Drillability of Rocks and the Construction of a Rational Cycle in Core Drilling with Cemented Carbides

Mining engineer V. A. Nudner.

INTRODUCTION

A properly constructed cycle is one of the basic conditions for the success of any work. This question is of decisive importance, in particular, in the core drilling of boreholes—an industry that in our country still stands at a very low production and technical level.

The aim of the present work is to attempt to develop a methodology for constructing a rational working cycle in core drilling, using as an example the work of the drilling crews of the Leninsk Geological-Prospecting Office of the “Kuzbassugol” Combine.

The main task that had to be solved in connection with the question posed consisted in establishing the correct relationship between the duration of the time of pure drilling, on the one hand, and the duration of all the other operations included in the cycle, on the other. It was necessary to determine the influence on this relationship of the various factors encountered in the drilling process—above all, the depth of the borehole and the character of the rocks being penetrated.

We were also compelled to investigate the influence of the duration of the bit’s work at the bottom on the rate of its advance. It should be noted that this latter question has long been within the sphere of attention of various investigators concerned with questions of rock drillability. It has long been established that the rate of advance of the bit falls as the duration of its work at the bottom increases. Clear examples of this may be found in the work of B. I. Vozdvizhensky and N. I. Kulichikhin (2), in the work of E. V. Borovsky and F. A. Shamshev (1), and in a number of other authors, in articles in the journal Razvedka nedr, etc. Unfortunately, none of these authors gives a general law for the process under consideration or establishes quantitative regularities in it.

An attempt to establish quantitative regularities and to express in mathematical form the law of variation of the rate of advance—

The dependence of bit penetration on the time of its operation at the face was made by I. M. Mlodik (5), who proposed the formula

\[ S = ae^{bt}. \]

Here
\(t\) — time of operation of the bit,
\(a\) and \(b\) — constant coefficients,
\(s\) — path traveled by the bit during time \(t\).

As M. V. Machinsky (5) correctly noted, this formula is not satisfactory, since the rate of penetration of the bit, which is determined from this formula in the form of the expression

\[ v = \frac{ds}{dt} = abe^{bt}, \]

turns out to increase with an increase in the duration of operation of the bit (time \(t\)), which is clearly contrary to reality.

M. V. Machinsky (5), in a work devoted to the theory of core drilling, also gives a formula for determining the rate of penetration of the bit, but this formula is very cumbersome and contains a large number of different physical constants and coefficients requiring preliminary experimental determination. Therefore, this formula cannot yet be used for our purposes.

Finally, the work of E. F. Epstein (8) should be noted, in which he proposed the following formula for determining the advance of the bit during time \(t\):

\[ S = \frac{Cnt}{\sigma_p (R-r)\operatorname{tg}\beta} - \frac{2}{3} \sqrt{ \frac{2\omega C\pi f(R+r)n^3 m}{(R-r)\operatorname{tg}\beta} } \cdot Vt^3 \]

Here
\(C\) — axial pressure on the bit,
\(n\) — number of revolutions of the bit per minute,
\(\omega\) — coefficient of volumetric wear of the cutter in \(\text{cm}^3/\text{kgm}\),
\(R\) — outer radius of the bit,
\(r\) — inner radius of the bit,
\(f\) — coefficient of friction,
\(m\) — number of cutters in the bit,
\(\sigma_p\) — destructive stress of the rock,
\(\beta\) — angle of sharpening of the cutter,

It is now difficult to say how suitable this formula is. It requires experimental verification under various drilling conditions and the establishment of \(\sigma_p\) and the coefficient \(\omega\).

M. V. Machinsky and E. F. Epstein obtained their formulas by proceeding from purely theoretical constructions. We have taken, as will be seen below, a different path, using the method of statistical processing of time-study observations of the rate of penetration of the bit under actual drilling conditions.

As regards determining the optimal duration of pure drilling in a cycle, this question, as far as we know, has not been covered at all in the literature, apart from the formulation of this problem by A. M. Golyanitskii (7), who constructed diagrams of the average drilling speed as a function of borehole depth for 9- and 12-meter drilling derricks. But this author does not give the basic dependence of the duration of the bit’s stay at the bottom on the character of the rocks being penetrated, and does not draw generalizing quantitative conclusions.

In the present work we have attempted to obtain a general formula for determining the optimal duration of operation of the bit at the bottom as a function of borehole depth, the length of the rods used, and the decrease in the rate of advance of the bit with time.

I consider it my pleasant duty to express here my deep gratitude to Prof. I. A. Molchanov, who repeatedly helped me with his advice, and to B. A. Fortes, a student of the Tomsk Industrial Institute named after S. M. Kirov, who performed a substantial part of the relatively very large amount of computational work connected with the topic under consideration.

Chapter I

Operations of the Drilling Process and Their Classification

In the study of any complex phenomenon, in order to reveal the laws contained in it, that phenomenon must be broken down into separate elements. Therefore, naturally, when undertaking an analysis of core drilling—a very complex process that contains a whole complex of different, extremely diverse factors, often not accessible to direct observation—it is necessary first of all to break this process down into its constituent elements, the totality of which, taken in a definite sequence, constitutes core drilling proper. Attention must be paid to the study of the elements of the process and to the development of their rational classification, since the entire subsequent analysis, and then the synthesis of the process, is built upon this.

The entire set of operations of which the sinking of a core borehole is composed has long been customarily divided into three main groups:

1. Preparation for drilling a borehole, which includes work on erecting the drilling derrick, digging pits at the collar of the borehole and for settling tanks, assembling mechanisms, and other preparatory drilling operations.

2. Drilling—a group of operations comprising the work of actually sinking and casing the borehole, as well as various geological and hydrogeological observations and work at the borehole that take place during its sinking.

3. Abandonment of the borehole, including operations for extracting casing pipes, final cementation of the borehole performed after completion of its sinking, operations for dismantling mechanisms, dismantling the drilling derrick, securing the borehole collar, etc.

In the present work we dwell only on an analysis of the most important operations of the second group, which are of primary significance in the overall process of sinking a borehole and are directly connected with the subject under consideration. For this reason the classification given below (Table 1) covers only the operations of this second group from the entire complex of work associated with sinking a borehole.

Looking closely at the drilling process, it is easy to notice in it, first, a definite sequence of operations and, second, their regular periodic recurrence. Thus, during drilling waterways...

Table 1.

Classification of operations in the drilling process.

Large cycle	Small cycle	Type of operations	Duration category	Operation
Large cycle	Small cycle	Main operations	Duration depends on the amount of advance per small cycle.	Pure drilling.
Large cycle	Small cycle	Auxiliary operations	Duration depends on the depth of the borehole.	Running drill-rod stands into the borehole.
Large cycle	Small cycle	Auxiliary operations	Duration depends on the depth of the borehole.	Hoisting drill-rod stands from the borehole.
Large cycle	Small cycle	Auxiliary operations	Duration is constant within the small cycle.	Other operations when lowering the tool into the borehole.
Large cycle	Small cycle	Auxiliary operations	Duration is constant within the small cycle.	Other operations when hoisting the tool from the borehole.
Large cycle	Small cycle	Auxiliary operations	Duration is constant within the small cycle.	Placing the tool on the bottom.
Large cycle	Small cycle	Auxiliary operations	Duration is constant within the small cycle.	Operations for wedging and breaking off the core.
Large cycle	Small cycle	Auxiliary operations	Duration is constant within the small cycle.	Operations with the core-barrel assembly.
Large cycle	Additional cycle	Spudding the borehole and operations for changing the diameter	Spudding the borehole and operations for changing the diameter	Spudding the borehole.
Large cycle	Additional cycle	Spudding the borehole and operations for changing the diameter	Spudding the borehole and operations for changing the diameter	Operations for changing the borehole diameter.
Large cycle	Additional cycle	Casing the borehole	Casing the borehole	Preparation for casing.
Large cycle	Additional cycle	Casing the borehole	Casing the borehole	Running casing pipes.
Large cycle	Additional cycle	Casing the borehole	Casing the borehole	Other operations during casing.
Large cycle	Additional cycle	Miscellaneous work	Miscellaneous work	Minor operations occurring during drilling.
Work not relating to drilling proper	Work not relating to drilling proper	Episodic work	Episodic work	Cementation and grouting.
Work not relating to drilling proper	Work not relating to drilling proper	Episodic work	Episodic work	Artificial deviation of the borehole.
Work not relating to drilling proper	Work not relating to drilling proper	Episodic work	Episodic work	Reaming (enlarging) the borehole.
Work not relating to drilling proper	Work not relating to drilling proper	Episodic work	Episodic work	Other episodic work.
Work not relating to drilling proper	Borehole documentation	Geological observations	Geological observations	Measurements of deviations.
Work not relating to drilling proper	Borehole documentation	Geological observations	Geological observations	Check measurements of borehole depth.
Work not relating to drilling proper	Borehole documentation	Hydrogeological observations	Hydrogeological observations	Measurements of the water level in the borehole.
Work not relating to drilling proper	Borehole documentation	Hydrogeological observations	Hydrogeological observations	Pumping and injection.

for each core borehole we have the following sequence of operations.

1. Lowering the tool into the borehole.
2. Setting the tool on the bottom.
3. Drilling proper.
4. Wedging and breaking off the core.
5. Raising the tool from the borehole.
6. Operations with the core barrel at the surface: extracting the core and preparing the barrel for a new lowering into the borehole in order to continue drilling.

This sequence begins with lowering the tool into the borehole and ends with preparing the core barrel for a new lowering, after which all the operations listed are repeated again in the same order. Thus, here we have a quite definite and clearly expressed cyclicity of the process. Therefore, the given sequence of operations is singled out in the classification as an independent group, which we call the small cycle.

Among the operations of the small cycle, some produce the actual productive effect—the deepening of the borehole—and directly accompany it, while others service this deepening. The former operations are combined into the subgroup of principal operations, the latter into the subgroup of auxiliary operations.

After the borehole has been drilled to a certain, generally varying, depth (which requires a known number of small cycles), drilling is suspended. The borehole is cased with casing pipes, another, smaller diameter of core barrel is taken, and drilling is resumed. Sometimes the borehole is not cased with pipes (clay flushing), but a change of diameters after certain intervals occurs almost always.

Operations for changing the diameter of the core barrel (centering a borehole of smaller diameter at the bottom of the borehole of the preceding diameter), collaring the borehole, and casing it with pipes are singled out in the classification as an independent group, which we call the additional cycle.

Operations connected with possible, but not necessarily encountered in drilling every borehole, works (cementation and plugging, artificial deviation, reaming—enlargement of the borehole, etc.) are singled out into an independent group of episodic works.

Operations that take place in connection with geological and hydrogeological observations and documentation are combined into the group borehole documentation. These operations may, generally speaking, be performed at various moments during drilling of the borehole, between individual small cycles.

Finally, the classification provides for a separate group of miscellaneous works. This group should include minor and sporadic works and time expenditures often necessary in

in the drilling process, but by their nature not falling into any of the main divisions established above (taking over and handing over the shift, lubricating the gears of the machine, etc.).

A series of small cycles spent on deepening the borehole from one shift to the next change of diameters (or from one casing operation to the next casing operation), the corresponding additional cycle, and a group of various operations are combined by us under the concept of the large cycle.

A known number of large cycles (their number depends on the construction of the borehole), episodic operations, if any were performed, the time expenditures for operations connected with borehole documentation and for other operations necessary in drilling, as well as certain losses of time in connection with possible stoppages and accidents in the course of the work, together form the complex of operations and time expenditures for the actual sinking of the borehole.

We shall now dwell in greater detail on the analysis of the operations of the small cycle.

Pure drilling is the time during which the bit works at the bottom, i.e., the actual deepening of the borehole is being carried out. This time, which we shall denote by \(t\), depends on the magnitude of the advance per cycle and on the rate of pure drilling, which, in turn, is a function of the drillability of the rocks and of the drilling regime (number of revolutions of the bit, pressure on the bottom, shape and character of sharpening, and number of cutters, etc.). Pure drilling is the most important element of the process, and the correct determination of its duration predetermines the success of drilling operations in general.

Operations accompanying pure drilling—in other words, operations connected with ensuring the progressive motion of the core barrel assembly located in the borehole. Consequently, this includes such operations as raising the core barrel (stopping the machine, releasing the chuck, the actual raising of the core barrel, fastening the chuck, starting the machine) and adding rods (stopping the machine and pump, unscrewing the stuffing box, screwing on the next rod, screwing on the stuffing box, starting the pump and the machine).

Together with pure drilling, these operations give what in our classification we called the basic operations and what in practice is mistakenly commonly called pure drilling.

The time spent within the limits of the small cycle on operations accompanying pure drilling is directly dependent on the magnitude of the advance for the given small cycle, for the greater the advance per cycle, the greater the number of times we shall be forced to raise the core barrel and add rods.

Let us introduce the following notation:

\(S\) — advance per small cycle, in mm;

\(n\) — useful length of the core barrel, i.e., its travel, in mm;

\(m\) — length of the rod being added, in mm;

\(t_k\) — time spent on raising one core barrel, in min;

\(t_{\sh}\) — time spent on adding one rod, in min;

$t_1$ is the time expended in the small cycle on raising the swivel and adding rods, i.e., on operations whose total duration in the cycle depends on the amount of advance per small cycle.

Then the total duration of these operations per cycle will be expressed as:

\[ t_1=\frac{S}{n}\cdot t_k+\frac{S}{m}\cdot t_{\mathrm{ш}}\text{ minutes.} \]

Lowering the tool into the borehole. This concept usually combines all the operations that make up the process of lowering the tool, beginning with the lowering into the borehole of the core-barrel assembly and ending with screwing the gland onto the upper rod and starting the pump and the rig after the lowering has been completed. Thus, here we have the following operations: raising the core-barrel assembly above the mouth of the borehole after screwing the bit onto it (or, if work is being carried out with two core-barrel assemblies, after screwing the swivel onto it), opening the borehole, lowering the core-barrel assembly into the borehole, placing the core-barrel assembly on the wrenches or rod holder, unscrewing the swivel, screwing on the next stand, raising the swivel, screwing on the swivel, removing the assembly from the wrenches or rod holder, the actual lowering of the stand, placing the stand on the wrenches or rod holder, unscrewing the swivel, screwing on the next stand, and so on. After the last stand has been lowered—removing the safety cover from the horizontal box, closing the rig box, lowering the upper rod through the spindle, screwing on the rod, screwing on the gland, fastening the chuck, and starting the pump and the rig.

Considering the operations listed, it may be noted that some of them, during the process of lowering the tool, are repeated a greater or lesser number of times depending on the depth of the borehole, while other operations of the process occur during the entire lowering of the tool into the borehole only once, regardless of the depth of the borehole. Consequently, the time occupied in the small cycle by operations of the first category is not constant; it differs for different cycles, increasing with the depth of the borehole. The time occupied by operations of the second category, however, is constant, does not depend on the depth of the borehole, and is therefore the same in every small cycle.

This circumstance made it necessary to divide the operations for lowering the tool into the borehole into two groups. The first group includes operations whose total duration within the small cycle depends on the depth of the borehole. They are combined under the term “operations for lowering stands.” This group includes operations beginning with screwing the first stand onto the core-barrel assembly and ending with unscrewing the swivel from the last stand lowered into the borehole, i.e., up to removal of the safety cover from the horizontal box of the rig. All the remaining

The operations listed above, encountered when lowering the tool into the borehole, belong to the second category. Their duration does not depend on the depth of the borehole, and they have been assigned to the group “other operations in lowering the tool.”

If we denote by

\(L\) — the depth of the borehole at the beginning of the cycle, in mm;

\(l\) — the length of a stand, in mm;

\(t_c\) — the duration of the operations for lowering one stand, in min.;

\(t'_2\) — the time required to lower all stands into the borehole in one small cycle, in min.;

\(t''_2\) — the duration of other operations for lowering stands in the small cycle, in min.;

\(t_2\) — the total duration of the operations for lowering the tool into the borehole in the small cycle,

then

\[ t'_2=\frac{L}{l}\cdot t_c \]

and

\[ t_2=t'_2+t''_2 \]

or

\[ t_2=\frac{L}{l}\cdot t_c+t''_2 \text{ minutes.} \]

Raising the tool from the borehole. Here, as in lowering the tool into the borehole, we have two kinds of operations. Some operations depend neither on the depth of the borehole nor on the amount of advance in the small cycle, and, for a given qualification of the drilling crew working on specified equipment, their total duration is a constant value in any small cycle. The total duration of the other operations is a function of the depth of the borehole, increasing with an increase in the depth of the drilling interval. By analogy with the operations for lowering the tool into the borehole, these latter operations have been assigned to the group “operations for raising stands.” The operations of the first category are combined in the group “other operations in raising the tool.” This group includes the following operations: unscrewing the stuffing box before the start of raising, loosening the chuck of the drilling machine, unscrewing the upper rod, opening the box and moving the vertical box aside together with the upper rod located in it, closing the horizontal box with the safety cover, then, after raising the last stand, screwing the swivel onto the rod of the core barrel assembly, raising the core barrel assembly, moving the assembly aside, and closing the mouth of the borehole with a plug.

The group of operations for raising stands consists of the following operations: screwing the swivel onto the upper stand, removing

stands from the rod holder or wrenches, the actual lifting of the stand, placing the stand on the rod holder or wrenches, unscrewing the auger, unscrewing the stand, removing the stand and lowering the auger, screwing the auger onto the next stand, etc., up to the lifting of the last stand, its removal to the side, and lowering the auger in order to lift the core barrel.

Let us denote:

\(t_n\) — the duration of the operations for lifting one stand, in min.;

\(t'_3\) — the time required to lift all the stands from the borehole in the given small cycle, in min.;

\(t''_3\) — the duration of the other operations for lifting the tool in the small cycle, in min.;

\(t_3\) — the total duration of the operations for lifting the tool from the borehole in the small cycle.

Then

\[ t'_3=\frac{L+S}{l}\cdot t_n . \]

\[ t_3=t'_3+t''_3 ; \]

\[ t_3=\frac{L+S}{l}\cdot t_n+t''_3 \text{ minutes.} \]

Setting the tool on the bottom. This category includes all operations connected with bringing the core barrel to the bottom of the borehole, i.e., breaking up the core remaining in the borehole, flushing the borehole to clean the bottom of sludge, etc. Under an ideally proceeding process, these operations, generally speaking, should not exist, and the barrel, as a result of normal lowering, should come directly to rest on the bottom; but since in reality these operations do occur and take a certain amount of time in the small cycle, it appears necessary to allocate them to an independent group.

The duration of these operations in the small cycle depends neither on the footage per cycle nor on the depth of the borehole. At the same time, however, its fluctuations over individual cycles can be, and in reality are, quite significant; this is explained by a whole series of attendant causes, of which the chief one is the character of the rocks being drilled, from the standpoint of their strength, stability, ability to produce sludge of one or another coarseness, etc.

For the time being we have no possibility of taking this influence into account numerically, and therefore, for these operations, within the limits of one small cycle, it is necessary to take their average duration, determined by time studies over a number of cycles, and thus to regard this duration as identical for all small cycles.

Let us denote the time in minutes required for these operations in the small cycle by \(t_4\).

Operations for wedging and breaking off the core. The following operations have their place here: stopping the machine and the pump, unscrewing the stuffing box, filling in the wedge, screwing on the stuffing box, starting the pump, pressing in the wedge, stopping the pump, and breaking off the core.

These operations are obligatory in core drilling; moreover, the time occupied by them in the small cycle is not, strictly speaking, constant. But taking into account that the deviations from the norm are random in character and depend neither on the depth of the borehole nor on the advance per cycle, we may take the duration of these operations in the cycle as constant, equal to the arithmetic mean from the series of observations.

Let us denote the time expended on these operations in the small cycle by \(t_5\).

Operations with the core barrel. This group combines operations relating to manipulations with the core barrel at the surface. If work is being carried out with interchangeable core barrels—which, generally speaking, must be considered obligatory—then this group includes the operations of unscrewing the swivel from the raised core barrel and screwing the swivel onto the core barrel being lowered into the borehole. If work is being carried out with a single core barrel, which still often occurs in practice, then this group includes the operations of unscrewing the bit after extracting the core barrel from the borehole, extracting the core from the core tube, screwing on a new bit, and cleaning the sludge tube.

It is clear that the time expended on these operations does not depend on the depth of the borehole. The influence of the magnitude of the advance per cycle is also practically not manifested here (extraction of a somewhat larger or somewhat smaller amount of core). Therefore the duration of operations with the core barrel may be taken as constant in the small cycle.

Let us denote the duration of these operations in the small cycle by \(t_6\).

Let us now write the formula for determining the total duration of the small cycle \(T\).

\[ T = t + t_1 + t_2 + t_3 + t_4 + t_5 + t_6 \]

or

\[ T = t + \frac{S}{n}\cdot t_k + \frac{S}{m}\cdot t_{\sh} + \frac{L}{l}\cdot t_c + t_2'' + \frac{L+S}{l}\cdot t_n + t_3'' + t_4 + t_5 + t_6 \]

Here the total duration of the operations on which the times \(t_2''\), \(t_3''\), \(t_4\), \(t_5\), and \(t_6\) are expended is constant for any small cycle. Let us denote the total duration of these operations by \(K\), i.e., let us put:

\[ t_2'' + t_3'' + t_4 + t_5 + t_6 = K \tag{1} \]

Then

\[ T=t+\frac{S}{n}t_k+\frac{S}{m}t_{\sh}+\frac{L}{l}t_c+\frac{L}{l}t_p+\frac{S}{l}t_p+K \]

or

\[ T=t+S\left(\frac{t_k}{n}+\frac{t_{\sh}}{m}+\frac{t_p}{l}\right)+\frac{L}{l}(t_c+t_p)+K \tag{2} \]

The quantities \(t_c\) and \(t_p\), i.e., the time spent on lowering one rod and on hoisting one rod, are functions of the rod length, for each of them consists of a certain constant time (\(t'_c\) and \(t'_p\)), independent of the rod length (to a first approximation, with accuracy sufficient for practice), and a certain amount of time spent on the actual lowering (\(t''_c\)) and the actual hoisting (\(t''_p\)) of the rod. These latter quantities depend on the length of the rod, the number of revolutions of the hoist drum, and its diameter. Since, in drilling a given borehole, the rod length, the diameter of the hoist drum, and the number of revolutions of this drum are usually constant, it follows that, for the given specific conditions, \(t'_c\) and \(t'_p\) will be constant, and therefore so will \(t_c\) and \(t_p\).

The quantities \(\frac{t_k}{n}\), \(\frac{t_{\sh}}{m}\), and \(\frac{t_p}{l}\) are also constant for the specific conditions of drilling a given borehole.

Thus, the total duration of the small cycle depends on two variable quantities: the duration of net drilling in the cycle \(t\) and the advance per cycle “\(S\),” with the advance per cycle itself being a function of the duration of net drilling in the cycle and of the nature of the rocks penetrated by the borehole in that cycle. We shall turn to the analysis of the duration of net drilling in the small cycle and of the advance during this cycle somewhat later.

Chapter II

Drillability Curves of the Rocks of the Lenin District of Kuzbass and Their Analytical Expression

The material for the present chapter consisted of observations made in the summer of 1937 in the course of time-study work in the drilling crews of the Lenin Geological Prospecting Office of Coal Exploration of the Kuzbass Combine, carried out under the author’s direction. In carrying out this work, along with other observations, the time of net drilling expended on deepening the borehole over the interval of each reamer was recorded. It proved impossible for us, for a whole series of reasons, to arrange the recording of smaller intervals of advance, as is recommended by some authors (4). Nor was there any particular need for this, since the rocks of the productive strata of Kuzbass are not distinguished by considerable hardness, and the time expended on deepening the borehole over the interval of one reamer was therefore not great (10–15, rarely more, minutes). On the other hand, recording the advance by reamers presents obvious advantages.

The observations were recorded in the following form (Table 2).

Table 2.

Rate of net drilling by intervals in borehole No.

Reamer No.	Cycle date: start and end of net drilling, h	Cycle date: start and end of net drilling, ′	Cycle date: start and end of net drilling, ″	Measurement in mm at the start of drilling	Deepening in mm	Downtime, ′	Downtime, ″	Net drilling, ′	Net drilling, ″	Rate of net drilling, mm/min	Name and characteristics of the rocks
1	7	4 11	14 40	1080	115	—	—	7	26	15,5
2	7	12 32	51 22	965	260	—	—	19	31	13,3
3	7	33 50	36 58	705	210	—	—	17	22	12,1

The conditions under which drilling was carried out were noted in the basic time chart. Their brief characterization is as follows.

Drilling was carried out mainly with bits of 65, 75, and 85 mm diameter. Prismatic hard alloys of grades RE—6 and RE—8 were used as cutters. Six cutters each were inserted into bits 65 and 75 mm in diameter, and eight cutters into 85 mm bits. The cutters were fixed perpendicular to the face of the bit and were not sharpened. Peening was done carelessly.

The number of revolutions of the bit at the bottom ranged from 90 to 110–120 per minute, rarely more. The pressure on the bottom was regulated manually by the shift foreman. A load on the lever was not used. As an approximation, it may be assumed that the pressure ranged from 300–350 to 400–450–500 kg. The depth of the borehole ranged within 200–250 m.

As can be seen, both the drilling regime and the bits used leave much to be desired, but we shall not dwell here on a critical analysis of the drilling conditions. This has been done by us in another work (6).

After the drill string was raised to the surface and the core was extracted, in Table 2, opposite each core lifter, the name of the rock passed through in that interval was entered, and a brief characterization of its physical properties was given. Subsequently, only those short cycles within which there was no change of rock were used. This restriction had to be introduced in order to make it possible to find the law of variation of the rate of net drilling with time for each category of rock.

Knowing the net drilling time expended on deepening the borehole within each core lifter of the short cycle, and the average rate of net drilling in the given rock over the interval of each core lifter, one can determine to what moment in time, counted from the start of net drilling in the cycle, this average rate of net drilling refers. It is obvious that, without large error, the average rate of net drilling in the given borehole interval may be taken as the true rate of net drilling in the middle of the given time interval, since within the comparatively small intervals of time and intervals of advance that we have singled out (not more than the useful length of one core lifter, i.e., about 200–250 mm and 10–20 minutes), the change in rate will closely correspond to the law of a straight line.

Using this premise, tables were compiled for each of the selected categories of rocks; in their columns were placed the numbers of the cycles, the ordinal numbers of the core lifters (intervals) in the cycle, the duration of net drilling in the interval of the given core lifter, the time of net operation of the bit from the beginning of the cycle to the middle of the corresponding interval, and the rate of net drilling in the given interval. The form of the record is illustrated by Table 3.

Table 3.

Net drilling rate over individual intervals of the small cycle by rocks of … category.

Logging No.	Small cycle No.	Sequence No. of the drill bit	Duration of net drilling over the interval of one bit	Duration of net drilling over the interval of one bit	Time of operation of the bit from the beginning of the small cycle to the midpoint of the given interval	Time of operation of the bit from the beginning of the small cycle to the midpoint of the given interval	Net drilling rate, mm/min, over the given interval	Note
			I′	II″	I′	II″
253	23	1	7	26	3	43	15.5
253	23	2	19	31	17	11	13.3
253	23	3	17	22	35	38	12.1

Thus, for each category of rocks, the rates of net drilling at different moments of time were calculated, counting from the beginning of operation of the bit in the given cycle.

After this, the time occupied in the small cycle by net drilling is divided into arbitrary but equal intervals. In choosing the size of the interval, it should be borne in mind that excessively small intervals, and consequently a large number of them in the cycle, will greatly increase the amount of subsequent computational work, without increasing the accuracy of the subsequent constructions and conclusions. Conversely, excessively large intervals will not provide a sufficient number of mean points and will not ensure convincing conclusions.

In processing the materials collected in Leninsk, we settled on 10-minute intervals. This interval size proved to be the most suitable.

Thus, we divided the time of net drilling in the small cycle into intervals:

0–10 minutes,
10–20 minutes,
20–30 minutes,
30–40 minutes,
etc., to the end of the cycle.

The data of columns 5 and 6 of Table 3 are broken down according to these intervals, and within each interval the following are calculated: the average net drilling rate and the corresponding moment of time, counted from the beginning of net drilling in the cycle.

Assuming, just as was done above, that the change in rate within the 10-minute intervals we adopted obeys the law of a straight line, it may be taken that the average drilling rate within the given interval represents the true net drilling rate at the moment of time that is the mean of the individual moments of time included in this interval.

The average rate of net drilling in each interval is obtained by the arithmetic-mean method from the rates corresponding to the moments of time that fall within the given interval. The moments of time corresponding to the average rates are also obtained by the arithmetic-mean method from the individual moments of time lying within the limits of each of the selected intervals.

As an example of these calculations, we shall give the determination of the average rate and the corresponding moment of time for rock category IV in the interval from 40 to 50 minutes.

$t$	$v$	$t$	$v$
42′30″	26.0	41′23″	27.6
43′42″	21.6	43′26″	29.2
47′30″	22.0	40′32″	14.1
45′05″	18.0	47′12″	23.7
40′05″	19.0	42′50″	22.2
49′51″	19.7	42′38″	19.2
47′20″	18.1	46′00″	36.0
42′34″	26.9	44′03″	21.4
46′04″	22.4	46′38″	9.9
40′44″	46.0	40′59″	24.0
40′07″	33.7
40′29″	18.2
47′05″	22.8
		1008′38″	541.7

\[ t=\frac{1008'38''}{23}=43'51'' \]

\[ v=\frac{541.7}{23}=23.5\ \mathrm{mm/min}. \]

Thus we have found that, 43′51″ after the beginning of net drilling, the bit will have, in category IV rocks, a rate equal to 23.5 mm/min.

Using the method set forth above, we found the average rates of net drilling and the corresponding moments of time for each of the ten-minute intervals separately for each of the three categories of rocks for which it was possible to collect sufficient material. These data are given in Table 4.

It should be noted that, depending on drillability in the Lenin district, we distinguished six categories of rocks (from I to VI); but category I (overburden) and category II (weathered rocks) were of no interest for the purposes of the present work and therefore do not appear here. As for category VI (hard ...

Table 4

Average net drilling rates over individual intervals of the cycle \((v)\) and the corresponding moments of time \((t)\).

Interval	III \(v\)	III \(t\)	III \(n\)	IV \(v\)	IV \(t\)	IV \(n\)	V \(v\)	V \(t\)	V \(n\)	Note
0—10	4,9	37,5	51	4,5	31,5	33	4,9	21,7	70	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
10—20	14,8	36,7	43	14,4	27,5	30	14,8	18,1	49	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
20—30	24,8	32,0	41	25,1	24,5	29	25,8	17,9	46	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
30—40	35,2	32,2	34	35,1	24,7	22	35,5	17,6	35	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
40—50	44,6	31,9	35	43,9	23,5	23	45,1	16,4	37	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
50—60	54,4	29,3	26	55,1	21,1	22	54,6	15,2	32	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
60—70	64,8	27,8	27	64,6	23,3	18	65,0	16,8	38	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
70—80	75,3	27,2	18	74,7	21,9	10	75,5	16,2	29	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
80—90	86,5	21,6	10	84,9	18,5	12	84,3	13,9	35	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
90—100	97,0	23,9	8	92,9	20,9	6	95,8	14,3	27	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
100—110	106,7	26,4	9	106,1	18,5	5	104,9	13,3	24	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
110—120	115,5	26,3	5	113,1	20,1	1	115,0	14,7	26	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
120—130	124,7	24,0	10	122,9	20,2	3	124,4	14,1	17	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
130—140	140,4	23,1	6	134,6	18,4	4	134,9	15,0	22	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
140—150				147,8	18,7	3	144,9	13,6	16	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
150—160							155,1	13,5	15	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
160—170							163,6	15,0	10	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
170—180							175,5	12,9	11	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
180—190							185,7	11,0	3	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min
190—200							193,7	11,5	7	\(n\)—number of observations by which \(v\) and \(t\) were determined; \(t\) in minutes; \(v\) in mm/min

silicified sandstones), it occurs in the Leninskii district comparatively rarely, and we were unable to obtain from it a sufficient number of observations for drawing conclusions.

Thus, according to the method set forth above, observations on the drillability of rocks of categories III, IV, and V have been processed. The characteristics of these rocks are given in Table 5.

Let us now construct graphs for each category of rock, using for this purpose the data given in Table 4. To do this, in a rectangular coordinate system we shall plot along the abscissa axis the time of net drilling, counting from the beginning of the bit’s work at the face in the small cycle, and along the ordinate axis—the rates of net drilling corresponding to the respective moments of time. By connecting the points thus obtained with straight lines, for each category of rock we shall have broken lines character-

Table 5.

Characteristics of rocks of categories III, IV, and V of the Lenin district of Kuzbass.

Category	Characteristics of rocks	Coefficients a	Coefficients b
III	Medium- and fine-grained sandstones of medium strength with calcareous-clayey cement (effervesce with $HCl$), often with clearly expressed banding. Clayey fine- and very fine-grained sandstones, dense, sometimes slightly weathered. Dense coaly sandstones with calcareous-clayey cement. Argillites, slightly affected by weathering, sometimes somewhat softened. Crumpled and fractured argillites of tectonic zones.	195	3
IV	Argillites, dense, with a characteristic conchoidal fracture, usually gray or dark-gray in color. Weakly sandy argillites.	152	3
V	Fine- and very fine-grained sandstones, dense, well cemented by calcareous-clayey material (effervesce with $HCl$), strong, often banded. Strongly sandy dense, strong argillites, sometimes with thin interbeds of sandstone. Dense, viscous, rubber-like argillites without admixture of sandy material, almost always black or dark-gray in color.	109	3

drilling rates at separate time intervals (Figs. 1, 2, 3). It is clear that these broken lines by no means reflect with sufficient accuracy the actual process of change in the rate of net drilling over time, a process which, by its very nature, proceeds smoothly, gradually, and consequently should be represented in the drawing by a smooth curve, just as are most quantitative changes of various physical quantities, whose abrupt changes, jumps, etc., already characterize a qualitative change in the system.

In the present case, broken lines instead of smooth curves were obtained as a result of a whole series of attendant causes. Among the chief of these causes are the following.

a) Fluctuations in the magnitude of the pressure on the bottomhole, the accounting of which, in the absence of the appropriate measuring instruments, could be made only very roughly.

b) Fluctuations in the number of revolutions of the bit at the bottomhole.

c) Fluctuations of strength in rocks of one and the same category.

d) The presence of fractures in the rocks.

e) Different intensity of flushing of the borehole.

f) Not strictly uniform setting of the cutters in the bits, etc.

Of course, when calculating average rates by intervals, the influence of these factors was somewhat smoothed out (especially for those

Drawing 1.

Drawing 2.

Rock drillability curve for category V rocks
(Leninsky District)

$$ v = \frac{109}{t/3 + 6}\ \text{mm/min}. $$

Vertical axis: \(v\), mm/min.
Horizontal axis: \(t\), min.

Curve labels: 1, 2.

Drawing 3.

intervals within which a considerable number of observations fell), but it is, of course, impossible to exclude their influence entirely. Therefore, in order to obtain a conception closer to the truth of the law governing the change in the net drilling rate, it is necessary to construct so-called smoothing curves.

These curves are first drawn by eye, in such a way that the experimental points are distributed uniformly on both sides of the smoothing curve, as far as possible at an equal distance from it, and so that the profile of the curve most closely approaches the general direction indicated by the experimental points. The curves constructed are recommended by A. Vigneron (3) to be improved by using for this purpose the so-called method of differences.

The essence of the method of differences is as follows. The axis of abscissas, in our case, and therefore the time axis, is divided into arbitrary but equal intervals, and for each point of division the ordinate of the corresponding point on the curve is taken. Then the differences of adjacent ordinates are calculated (first-order differences). If the curve has been drawn correctly, the change in the differences should proceed monotonically, i.e., the differences should either increase smoothly or, as in our case, decrease smoothly, or, finally, they may be equal to one another. If the differences do not obey this and irregular jumps occur among them, the curve in the corresponding section must be corrected so as to ensure a smooth change of the differences in one direction.

Obviously, the second-order differences, i.e., the differences between adjacent first-order differences, and so on, must satisfy the same condition.

Having thus corrected our curves, initially drawn by eye, we obtain a graphical expression of the change in the net drilling rate for each category of rock over time, i.e., a graphical representation of the law of change of the net drilling rate.

It is now of interest to find an analytical expression for our law. To do this, we shall analyze the nature of the curves obtained, which we have called drillability curves, and shall try to obtain a general idea of the construction that the analytical expression of the law of interest to us should have.

Analysis of the drillability curves makes it possible to note the following features.

1. The ordinates of the curves decrease regularly as the abscissas increase, i.e., as the duration of operation of the core bit at the bottom increases, the net drilling rate falls. Consequently, the relationship between \(t\) and \(v\) is inverse.

2. The curve is located higher relative to the axis of abscissas the lower the strength of the rock category whose drillability it represents.

Hence, also taking into account the essence of the process of operation of the bit at the face, the following conclusions may be drawn:

1. The relationship between the time of operation of the bit at the face \(t\) and the rate of net drilling \(v\) is inverse.

2. At \(t = 0\), \(v\) must be a maximum finite value.

3. As \(t\) increases, the curve must asymptotically approach the abscissa axis, and at \(t = \infty\), \(v = 0\), since otherwise we would come into contradiction with the principles of thermodynamics.

The simplest equation satisfying all the conditions stated above will be an equation of the form:

\[ v = \frac{a}{t^n + b} \]

Indeed:

1. At \(t = 0\), \(v = \frac{a}{b}\)—the maximum finite value

2. At \(t = \infty\), \(v = 0\)

3. As \(t\) increases, the rate \(v\) falls.

The stronger the category of rock whose drillability is expressed by this equation, the smaller, evidently, the ratio

\[ \frac{a}{b}. \]

As for the exponent \(n\), it may be an integer or a fractional number and is found by selection in such a way as to ensure the character of the curve that most closely corresponds to the character of the experimental curves.

As a result of such selection we found that the character of the curves is maintained quite well at \(n = 1/3\), and, consequently, the general equation of rock drillability takes the form

\[ v = \frac{a}{t^{1/3} + b}\ \text{mm/min} \tag{3} \]

The next task consists in determining the coefficients \(a\) and \(b\) for each of the rock categories. This problem can be solved by the method of squares, the essence of which is as follows.

In equation (3), which is more conveniently represented in the form

\[ a = vb + vt^{1/3}, \tag{4} \]

instead of \(v\) and \(t\), their experimental values obtained for the corresponding rock category from Table 4 are substituted. In this way we obtain, generally speaking, \(n\) equations with two unknowns \(a\) and \(b\):

\[ \begin{aligned} a &= v_1 b + v_1 t_1^{1/3} \\ a &= v_2 b + v_2 t_2^{1/3} \\ a &= v_3 b + v_3 t_3^{1/3} \\ &\cdot \quad \cdot \quad \cdot \\ &\cdot \quad \cdot \quad \cdot \\ a &= v_n b + v_n t_n^{1/3} \end{aligned} \]

From these equations two series of equations are formed. One series is obtained by multiplying each of the original equations by the coefficient standing in the corresponding equation before one of the unknowns (for example, before \(a\)), and the second series by multiplying each of the same original equations by the coefficient standing in the corresponding equation before the other unknown (before \(b\)).

In our case the coefficient before the unknown “\(a\)” is unity; consequently, the original equations themselves constitute the first series. The second series will be:

\[ \begin{aligned} v_1 a &= v_1^2 b + v_1^2 t_1^{1/3} \\ v_2 a &= v_2^2 b + v_2^2 t_2^{1/3} \\ v_3 a &= v_3^2 b + v_3^2 t_3^{1/3} \\ &\cdot \qquad \cdot \qquad \cdot \\ &\cdot \qquad \cdot \qquad \cdot \\ &\cdot \qquad \cdot \qquad \cdot \\ v_n a &= v_n^2 b + v_n^2 t_n^{1/3} \end{aligned} \]

Let us now find the sum of each of the series. We obtain

\[ n a = \left(\sum_{i=1}^{i=n} v_i\right)b + \sum_{i=1}^{i=n} v_i t_i^{1/3} \quad \text{(from the first series)} \]

\[ \left(\sum_{i=1}^{i=n} v_i\right)a = \left(\sum_{i=1}^{i=n} v_i^2\right)b + \sum_{i=1}^{i=n} v_i^2 t_i^{1/3} \quad \text{(from the second series)} \]

Solving these two equations, we find the values of \(a\) and \(b\) that are of interest to us; substituting these values into equation (3), we obtain the drillability equation for each category of rock for which these values were determined. Table 6 gives the coefficients \(a\) and \(b\) found in this manner and the drillability equations.

Table 6.

Values of the coefficients \(a\) and \(b\) for the rocks of the Lenin District of the Kuzbass and the drillability equations for these rocks.

Rock category	Coefficients	Coefficients	Drillability equation
	\(a\)	\(b\)
III	195	3	\(v=\dfrac{195}{t^{1/3}+3}\ \text{mm/min}\)
IV	152	3	\(v=\dfrac{152}{t^{1/3}+3}\ \text{mm/min}\)
V	109	3	\(v=\dfrac{109}{t^{1/3}+3}\ \text{mm/min}\)

Using the drillability equations found, final drillability curves are plotted for each category of rocks (Figs. 1, 2, and 3). It should be noted that, when this method is applied, there is no need for a preliminary improvement, by the method of differences, of the curves constructed by eye. But such improvement of the curves is obligatory if, for determining the coefficients \(a\) and \(b\), one uses the approximate method, which, generally speaking, gives less accuracy (although in our case practically almost complete agreement was obtained), while at the same time it is considerably less laborious.

This simplified method is as follows. Three or four pairs of equations are formed by substituting into the equation

\[ a = vb + vt^{1/3} \]

the variables \(v\) and \(t\) with their values obtained from the graph of the curves improved by the method of differences.

Then each pair of equations is solved with respect to \(a\) and \(b\), and as their final values the mean values are taken of the quantities found from the separate pairs of equations.

As an example of this method, let us give the determination of the coefficients \(a\) and \(b\) for rocks of category V, limiting ourselves, for simplicity, to only two pairs of equations.

We compose the equations and solve them.

\[ \begin{aligned} a_1 &= 18.3 b_1 + 27^{1/3}\cdot 18.3 \\ a_1 &= 14.5 b_1 + 98^{1/3}\cdot 14.5 \end{aligned} \qquad \begin{aligned} a_2 &= 15.7 b_2 + 64^{1/3}\cdot 15.7 \\ a_2 &= 13.7 b_2 + 125^{1/3}\cdot 13.7 \end{aligned} \]

\[ \begin{aligned} a_1 &= 18.3 b_1 + 54.9 \\ -\,a_1 &= 14.5 b_1 + 66.7 \\ \hline 0 &= 3.8 b_1 - 11.8 \\ b_1 &= 3.1 \\ a_1 &= 18.3\cdot 3.1 + 54.9 \\ a_1 &= 111.6 \end{aligned} \qquad \begin{aligned} a_2 &= 15.7 b_2 + 62.8 \\ -\,a_2 &= 13.7 b_2 + 68.5 \\ \hline 0 &= 2.0 b_2 - 5.7 \\ b_2 &= 2.9 \\ a_2 &= 15.7\cdot 2.9 + 62.8 \\ a_2 &= 108.3 \end{aligned} \]

\[ b = \frac{3.1 + 2.9}{2} = 3 \]

\[ a = \frac{111.6 + 108.3}{2} = 110 \]

Using the method of squares to calculate the coefficients \(a\) and \(b\), we obtain the equation of a smoothing curve that most accurately corresponds to the experimental points. As is known, in this case the sum of the squares of the deviations of the individual points from the curve will be minimal.

It is now of interest to determine the root-mean-square deviation, the mean absolute and relative errors, and also the coefficient of variation for each of the rock categories.

In order not to overburden the work with excessive calculations, we shall carry out the computation of these quantities for only one;

of the curves, and for the other two curves in Table 8 we give the already calculated values.

As an example, we shall use the curve for rocks of category III. Let us note, incidentally, that this is the least “successful” curve, since one of its experimental points \((t = 86.5,\ v = 21.6)\) lies comparatively far from the curve. In determining the coefficients \(a\) and \(b\) of the curve, we rejected this point as clearly defective. We shall also reject this point when calculating the errors. As for the remaining experimental points, both for this category of rocks and for the other two, the smoothing curves on the whole correspond to them very closely.

Table 7.

Calculation of the squares of deviations for category III rocks.

Time in min from the beginning of the cycle \(t\)	Drilling speed, mm/min — experimental \(v_e\)	Drilling speed, mm/min — according to the drillability equation \(v_m\)	Deviation \(\alpha\)	Square of deviation \(\alpha^2\)
4,9	37,5	41,5	+4,0	16,0
14,8	36,7	35,7	−1,0	1,0
24,8	32,0	32,9	+0,9	0,8
35,2	32,2	31,1	−1,1	1,2
44,6	31,9	29,8	−2,1	4,4
54,4	29,3	28,7	−0,6	0,4
64,8	27,8	27,8	0,0	0,0
75,3	27,2	27,1	−0,1	0,0
97,0	23,9	25,7	+1,8	3,2
106,7	26,4	25,2	−1,2	1,4
115,5	26,3	24,8	−1,5	2,2
124,7	24,0	24,4	+0,4	0,2
140,4	23,1	23,8	+0,7	0,5
	378,3		+0,2	31,3

Let us find the root-mean-square deviation \(\sigma\)

\[ \sigma = \sqrt{\frac{\sum \alpha^2}{n - 1}} = \pm \sqrt{\frac{31,3}{12}} \]

\[ \sigma = \pm 1,6\ \text{mm/min} \tag{5} \]

Let us determine the coefficient of variation

\[ W = \frac{\sigma}{M}\,100\% \]

\[ M = \frac{\sum v_e}{n} = \frac{378,3}{13} \]

\[ M = 29,1\ \text{mm/min}. \]

\[ W = \frac{1,6}{29,1}\cdot 100\% \]

\[ W = 5,5\% \tag{6} \]

Error of the arithmetic mean:

$$ m=\frac{\sigma}{\sqrt{n}}=\frac{1.6}{13} $$

$$ m=0.44\ \text{mm/min} \tag{7} $$

Relative error of the arithmetic mean

$$ \mu=\frac{m}{M}\cdot 100=\frac{0.44}{29.1}\cdot 100\% $$

$$ \mu=1.5\% \tag{8} $$

What, then, is the physical meaning of the quantities \(W\) and \(\mu\) found by us? The coefficient of variation \(W\) characterizes, to a certain degree, that relative error with which we, using the drillability equation, can determine the rate of net drilling at a given moment, or it represents the root-mean-square deviation of the experimental points from the drillability curve, referred to the mean value of the rate of net drilling.

The relative error of the arithmetic mean \(\mu\) characterizes the accuracy of determining the mean rate of net drilling over a given interval, and, consequently, it also characterizes the magnitude of the interval drilled during a certain period of time.

Indeed,

$$ v_{\mathrm{cp}}=\frac{S}{t}\pm m $$

$$ S=(v_{\mathrm{cp}}\pm m)t $$

$$ \mu=\frac{m}{v_{\mathrm{cp}}}\cdot 100;\quad m=\frac{\mu v_{\mathrm{cp}}}{100} $$

$$ S=v_{\mathrm{cp}}t\left(1\pm\frac{\mu}{100}\right) $$

Table 8.

Root-mean-square deviation, absolute and relative errors, and coefficient of variation of the drillability curves.

Rock category	Root-mean-square deviation in mm/min	Error of the arithmetic mean in mm/min	Relative error in %	Coefficient of variation in %
III	1.6	0.44	1.5	5.5
IV	1.1	0.28	1.3	5.0
V	1.1	0.26	1.7	7.3

As can be seen from Table 8, the relative error $\mu$ and the coefficient of variation $W$ for the different categories of rocks do not fluctuate especially strongly, and they may, without great error, be taken as having the following mean values:

for the relative error $\mu = \pm 1.5\%$

for the coefficient of variation $W = \pm 6\%$

In the drillability equation

\[ V=\frac{a}{t^{1/3}+b}\ \text{mm/min} \]

the following very interesting fact attracts attention. It turns out that the coefficient $b$ remains constant for all three categories of rocks (see Table 6).

This shows that the characteristic indicator of rock drillability is the coefficient $a$, which changes together with the change in the category of rocks by strength: the higher the strength, the smaller the coefficient $a$.

As for the magnitude of the coefficient $b$, insofar as it reflects to a certain degree the steepness of the curve, it will, as we have some grounds to think, characterize the technological drilling conditions, as well as the features of the bit in terms of the number of cutters, their shape and sharpening, the character of the insert, etc. This basis rests on the fact that for the Anzherka district, the processing of the materials for which has not yet been completed, the constancy of the coefficient $b$ is likewise indicated, although here it has a different value. Taking into account that the drilling conditions and the type of bits in the Anzherka district differ sharply from those in the Leninsk district, the conclusion drawn suggests itself. Of course, the interpretation of the coefficients $a$ and $b$ in the drillability equation still requires serious verification, and the considerations presented regarding their role in the drillability equation must be regarded as preliminary.

The proposed equation makes it possible to refine the characterization of rocks from the standpoint of their drillability. As is known, by the drillability of a rock is meant the average rate of pure drilling shown by a bit in the given rock over a certain interval of time under a definite drilling condition. This value of drillability is also used in practice. It is also usually given in norm-setting handbooks, often without indicating the parameters of the drilling conditions, the characteristics of the bit, and the time interval during which the bit worked in testing the rock and for which the average drilling rate is given. Often, too, in norm-setting handbooks, data obtained by processing production indicators by a statistical method are taken as the drillability of rocks.

We believe that it is now time to abandon completely the use of production reports as material for developing norms and indicators of rock drillability, in view of the very inaccurate (and sometimes outright tendentious) documentation of drilling…

working process, almost always observed in production, of which we, and everyone who has dealt directly with drilling, have of course had to be convinced more than once.

Nor can we be satisfied with the drillability of rocks in the form in which it is presently given, even if established from time-and-motion observation data.

Indeed, the drillability of a rock, i.e., the rate of advance of the bit per unit time, as we have shown above, is a variable quantity depending on the time during which the bit works at the face (at least for drilling with hard alloys), and also on the technological drilling conditions. There is no doubt that the drillability index of one rock or another must reflect these features of the process.

Using the currently accepted drillability index, however, we have no possibility of taking into account the actual average drilling rate per cycle. Thus, if the duration of net drilling in the cycle is taken to be greater than the time of bit operation for which the given drillability index was derived, then the average drilling rate obtained in this cycle will be less than the drillability index of the rock.

Conversely, with a shorter duration of net drilling in the cycle, the average rate of net drilling will be greater than the drillability index for the given rock. And this will naturally also affect the fulfillment of the shift norm.

Taking this circumstance into account, we propose, as the drillability index of a rock, to use the ratio (but not the quotient)

\[ v_0=\frac{a}{b} \tag{9} \]

obtained from the drillability equation (3), if in it one sets \(t=0\).

The drillability equation for each rock must in this case be established under a definite standard technological regime (number of revolutions, pressure on the face) and for a standard bit (number and shape of cutters, angle of their sharpening, setting angle). For deviations from the standard, the corresponding correction factors must be given, similar to those factors that we have at present, for example, for different bit diameters and for different angles of inclination of the borehole.

Under this condition, the drillability of rocks will have a sufficiently complete and accurate characterization. Using the drillability equation (3), we shall be able to determine the average drilling rate for any duration of bit operation at the face; we shall be able to find the rational duration of this operation in the cycle and, consequently, to construct the cycle itself correctly.

All these questions are discussed by us in the following chapter.

Chapter III.

Construction of a Rational Short Cycle in Core Drilling.

In Chapter 1 of the present work we established that the duration of the short cycle depends on the quantity \(K\), which is constant for any cycle (the total duration of the cycle operations, the time expenditures for which are constant), and on two variable quantities—the depth of the borehole \(L\) and the feed interval per cycle \(S\).

Let us find an expression for \(S\) from the drillability equation (3).

\[ S=\int V\,dt \tag{10} \]

\[ S=\int \frac{a\,dt}{t^{1/3}+b} \tag{11} \]

Denote \(t^{1/3}\) by \(Z\), i.e.

\[ t^{1/3}=Z, \]

then

\[ t=Z^3 \]

\[ dt=3Z^2\,dz \]

\[ S=\int \frac{3aZ^2\,dz}{Z+b} \]

\[ S=3a\int \frac{(Z+b)(Z-b)+b^2}{Z+b}\,dz \]

\[ S=3a\int (Z-b)\,dz+3ab^2\int \frac{dz}{Z+b} \]

\[ S=\frac{3}{2}aZ^2-3abZ+3ab^2\ln(Z+b)+C \]

Let us find the constant \(C\). When \(t=0\), obviously,

\[ Z=0 \quad \text{and} \]

\[ S=0 \]

\[ C=-3ab^2\ln b, \]

then

\[ S=\frac{3}{2}aZ^2-3abZ+3ab^2\ln\frac{Z+b}{b} \]

Replacing \(Z\) by \(t^{1/3}\), we obtain

\[ S=\frac{3}{2} a t^{2/3}-3ab t^{1/3}+3ab^2 \ln \frac{t^{1/3}+b}{b} \tag{12} \]

Let us expand in a series \(\ln \dfrac{t^{1/3}+b}{b}\).

The most convenient series, converging sufficiently rapidly, is the following:

\[ \frac{1}{2}\ln \frac{1+X}{1-X}=X+\frac{X^3}{3}+\frac{X^5}{5}+\ldots+Rn \]

Let us find what the value of \(X\) must be in this case. Obviously it must be:

\[ \frac{1+X}{1-X}=\frac{t^{1/3}+b}{b} \]

\[ b+bX=t^{1/3}+b-t^{1/3}X-bX \]

\[ X(2b+t^{1/3})=t^{1/3} \]

\[ X=\frac{t^{1/3}}{2b+t^{1/3}} \tag{13} \]

As is known, the series we have adopted will be convergent when

\[ -1<X<1 \]

In our case

\[ 0<\frac{t^{1/3}}{2b+t^{1/3}}<1, \]

i.e., the convergence condition is satisfied.

Thus,

\[ \ln \frac{t^{1/3}+b}{b} = 2\frac{t^{1/3}}{2b+t^{1/3}} + \frac{2}{3} \left( \frac{t^{1/3}}{2b+t^{1/3}} \right)^3 + \]

\[ + \frac{2}{5} \left( \frac{t^{1/3}}{2b+t^{1/3}} \right)^5 +\ldots+Rn. \]

We shall discard the third and subsequent terms of the series. As will be seen below, the error in determining the logarithm, and consequently also the value of \(S\), will not thereby exceed the practically permissible limits.

Thus,

\[ \ln \frac{t^{1/3}+b}{b} = \frac{2t^{1/3}}{2b+t^{1/3}} + \frac{2}{3} \left( \frac{t^{1/3}}{2b+t^{1/3}} \right)^3 \tag{14} \]

Raising the second member of the equality to the cube and bringing both members to a common denominator, we shall have

\[ \ln \frac{t^{1/3}+b}{b} = \frac{ 4b^2 t^{1/3}+4b t^{2/3}+\frac{4}{3}t }{ 4b^3+6b^2 t^{1/3}+3b t^{2/3}+\frac{1}{2}t } \tag{15} \]

Substitute the value obtained for \(\ln \dfrac{t^{1/3}+b}{b}\) into equality (12). Then we shall have:

\[ S=\frac{3}{2}a t^{2/3}-3ab t^{1/3}+3ab^2 \frac{ 4b^2 t^{1/3}+4b t^{2/3}+\frac{4}{3}t }{ 4b^3+6b^2 t^{1/3}+3b t^{2/3}+\frac{1}{2}t } \]

Bringing the right-hand side to a common denominator, we find:

\[ S= \frac{ \frac{3}{4}a t^{5/3}+3ab t^{4/3}+4ab^2t }{ \frac{1}{2}t+3bt^{2/3}+6b^2t^{1/3}+4b^3 } \]

\[ S= \frac{ \frac{3}{4}at\left(t^{2/3}+4bt^{1/3}+\frac{16}{3}b^2\right) }{ \frac{1}{2}\left(t+6bt^{2/3}+12b^2t^{1/3}+8b^3\right) } \]

\[ S= \frac{ \frac{3}{4}at\left[\left(2b+t^{1/3}\right)^2+\frac{4}{3}b^2\right] }{ \frac{1}{2}\left(2b+t^{1/3}\right)^3 } \]

\[ S= \frac{ \frac{3}{4}at\left(2b+t^{1/3}\right)^2+ab^2t }{ \frac{1}{2}\left(2b+t^{1/3}\right)^3 } \]

\[ S= \frac{\frac{3}{2}at}{2b+t^{1/3}} + \frac{2ab^2t}{\left(2b+t^{1/3}\right)^3} \tag{16} \]

finally

or

\[ S= \frac{at}{2b+t^{1/3}} \left[ 1.5+ \frac{2b^2}{\left(2b+t^{1/3}\right)^2} \right] \ \text{mm} \tag{17} \]

Applied to our case, i.e., for \(b=3\), the equation will take the form

\[ S=\frac{at}{6+t^{1/3}}\left[1.5+\frac{18}{(6+t^{1/3})^2}\right]\ \text{mm} \tag{18} \]

The graph of this equation is given in Fig. 4.

In expanding \(\ln\left(\frac{t^{1/3}+b}{b}\right)\) in a series and discarding the terms beginning with the fifth degree and higher, we have allowed an error equal to the remainder term, which for the series adopted by us has the form

\[ R_n=\int_0^x \frac{\tau^n}{1-\tau^2}\,d\tau \tag{19} \]

Moreover, for

\[ -1<X<+1 \]

the inequality will hold

\[ R_n=\int_0^x \frac{\tau^n}{1-\tau^2}\,dt<\frac{X^{n+1}}{n+1}\cdot \frac{1}{1-X^2}, \]

i.e., in our case

\[ R_n<\frac{X^5}{5}\cdot \frac{1}{1-X^2} \]

or, substituting for \(X\) its value from (13), we obtain

\[ R_n<\frac{t^{5/3}}{5(2b+t^{1/3})^5}\cdot \frac{1}{1-\left(\frac{t^{1/3}}{2b+t^{1/3}}\right)^2} \]

or

\[ R_n<\frac{t^{5/3}}{20b(2b+t^{1/3})^3(b+t^{1/3})} \tag{20} \]

Let us now find the limiting error in determining \(S\) for \(t=27\), for \(t=125\), and for \(t=343\) min.

First let us find the limiting value of \(R_n\)

\[ R_{(27)}<\frac{3^5}{20\cdot3\cdot(2\cdot3+3)^3\cdot(3+3)}=0.00093 \]

\[ R_{(125)}<\frac{5^5}{20\cdot3\cdot(2\cdot3+5)^3\cdot(3+5)}=0.00489 \]

\[ R_{(343)}<\frac{7^5}{20\cdot3\cdot(2\cdot3+7)^3\cdot(3+7)}=0.01275 \]

In order to find the possible limiting error in determining \(S\), the remainder term \(R_n\) must be multiplied by \(3ab^2\) (see Eq. 12).

\[ 3ab^2R_{(27)}<0.0009\cdot27a;\quad 3ab^2R_{(27)}<0.0251a \]

\[ 3ab^2R_{(125)}<0.0049\cdot27a;\quad 3ab^2R_{(125)}<0.1323a \]

\[ 3ab^2R_{(343)}<0.0127\cdot27a;\quad 3ab^2R_{(343)}<0.3443a \]

Graph of the equation

\[ S=\frac{\alpha t}{\delta+\alpha t/3}\left[1.5+\frac{18}{(\delta+\alpha t/3)^2}\right] \]

for rocks of categories III, IV, and V.

Figure 4.

Vertical axis: \(S\), mm.
Horizontal axis: \(t\), min.

Curve labels:

• rocks of category III, \(I_0=(80)\)
• rocks of category IV, \(I_0=(50)\)
• rocks of category V, \(I_0=(10)\)

Thus, the error increases with increasing duration of net drilling, and it will be greatest for rocks of category III, for which the value of the coefficient \(a\) is equal to 195. The limiting possible magnitude of this error will be:

\[ \begin{aligned} \text{for } t &= 27 \text{ minutes} &&—\ 4.9\ \text{mm} \\ t &= 125 \text{ minutes} &&—\ 25.8\ \text{mm} \\ t &= 343 \text{ minutes} &&—\ 67.1\ \text{mm}, \end{aligned} \]

i.e., the error does not exceed the limits of the accuracy of the observations.

As for the relative error

\[ \lambda=\frac{3ab^2R_n}{S}\cdot 100\%, \]

then its limiting possible magnitude for the same values of \(t\) will be:

\[ \begin{aligned} t &= 27 \text{ minutes} &&—\ 0.47\% \\ t &= 125 \text{ minutes} &&—\ 0.71\% \\ t &= 343 \text{ minutes} &&—\ 0.81\% \end{aligned} \]

and for all categories of rocks this error will be the same.

As can be seen, the maximum possible error does not exceed 1%, and it may be neglected without detriment to the matter.

Let us now determine the rational duration of net drilling in the short cycle. Obviously, for this we must find the average drilling speed within the cycle and establish at what duration of operation of the bit at the bottom this speed will be maximal.

Above, in Chapter I, we showed that the total duration of the cycle may be expressed by the formula:

\[ T=t+S\left(\frac{t_k}{n}+\frac{t_{\sh}}{m}+\frac{t_n}{l}\right)+\frac{L}{l}\left(t_c+t_n\right)+K \tag{20} \]

Let us adopt, for brevity of notation,

\[ \frac{t_k}{n}+\frac{t_{\sh}}{m}+\frac{t_n}{l}=A \tag{21} \]

and

\[ \frac{L}{l}\left(t_c+t_n\right)+K=B \tag{22} \]

Then

\[ T=t+AS+B \tag{23} \]

and the average speed is

\[ V_{cp}=\frac{S}{T}=\frac{S}{t+AS+B} \tag{24} \]

In order to find at what duration of net drilling \(t\) the average speed \(V_{cp}\) will be maximal, we

we must find the maximum of the function, i.e., take the first derivative of \(S\) with respect to \(t\) in equation (24) and, setting it equal to zero, determine \(t\).

\[ V_{\mathrm{av}}=\frac{S'(AS+B+t)-(AS+B+t)'S}{(AS+B+t)^2}=0 \]

\[ ASS' + BS' + tS' - ASS' - S = 0 \]

Since

\[ S'=V=\frac{a}{t^{1/3}+b}, \]

and, by equation (16),

\[ S=\frac{\frac{3}{2}at}{2b+t^{1/3}}+\frac{2a+b^2t}{(2b+t^{1/3})^3}, \]

then

\[ \frac{Ba}{b+t^{1/3}}+\frac{at}{b+t^{1/3}}-\frac{\frac{3}{2}at}{2b+t^{1/3}}-\frac{2ab^2t}{(2b+t^{1/3})^3}=0 \]

or, canceling \(a\) and transferring the negative terms to the right-hand side of the equation, we obtain

\[ \frac{B}{b+t^{1/3}}+\frac{t}{b+t^{1/3}} = \frac{\frac{3}{2}t}{2b+t^{1/3}} + \frac{2b^2t}{(2b+t^{1/3})^3} \]

Reducing the terms of the equation to a common denominator and eliminating it, we shall have

\[ B(2b+t^{1/3})^3=\frac{1}{2}t^2+\frac{3}{2}bt^{5/3}+2b^2t^{4/3} \]

or, multiplying both sides of the equation by 2 and taking \(t^{4/3}\) outside the brackets, we finally obtain

\[ 2B(2b+t^{1/3})^3=t^{4/3}\left[(2b+t^{1/3})^2-bt^{1/3}\right] \tag{25} \]

This equation is not solved analytically. Let us use a graphical method to solve it. For this purpose we rewrite equation (25) in the form

\[ \frac{2B}{t^{4/3}}=\frac{(2b+t^{1/3})^2-bt^{1/3}}{(2b+t^{1/3})^3} \tag{26} \]

and set

\[ \frac{2B}{t^{4/3}}=y \tag{27} \]

then, evidently,

\[ \frac{(2b+t^{1/3})^2-bt^{1/3}}{(2b+t^{1/3})^3}=y \tag{28} \]

Equation (27) represents a system of curves, and each curve of this system will correspond to a definite depth of the borehole, since the quantity \(B\) is a function of the borehole depth and of the depth only. \(B\) contains no other variables (see Eq. 22).

Conversely, equation (28) is a function of the character of the rocks penetrated and of the technological drilling regime, represented in the equation by the coefficient \(b\).

It should be noted that, in our specific case, for all three categories of rocks the coefficient \(b\) remains constant. This is understandable, since the technological drilling regime remained more or less constant, while the abrasive properties of the rocks evidently changed so insignificantly that this was not reflected in the intensity of cutter blunting. As for the strength of the rocks, it should have no influence on the magnitude of the relative rate of blunting.

Graphical solution

\[ \begin{cases} y=\dfrac{2B}{t^{1/3}},\\[6pt] y=\dfrac{(2b+t/3)^2\, b\,t^{1/3}}{(2b+t/3)^3}. \end{cases} \]

Drawing 5.

Thus, equation (28) gives us only one curve (for \(b=3\)); the intersection of this curve with the system of curves of equation (27) gives a series of points, whose coordinates along the time axis \(t\) will be the roots of our equation, i.e., will give the optimum duration of pure drilling in the cycle for the borehole depths corresponding to the intersected curves.

The graphical solution of equations (27–28), as applied to the specific example under consideration, is given in Drawing 5.

The optimum duration of pure drilling at different borehole depths is given in Table 9, and the advance per cycle for rocks of different categories is given in Table 11.

In determining the optimum duration of pure drilling \(t\), the values of the quantities entering equations (27) and (28) were taken as those indicated in Table 10.

Table 9.

Optimal duration of net drilling at various well depths for the Leninsk district of Kuzbass.

Depth in m $L$	50	100	150	200	250	300
Optimal duration of net drilling in min $t$	138	175	211	246	281	316

Table 10.

Duration of individual operations of the small cycle, in minutes.

Operation	$t_k^{1)}$	$t_{\sh}^{2)}$	$t_n^{3)}$	$t_c^{3)}$	$t''_3$	$t'''_3$	$t_4$	$t_5$	$t_6^{4)}$
Duration in minutes	0.65	1.45	0.61	0.47	3.8	3.2	5.9	5.6	0.3

Note:

1) Useful length of the core lifter, 0.25 m.
2) Length of rods added, 1.5 m.
3) Length of a stand, 6 m.
4) With replaceable core-barrel assemblies.

These values were established from time-study data of work carried out in the drilling crews of the Anzherka and Leninsk geological-prospecting offices of the Kuzbass Combine in 1937.

Table 11.

Optimal advance per small cycle at different well depths and for different rock categories (in meters).

Rock category	50	100	150	200	250	300
III	3.9	4.8	5.6	6.3	7.0	7.7
IV	3.1	3.7	4.4	4.9	5.5	6.0
V	2.2	2.7	3.1	3.5	3.9	4.3

Let us now calculate the total duration of the small cycle for different well depths and different rock categories. To do this, one should use the equations:

\[ t_1=\frac{S}{n}t_k+\frac{S}{m}t_{\sh} \qquad t'_2=\frac{L}{l}t_c \]

\[ t'_3=\frac{L+S}{l}t_n \]

as well as the data of Tables 9, 10, and 11.

The results of these calculations are summarized in Table 12.

Operations and time expenditures for short-cycle operations	50	50	50	100	100	100
Borehole depth, m / Rock category	III	IV	V	III	IV	V
Operations whose duration depends on penetration:
Pure drilling \(t\)	138	138	138	175	175	175
Operations accompanying pure drilling \(t_1\)	13,9	11,1	7,8	17,1	13,2	9,6
Total	151,9	149,1	145,8	192,1	188,2	184,6
Operations whose duration depends on borehole depth:
Lowering drill rods \(t'_2\)	3,9	3,9	3,9	7,8	7,8	7,8
Raising drill rods \(t'_3\)	5,5	5,4	5,3	10,7	10,5	10,4
Total	9,4	9,3	9,2	18,5	18,3	18,2
Operations whose duration in the cycle is constant:
Other operations when lowering the tool into the borehole \(t''_2\)	3,8	3,8	3,8	3,8	3,8	3,8
Other operations when raising the tool out of the borehole \(t''_3\)	3,2	3,2	3,2	3,2	3,2	3,2
Setting the tool on the bottom \(t_4\)	5,9	5,9	5,9	5,9	5,9	5,9
Operations for wedging and breaking off the core \(t_5\)	5,6	5,6	5,6	5,6	5,6	5,6
Operations with the core-barrel assembly \(t_6\)	0,3	0,3	0,3	0,3	0,3	0,3
Total	18,8	18,8	18,8	18,8	18,8	18,8
Grand total	180,1	177,2	173,8	229,4	225,3	221,6
Percentage of pure drilling	76,6	77,9	79,4	76,3	77,7	79,0

…as a function of borehole depth and rock category

Table 12

150 III	150 IV	150 V	200 III	200 IV	200 V	250 III	250 IV	250 V	300 III	300 IV	300 V
211	211	211	246	246	246	281	281	281	316	316	316
20,0	15,7	11,1	22,5	17,5	12,5	25,0	19,6	13,9	27,5	21,4	15,3
231,0	226,7	222,1	268,5	263,5	258,5	306,0	300,6	294,9	343,5	337,4	331,3
11,7	11,7	11,7	15,7	15,7	15,7	19,6	19,6	19,6	23,5	23,5	23,5
15,8	15,7	15,6	21,0	20,8	20,7	26,1	26,0	25,8	31,3	31,1	30,9
27,5	27,4	27,3	36,7	36,5	36,4	45,7	45,6	45,4	54,8	54,6	54,4
3,8	3,8	3,8	3,8	3,8	3,8	3,8	3,8	3,8	3,8	3,8	3,8
3,2	3,2	3,2	3,2	3,2	3,2	3,2	3,2	3,2	3,2	3,2	3,2
5,9	5,9	5,9	5,9	5,9	5,9	5,9	5,9	5,9	5,9	5,9	5,9
5,6	5,6	5,6	5,6	5,6	5,6	5,6	5,6	5,6	5,6	5,6	5,6
0,3	0,3	0,3	0,3	0,3	0,3	0,3	0,3	0,3	0,3	0,3	0,3
18,8	18,8	18,8	18,8	18,8	18,8	18,8	18,8	18,8	18,8	18,8	18,8
277,3	272,9	268,2	324,0	318,8	313,7	370,5	365,0	359,1	417,1	410,8	404,5
76,1	77,3	78,7	75,9	77,2	78,4	75,8	77,0	78,2	75,8	76,9	78,1

In Table 12 the following points attract attention.

1. The comparatively long duration of net drilling in the cycle and, accordingly, the high percentage of net drilling. Connected with this are also the significant allowances per cycle (Table 11), larger than is usually the case in practice.

The long duration of net drilling is explained by the fact that, in the Leninsk geological-prospecting office, bits with unsharpened cutters were used. This caused a low rate of blunting (more precisely, smoothing of the cutter angles) and consequently increased the time of rational operation of the bit at the bottom.

But, in addition, another conclusion follows from this same fact, namely: the duration of the bit’s operation at the bottom should be increased in comparison with what we had in Leninsk. This will bring about the use of long core-barrel assemblies, especially in deep boreholes and in weakly consolidated rocks.

1. The duration of the rational stay of the bit at the bottom increases with increasing borehole depth. This is understandable, since the deeper the borehole, the more time is spent on lowering and hoisting operations and, consequently, the less often the assembly should be raised to the surface.

2. The time of net drilling does not depend on the drillability category of the rocks. Obviously, this is connected with the fact that the abrasive properties of the rocks with which we dealt, and their hardness, differ not so greatly as to affect the rate of cutter blunting. It should be noted that the character of the rocks also has no noticeable effect on the rate of cutter blunting—both the argillites and the sandstones of the Leninsk district give one and the same rate of blunting.

3. The percentage of net drilling decreases with increasing borehole depth, but only very slightly (not more than 1.5%).

4. The percentage of net drilling increases with increasing rock strength (by 2.5–3% between the extreme categories).

Conclusion.

Thus, as a result of the investigation carried out, we obtained the following results.

1. A law has been established for the variation in the rate of advance of the bit as a function of the duration of its operation at the bottom.

2. A more rational characteristic of rock drillability has been found and is proposed.

3. A method has been developed for determining the optimum duration of bit operation at the bottom.

4. A method has been developed for constructing a rational working cycle in core drilling with cemented carbides.

Of course, the results of our investigation are far from exhausting the entire diversity of questions arising before the managers

…with the duration of drilling operations in connection with the construction of a rational working cycle. In particular, an extremely interesting problem—one that has remained unaddressed by us, but is of exceptionally great importance for increasing labor productivity in drilling operations—is the question of the proper choice of the cutting angle of the cutters, their number in the bit, and the determination of the influence of these factors on the rate at which the cutters become dull.

Another very important task is also to establish the optimum axial pressure on the bit and a rational number of its revolutions. All these are factors that directly affect the length of time the bit works at the bottom hole.

A considerable number of works have been devoted to resolving these questions, but unfortunately researchers have for the most part considered each of these factors in isolation, outside their mutual interrelationship. This has led to the fact that we now have more or less definite recommendations for each of the indicators of the drilling regime, but nowhere do we find indications concerning their influence on the rate at which the cutters become dull or on the duration of the bit’s work at the bottom hole. These indicators were developed without taking into account the need to ensure the maximum average drilling rate over a small cycle; yet this rate alone is decisive in increasing labor productivity. Therefore, future research in the development of optimum indicators of the drilling regime must be carried out with regard to the need to obtain the maximum average rate over the cycle.

References

1. E. V. Borovskoi and F. A. Shamshev—Study of the technological process of drilling. Transactions of TsNIGRI, issue 95, 1937.

2. B. I. Vozdvizhenskii and N. I. Kulichikhin—Materials for determining the influence of cemented-carbide quality, number of revolutions, and pressure on the productivity of core drilling. Transactions of MGRI, vol. III, 1936.

3. A. Vineron—Processing of the results of physicochemical observations. Moscow, 1936.

4. V. E. Doenin. How to observe the time of net drilling. Journal Razvedka Nedr, No. 17, 1936.

5. M. V. Machinskii—Theory of drilling with shot, as well as with surrogate and diamond bits. Transactions of TsNIGRI, issue 95, 1937.

6. V. A. Nudner—Analysis of the work of drilling crews of the Anzherka and Lenin geological-prospecting offices of Kuzbassugol. Materials for standardizing core drilling. Archives of Kuzbassugol. Tomsk, 1938.

7. F. A. Shamshev and E. V. Borovskoi. On the problem of the drillability of rocks. TsNIGRI, 1927.

8. E. F. Epshtein—Percussion-rotary electric drill for drilling blast holes. Collection The Application of Cemented Carbides in the Mining and Petroleum Industry. ONTI, 1937.