1, to the uniformity of variation of the composition of the vein; for case 2, to the uniformity of variation of its content per square unit of the vein area.

For the first case we can set up a series of equalities

\[ \left. \begin{aligned} T'_{(e_1,g)}&= \frac{2(N-2)} {\dfrac{g'_{e_1}+g'_{e_3}}{g'_{e_2}} +\dfrac{g'_{e_2}+g'_{e_4}}{g'_{e_3}} +\cdots+ \dfrac{g'_{e(N-2)}+g'_{eN}}{g'_{e(N-1)}}},\\ &\qquad \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots,\\ T'_{(l_1,g)}&= \frac{2(M-2)} {\dfrac{g'_{l_1}+g'_{l_3}}{g'_{l_2}} +\dfrac{g'_{l_2}+g'_{l_4}}{g'_{l_3}} +\cdots+ \dfrac{g'_{l(M-2)}+g'_{l(M)}}{g'_{l(M-2)}}},\\ &\qquad \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots; \end{aligned} \right\} \tag{50} \]

where all the notations are the same as in eq. (44), \(g'_{e_1}, \ldots, g^{s}_{l_1}, \ldots, g^{r}_{e_1}\) are known. In general the meaning of the equations is the same as in determining the volume. For case 2 one should proceed in exactly the same way, only instead of \(g_e\ldots\) one should place \(g_{e_1}h_{e_1}, \ldots\), so that equalities are obtained.

\[ I(e_g h_g)= \frac{N-2}{ \dfrac{g'_{e_1}h'_{e_1}+g'_{e_3}h'_{e_3}}{g'_{e_2}h'_{e_2}} + \dfrac{g'_{e_2}h'_{e_2}+g'_{e_4}h'_{e_4}}{g'_{e_3}h'_{e_3}} +\ldots+ \dfrac{h'_{e(N-2)}+g'_{e(N)}}{g'_{e(N-1)}} }. \]

Addendum

To explain formulas (6) and (7), I propose a purely geometrical derivation of them. We shall consider the mouths of the boreholes as reduced to one horizon (Drawing XVIII).

Through the point of intersection \(A\) of the shallowest borehole \(h_0\) with the stratum, we draw a horizontal plane, which intersects boreholes \(h_1\) and \(h_2\) at the points \(B_1\) and \(C_1\). The line of intersection of this horizontal plane with the plane of the stratum \((ABC)\) gives us the direction of the strike of the stratum \(QP\); the perpendicular to it \(AS\) is the direction of dip.

If through the boreholes \(h_1\) and \(h_2\) we draw planes perpendicular to \(QP\), then the angles \(B_1QB\) and \(C_1PC\) will give us the magnitude of the dip of the stratum \(\eta\).

From the similar right triangles \(BB_1Q\) and \(PC_1C\) we have:
\[ \frac{QB_1}{PC_1}=\frac{BB_1}{CC_1}. \]

The right triangles \(AB_1Q\) and \(PC_1A\) give us the relation:
\[ \begin{aligned} QB_1&=l_1\sin[180^\circ-(\alpha-\xi+90^\circ)] =l_1\cos(\xi-\alpha),\\ PC_1&=l_2\sin[\beta-\xi+90^\circ] =l_2\cos(\xi-\beta); \end{aligned} \]
and since, moreover, \(BB_1=h_1-h_0\) and \(CC_1=h_2-h_0\), one may write
\[ \frac{l_1\cos(\xi-\alpha)}{l_2\cos(\xi-\beta)} = \frac{h_1-h_0}{h_2-h_0}. \]

Let \(\xi-\beta=\varepsilon\), and consequently \(\xi-\alpha=\varepsilon+\beta-\alpha\); then our equation, in expanded form, may be written thus:
\[ \frac{\cos\varepsilon\cos(\beta-\alpha)-\sin\varepsilon\sin(\beta-\alpha)} {\cos\varepsilon} = \frac{l_2}{l_1}\cdot\frac{h_1-h_0}{h_2-h_0} =\mu \]
or else
\[ \cos(\beta-\alpha)-\tg\varepsilon\,\sin(\beta-\alpha)=\mu, \]

whence

\[ \operatorname{tg}\varepsilon=\frac{\cos(\beta-\alpha)-\mu}{\sin(\beta-\alpha)}. \]

If we substitute the former notation, then

\[ \operatorname{tg}(\xi-\beta)=\operatorname{cot}(\beta-\alpha)-\frac{l_2}{l_1}\cdot\frac{h_1-h_0}{h_2-h_0}\cdot\frac{1}{\sin(\alpha-\beta)}. \tag{6} \]

The dip of the bed is obtained from the same triangles \(BB_1Q\) and \(CC_1P\), namely

\[ \operatorname{tg}\eta=\frac{BB_1}{B_1Q}=\frac{CC_1}{C_1P}, \]

and consequently

\[ \operatorname{tg}\eta=\frac{h_1-h_0}{l_1}\,\frac{1}{\cos(\xi-\alpha)} =\frac{h_2-h_0}{l_2\cos(\xi-\beta)}. \tag{7} \]

[Table of drawings]

Drawing 16. Plan grid with square blocks. Visible labels: \(A\), \(B\), \(C\), \(D\); \(a\), \(b\), \(c\), \(d\); \(x\), \(y\); \(n\); \(l\), \(e\), \(h\).

Drawing 10. Section/profile with inclined lines. Visible labels include \(a\), \(b\), \(c\), \(d\), \(e\), \(f\), \(g\), \(h\); \(10\) sazhen.

Drawing 11. Small rounded sketch with crossing lines.

Drawing 14. Area plan with irregular shaded patches and point marks. Visible labels include \(a\), \(b\), \(c\), \(d\), \(e\), \(f\).

Drawing 15. Section with horizontal strata and a shaded body.

Drawing 12. Section/profile with inclined lines and vertical divisions. Visible labels include \(a\), \(b\), \(c\), \(d\), \(e\), \(f\), \(g\), \(h\).

Drawing 13. Plan diagrams with point marks and square/hexagonal/triangular grids.

Drawing 9. Section/profile with fan-like lines.

Drawing 7. Small section/profile with an arrow and label \(y\).

Drawing 4. Large geological section. Visible labels include \(A\), \(B\), \(C\), \(D\), \(O\); \(1\)-st, \(2\)-nd, \(3\)-rd, \(4\)-th; and dotted construction lines.

Drawing 2. Inclined body/vein diagram. Visible labels include \(c\), \(d\), \(g\), \(O\); \(2\)-nd.

Drawing 3. Inclined body/vein diagram. Visible labels include \(A\), \(c\); \(2\)-nd; \(h\).

Drawing 8. Plan/section construction with visible labels \(x\), \(y\), \(a\), \(b\), \(c\), \(d\), \(e\).

1. See Wysław, Investigation of the ground and exploration of useful minerals by means of a hand drill, and the supplement. ↩↩