KEY RESULTS OF A STUDY OF FRACTIONAL DRILLING REGIMES

V. I. MOLCHANOV

This article was written on the basis of the results of two years of research into the drilling process, carried out under production and laboratory conditions by a team from the institute. The author took direct part in the setup and conduct of experimental drilling in the summers of 1952 and 1953, and in the processing of all observations obtained. At present we have at our disposal materials from 320 run-observations conducted in the summer of 1952, from 200 run-observations in the summer of 1953, and from laboratory studies comprising approximately 100 run-observations.

In addition to the author’s materials, the article uses the results of observations carried out in teams working under the direction of P. M. Lopushinsky and P. F. Palyanov.

The principal task of the investigations was the development of a rational technology for fractional drilling as applied to specific conditions. The optimal technological regime should be worked out on the basis of an in-depth study of the process of rock destruction and of the influence exerted on this process by the principal factors. The fractional-drilling regime should first of all be characterized by the following parameters:

1) pressure on the bottom hole,
2) number of revolutions of the crown,
3) method of feeding and dosage of shot,
4) quantity of flushing fluid.

The regime developed, along with increasing productivity, must ensure a good core recovery.

Methodology of Field and Laboratory Investigations

At the experimental drilling rig, the shift included a timekeeper, who recorded all operations connected with preparation for drilling and, during drilling, every 30 minutes took the following readings:

1) advance per 30 minutes of net drilling,
2) water-meter readings,
3) ammeter and voltmeter readings.

Before lowering the tool and after hoisting it, the timekeeper measured the inside and outside diameter of the crown, the total height, and the height of the crown slot.

The pressure on the bottom hole, the number of spindle revolutions, and the rate and method of charging shot were assigned to the shift foreman in accordance with the general plan of investigations. The timekeeper was charged with supervising compliance with the specified parameters; he also made the necessary calculations to determine the weight of the drill assembly and the required load on the lever or on the Volkov regulator.

The general plan of the experimental drilling was drawn up with the aim of successively studying the influence of individual regime parameters on the rate of deepening. With the drilling conditions unchanged and the other parameters constant, the pressure on the bit was varied. In 1952 the unchanged regime was maintained for 9 shifts, after which only the pressure on the bit was changed by steps of 5 kg/cm² and the new regime was maintained for 9 shifts. Thus the range from 10 to 30 kg/cm² was studied, and at 90 rpm—to 40 kg/cm².

Such a procedure is justified only under conditions of very slow deepening of the borehole and homogeneity of the geological section. Observations plotted in coordinates “rate of deepening—pressure on the bit” give a clear picture of this dependence, obtained for different numbers of revolutions (Fig. 1). It is quite impossible to use this graph to study the influence of the number of revolutions of the bit on the rate of deepening, since the curves are not comparable with one another. Studies of the influence of pressure at 90 rpm continued for 36 rig-shifts. During this time about 11 meters were drilled, the borehole diameter changed, and comparison of the rates of deepening at 90 and 150 rpm loses all meaning.

From consideration of these graphs, a conclusion was drawn regarding the dependence of the optimum pressure on the number of revolutions, other conditions being equal (at 90 rpm \(C_{\text{opt}} = 32\) kg/cm²; at 150 rpm \(C_{\text{opt}} = 28\) kg/cm²; at 240 rpm \(C_{\text{opt}} = 25\) kg/cm²). This regular decrease of the optimum pressure was directly related by us to the torque and the power of the motor. Proceeding from these considerations, the research procedure in 1953 was changed somewhat. A definite combination of parameters was maintained for only one run. The next run (sometimes in the same shift) was carried out with the same regime parameters, but with another engine. The next pair of runs was made already at another pressure on the bit. A series of experiments was considered complete after paired observational runs had been obtained at all pressure steps—from 20 to 35 kg/cm². The next series of the same observations was carried out at another number of bit revolutions. The change in procedure fully justified itself. Two adjacent series are so close to each other that their comparison is entirely permissible. Consequently, using the graphs shown in Figs. 2, 3, 4, 5, and 6 as source material, we have the right to construct a graph of the dependence of the rate of deepening of the borehole on the number of revolutions of the bit.

The procedure for studying the influence of other parameters (method of feeding shot, flushing, etc.) on the rate of deepening is not set forth in the present article.

Laboratory investigations were carried out essentially by the same procedure. The investigations were conducted on a training drilling derrick equipped with a KA-2M-300 rig at a borehole depth of about 50 m, and on a specially equipped drilling machine. The training drilling derrick has the most ordinary equipment, and there is no point in dwelling on its description. Since the main conclusions are based on the results of bench drilling carried out on the drilling machine, the equipment of the latter and the research procedure should be considered separately. In the spindle of the drilling machine, by means of an ordinary cone, a stuffing box is fastened, to which a shot bit is attached. Drilling is carried out in a block of rock about 1 m high. As the borehole is deepened, 33-millimeter rods are added. Drilling is performed with bits made from rods of diameter \(D = 50\) mm, with the finest shot (“reps”) or with screened shot of diameter \(D = 1.5\) mm.

Flushing is carried out by means of a water pump driven by an A-22 engine, having an individual electric drive. Rotation of the machine is produced by an electric motor of 7.8 kW power. The electrical circuit of the stand consists of a complete set of electrical measuring instruments.

Graph
of the dependence of the rate of penetration on bottom-hole pressure

Vertical axis: Rate of penetration \(u\), cm/min
Horizontal axis: Pressure \(C\), kg/cm²

Table of reliability of points

Table of values of \(C, u, v\)

Fig. 1

Series III

Depth interval of the well: 85.58–92.25 m
Number of bit revolutions: 90 rpm
Quantity of flushing fluid: 25–14 l/min

Observation table

Vertical axis: drilling speed \(U\), cm/min.
Horizontal axis: pressure \(C\), kg/cm².

Fig. 2

Series IV

Depth interval of the well: 92.33–98.86 m
Number of bit revolutions: 150 rpm
Quantity of flushing fluid: 21–11 l/min

Observation table

Vertical axis: drilling speed \(U\), cm/min.
Horizontal axis: pressure \(C\), kg/cm².

Fig. 3

Observation table

Observation table

Rate of advance \(U\), cm/min
Pressure \(C\), kg/cm²

Fig. 4

Interval of borehole depth 103.15–150.47 m. Run No. 4. Number of bit revolutions \(n = 90\ \mathrm{rev/min}\). Motor power \(N = 20\ \mathrm{kW}\).

Observation table

Observation table

Rate of advance \(U\), cm/min
Pressure \(C\), kg/cm²

Fig. 5

Experimental drilling proceeded as follows. A rock block was set in place and fastened under the spindle of the machine. The borehole was started with a pobedit bit of diameter \(D = 55\) mm. After deepening by \(2\)--\(3\) cm, the work was changed over to shot drilling. The experiments began after the bit’s magazine slot had been covered. Before the start of an experimental run, a portion of shot weighing 50 g was poured into the borehole. The shot bit was carefully measured, the working area of the end face of the bit was calculated, the bit was weighed, the weight of the bit was added to the weight of the other parts of the tool, and the weight of the load necessary to create the required pressure on the bottom was determined. Pressure on the bottom was applied through Volkov’s regulator. Drilling with the given combination of parameters continued until five identical readings of the rate of deepening were obtained, taken every

Observation table

Depth interval of the borehole: \(150.99\)--\(156.77\) m
Number of bit revolutions: \(n = 140\) rev/min
Motor power: \(N = 20\) kW

Vertical axis: rate of deepening, \(v\), cm/min.
Horizontal axis: pressure \(C\), kg/cm\(^2\).

Fig. 6

5 minutes of net drilling. Before the start of drilling, in the middle, and at the end of the run, the power of idle rotation of the tool was determined. During drilling, at the same 5-minute intervals, the power of the working stroke was recorded. The power expended on drilling was determined as the difference between the power during drilling and the idle-stroke power.

Using this procedure, the dependences of the rate of deepening on the pressure on the bit were obtained for different numbers of revolutions (Fig. 7), and the dependence of the rate of deepening on the number of revolutions was revealed at pressures optimal for each number of revolutions.

On the nature of the action of shot

At present there is no unified view of the mechanism of rock destruction in shot drilling. Hypothetically, three kinds of action of shot on the rock are possible.

1. Cutting, planing by sharp-angled small but countless “cutters”—fragments of shot, which are fixed only briefly in the end of the bit and continuously change their shape and position. In the case of such action, a certain analogy emerges between shot drilling and the cutting of materials by cutters.

Fig. 7

1. Crushing of the rock by round shot under the action of loads exceeding the compressive strength limit of the rock. During rotation of the tool and rolling of the shot, the crushing loads move around the ring of the bottom hole, performing the useful work of drilling. In the case of such action, an analogy emerges between shot drilling and the wear of rolling-bearing races.

2. Chipping of the rock under the action of impact-induced spalling forces arising when bodies of non-spherical shape roll over an uneven surface under large loads. The process of destruction proceeds analogously to that in drilling with ball-type bits, or in such impact drilling when the drill turns simultaneously with the impact.

In the specialized literature on shot drilling, the first or the second hypothesis is set forth. The principal divergence between these hypotheses consists in two questions:

a) does angular or round shot operate?
b) does it operate by cutting or by crushing?

In the course of the discussion that took place in 1941 in the pages of the journal Razvedka nedr, the following became clear: the first hypothesis is untenable, first, because it assumes satisfactory operation of cast-iron “cutters” in rocks in which cutters made of hard alloys do not operate satisfactorily; second, in order for the “cutter” to move through the rock, there must be a friction force at the point of contact of the bit with the shot particles that is considerably greater than the friction force when the shot particles slide over the rock. In other words, the coefficient of friction in the sliding of steel over cast iron must be greater than in the sliding of cast iron over rock, which is unnatural.

A whole series of facts speaks against the second hypothesis. First, when drilling with cast-iron shot, round shot grains exist at the bottom for only a very short time (up to two hours), and only after all the shot has split under the end of the bit does intensive deepening of the borehole begin. Second, if one admits such high stress concentrations in the rock as lead to destruction of the rock, then, evidently, equally high stresses will also occur in the shot; and cast iron, for the most part, is less strong than the rock and will be destroyed first.

The author takes the position of a third hypothesis and considers that destruction of the rock under the end of a shot bit proceeds in the following way, depending on the quality of the shot.

1. Drilling with cast-iron shot. Cast-iron shot is not spherical in the geometrical sense of the word and, when the bit rotates, rolls over the uneven surface of the bottom. In this process “beating” of the tool inevitably arises, its vibration occurs, and impact loads will be transmitted through the shot to the rock, as a result of which both the shot and the rock will be crushed. The rolling angular particles create more favorable conditions for exciting oscillations of the tool and for destroying the rock with their sharp edges (Fig. 8). It is necessary to note that the oscillations of the tool will be of two kinds: axial and torsional.

Two forces act on each shot particle: the force of pressure on the bit, oriented along the normal to the bottom, determining the force tending to embed the shot particle in the rock; and the friction force, acting in the plane of the bottom and tending to tear away the part of the rock adjoining the shot grain. The combined action of these forces may be regarded as an impact directed at some angle to the surface of the bottom. The angular particles produced when cast-iron shot is split intensively destroy the rock at a comparatively small pressure on the bit; at the same time, because of the low strength of cast iron, they themselves wear out rapidly. A favorably oriented particle, under the action of impact, will produce destruction of the rock and will probably itself crumble into smaller, but likewise angular, particles; and at the bottom there will always be a sufficient number of “shot grains” favorably oriented with respect to the bottom and the bit.

1. Drilling with steel ball shot. In its main features the process proceeds analogously to the one just described, with the sole difference that the horizontal component of the impact is small; the tool is blunt, and it may be said that destruction of the rock proceeds under the action of a blunt tool, with impacts directed almost along the normal to the surface of the bottom (Fig. 9). Obviously, to achieve appreciable destruction of the rock the force “C” must be very large, and the productivity of such drilling, in general, is very insignif-

significant. Confirmation of this can be found in the experiments of E. V. Borovsky [1] and I. A. Ostroushko [9].

1. Drilling with steel cuttings. The general scheme of the action of forces remains unchanged (Fig. 10), but the process differs substantially from both the first and the second. The cutting arrives at the bottom in a form ready for work, having sharp edges, and at once produces effective deepening. The cutting is made of high-quality steel and for quite a long time retains its sharp edges at the same particle size, which is the reason for its highly productive operation. As it becomes rounded, the character of the cutting’s work approaches that of round shot; the cutting gradually loses its drilling properties, and the rate of deepening of the borehole falls.

Fig. 8

Fig. 9

Fig. 10

The drilling process differs substantially depending on the type of abrading material, but the common principle for all three is the destruction of the rock under the action of an obliquely directed blow. Let us consider the forces applied to an elementary particle (Fig. 11).

Fig. 11

Fig. 12

According to Coulomb’s law, the force of resistance to the relative displacement of bodies in contact is determined as \(P = F_{fr} + A\), where \(F_{fr} = fC\) is the friction force (\(f\) is the coefficient of friction, \(C\) is the force of normal pressure), and \(A\) is the cohesion of the rubbing pair [8].

The horizontal component of the force of the oblique blow is determined by the second term (cohesion). Observations of the operation of a chisel in surface treatment made it possible to conclude that the maximum cohesion and the greatest destruction of the rock are obtained at a chisel inclination angle \(\rho = 90 - \varphi\), where \(\varphi\) is the angle of friction between the chisel material and the surface being worked (Fig. 12). Indeed, if an ideally smooth surface has to be worked with complete absence of friction between the tool, then the only direction of the chisel that can cause destruction is along the normal to the surface (\(\varphi = 0\) and \(\rho = 90^\circ\)); conversely, with infinitely large friction, the correct direction of the chisel is along the tangent (\(\varphi = 90^\circ\) and \(\rho = 0\)).

To determine the most advantageous angle of application of the destructive force, it is obviously necessary to determine the coefficient of friction and the angle of friction. The determination of the friction force beneath the end face of the shot crown was carried out on a stand, the basic layout of which is shown in Fig. 13.

Fig. 13

1 Crown
2 Rock specimen
3 Table
4 Table bearings
5 Dynamometer

The friction force arising during rotation of the crown tended to set the rock specimen 2 in rotation together with table 3. Table 3 was held against rotation by the spring of dynamometer 5, connected to the table by a wire encircling it. The moment of friction between the crown and the rock was \(M_{tr} = P_{din} \cdot R_{stola}\).

On the other hand, the moment of friction will be equal to \(M_{tr} = f \dfrac{r_1 + r_2}{2} c\),

where \(f\) is the required coefficient of friction, \(r_1\) and \(r_2\) are the outer and inner radii of the crown, and \(c\) is the load on the crown.

Determinations of \(f = \tg \varphi\) were made under the following conditions: crown—\(r_1 = 24.5\) and \(r_2 = 19.5\); \(R_{stola} = 92.5\); number of revolutions of the crown—\(n = 20\) rpm; flushing—with water; size of shot particles up to 1 mm; rock—coarse-grained hornblende granite. The results of the determination are presented in Table 1.

Table 1

The obtained values of the coefficients of friction and the basic proposition concerning the most advantageous value of the angle of application of the destructive force are confirmed by other experiments, which will be presented in the next section.

Dependence of the rate of penetration on the pressure on the bit

In all works devoted to solving the question of the influence of pressure on the rate of penetration of a borehole, a graph of the dependence is given, representing a curve with a characteristic maximum at some optimum pressure. This maximum is explained differently by different authors: some [3] believe that at pressures above the optimum there occurs “too rapid crushing of the shot, as a result of which the rate of penetration decreases”; others [9] believe that at pressures on the bit above the optimum the shot splits and ceases to work (since only round shot works); still others [4] believe that the shot is squeezed out from under the end face of the bit and penetration of the borehole stops. In our view, the matter stands otherwise. The optimum pressure creates conditions most closely corresponding to the requirement of the most advantageous application of force \((\rho = 90 - \varphi)\) and ensuring maximum penetration¹. The force diagram corresponding to these conditions is shown in Fig. 11.

From the diagram it follows that \(P_{vr} = A + fC = \operatorname{ctg}\rho \cdot C + \operatorname{tg}\varphi \cdot C\), but \(\rho = 90 - \varphi\) and \(\operatorname{ctg}\rho = \operatorname{tg}\varphi\); then \(P_{vr} = 2 \operatorname{tg}\varphi \cdot c\), or

\[ \operatorname{tg}\varphi = \frac{P_{vr}}{2C}. \]

Laboratory determinations of \(\operatorname{tg}\varphi\) from drilling conditions at different numbers of revolutions gave the following values:

\[ \begin{aligned} \text{at } n &= 380\ \text{rpm}, && \operatorname{tg}\varphi = 0.127, && \varphi = 7^\circ,\\ \text{at } n &= 240\ \text{rpm}, && \operatorname{tg}\varphi = 0.129, && \varphi = 7^\circ,\\ \text{at } n &= 160\ \text{rpm}, && \operatorname{tg}\varphi = 0.157, && \varphi = 9^\circ,\\ \text{at } n &= \phantom{0}90\ \text{rpm}, && \operatorname{tg}\varphi = 0.127, && \varphi = 7^\circ, \end{aligned} \]

which agrees well with the values of \(\varphi\) obtained directly (Table 1).

Interesting conclusions of great practical importance follow from the experiments. If the coefficient of friction changes, then the position of the maximum and the value of the optimum pressure must change. And indeed, when drilling with a supply of shot, the pressure was increased to \(40\ \text{kg}/\text{cm}^2\), and the maximum had still not been reached (Fig. 7, dashed line). Exactly the same phenomenon is observed when drilling with steel cuttings. The coefficient of friction under the end face of the bit when drilling with steel cuttings is smaller, which makes it possible to increase the pressure on the bit and to achieve higher rates of penetration.

The second conclusion, which follows from consideration of the graph presented in Fig. 7, may be formulated as follows: for each number of revolutions of the bit there is its own optimum pressure, and its value decreases regularly with an increase in the number of revolutions.

Dependence of the rate of penetration on the number of revolutions

The results of investigations of the influence of the number of revolutions of the bit on the rate of penetration are presented by various authors [1, 9, etc.], with some researchers considering that the rate of penetration is directly proportional to the number

rotational speeds of the bit [1], while others [9] present data from which it is evident that increasing the number of revolutions by 280% increases the penetration rate by 450%.

Our investigations, carried out under the most varied conditions, give a different picture of this dependence. Experimental drilling under production conditions showed that the increase in penetration rate is not directly proportional to the number of revolutions of the bit, if the penetration rates are compared at optimum pressures. Special tests conducted under laboratory conditions make it possible to speak quite definitely about the general regularity of the change in penetration rate in connection with a change in the number of revolutions. This dependence is expressed by a curve of parabolic order (the graphs in Figs. 14 and 15).

Fig. 14

The results obtained make it possible to calculate the exponent \(k\), which characterizes the dependence of the penetration rate on the number of revolutions. The exponent \(k\) is a constant of the rock; therefore, we shall discuss methods for determining it below, after considering the resistance of rocks to drilling.

Fig. 15

On the resistance of rocks to drilling

In shot drilling, rocks are destroyed in a brittle manner; however, brittle deformations are always preceded by plastic ones; consequently, the ultimate strength of the rock still does not determine the resistance of the rock to drilling. For a complete characterization of the drillability of a rock, it is necessary to determine not only the ultimate compressive strength of the rock, but also the degree of its plasticity, that is, the rock’s capacity for flow. The behavior of a material beyond the elastic limit is expressed quite accurately by the polytropic equation [6, 10]:

\[ \sigma = \sigma_0 \left( \frac{l}{l_0} \right)^k, \]

where \(\sigma\) is the tensile (compressive) stress, \(\sigma_0\) is the conventional yield limit, \(l_0\) is the initial length (height) of the specimen, \(l\) is the final length (height) of the specimen, and \(k\) is the polytropic exponent.

The equation of the polytrope in logarithmic coordinates (Fig. 16) is a straight line inclined at an angle whose tangent is equal to \(k\), and which, when extended, cuts off on the stress axis the conditional yield limit \(\sigma_0\).

Fig. 16.

Fig. 17. 1 Absolutely rigid body.
2 Ideal elastic body. 3 Elastic-brittle body. 4 Elastic-plastic body. 5 Plastic body.
6 Real liquid. 7 Ideal liquid

All materials, including rocks, are capable of flowing, but the degree of their flow differs. Polytropes of various materials are shown schematically in Fig. 17. Deformations of rocks should be represented by a polytrope whose straight portion is inclined at an angle of \(40\text{–}70^\circ\), which corresponds to values of \(k = 0.7\text{–}2.75\). The greater the exponent of the polytrope, the smaller the capacity of the material for plastic changes. The compressive strength of rocks and the degree of flow can be determined by compressing specimens on Gagarin’s press, which is the first task of our further investigations. The polytropic exponent \((k)\) has also been determined by us from drilling conditions, using the general formula for the rate of penetration derived below.

Derivation of a general formula relating the principal regime parameters

There are several formulas in the literature relating drilling parameters to the power of the machine or to drilling productivity, derived for diamond drilling or for drilling with hard alloys [2, 5]. We have also made an attempt, on the basis of the views set forth, to derive an equation for the dependence of the rate of penetration on the principal parameters of the shot-drilling regime.

Let us introduce the necessary notation:

\(K_c\) — resistance of the rock to drilling \((\text{kg}/\text{cm}^2)\),
\(\Delta h\) — penetration per one revolution of the bit \((\text{cm})\),
\(R\) — radius of the borehole \((\text{cm})\),
\(r\) — radius of the core \((\text{cm})\),
\(n\) — number of revolutions of the bit per minute \((\text{rev}/\text{min})\),
\(U\) — rate of borehole penetration \((\text{cm}/\text{sec})\),
\(C\) — load on the bit \((\text{kg})\).

The work of rock destruction performed by the bit in one revolution is equal to:

\[ \Delta W=\pi(R^2-r^2)K_c\Delta h. \tag{1} \]

The work per second is equal to:

\[ W_{\mathrm{sec}}=\pi(R^2-r^2)K_c h_{\mathrm{sec}}, \tag{1a} \]

but the work per second is power, and the deepening per second is the rate of deepening; therefore

\[ N=\pi(R^2-r^2)K_c U. \tag{2} \]

Let us determine the torque at the face of the bit:

\[ M_{\mathrm{tor}}= \frac{71620\pi(R^2-r^2)K_cU}{n\cdot100\cdot75}. \tag{3} \]

After division by the mean radius \(\left(\dfrac{R+r}{2}\right)\) and calculation of the coefficients, we obtain an expression for the horizontal component of the force producing destruction:

\[ P_p=\frac{60}{n}K_cU(R-r). \tag{4} \]

Replacing the difference between the radii of the borehole and the core by the width of the annular face \((R-r=b)\), we obtain:

\[ P_p=\frac{60}{n}K_cUb\;(\mathrm{kg}). \tag{4a} \]

The resistance of the rock to drilling \((K_c)\), as a complex resistance to two kinds of deformation, is determined with sufficient accuracy by the equation [2].

\[ K_c\Delta h=\sigma_{\mathrm{cж}}\Delta h^k, \]

where \(\sigma_{\mathrm{cж}}\) is the ultimate compressive strength of the rocks, \(\Delta h\) is the absolute deformation, and \(k\) is the polytropic exponent. It is quite appropriate to assume that the deformation time varies to the same power \(k\); then \(K_cU=\sigma_{\mathrm{cж}}U^k\), and equation (4a) is represented in the following form:

\[ P_p=\frac{60}{n}\sigma_{\mathrm{cж}}U^k b\;(\mathrm{kg}). \tag{5} \]

In addition to this force, a friction force \(F_{\mathrm{fr}}\) will act in the same direction; it is determined by dividing the friction moment of the annular heel by the mean radius. The friction moment of the run-in annular heel is expressed [7] as:

\[ M_{\mathrm{fr}}=f\frac{r_1+r_2}{2}\cdot C, \]

where \(f\) is the coefficient of friction,
\(r_1\) and \(r_2\) are the inner and outer radii of the heel (bit),
\(C\) is the load on the heel (on the bit), kg.

Dividing by the mean radius, we obtain the friction force

\[ F_{\mathrm{fr}}=fC\ \mathrm{kg}. \tag{6} \]

Summing the reactive forces expressed by equations (5) and (6),

\[ P_{\mathrm{total}}=P_p+F_{\mathrm{fr}}= \frac{60}{n}\sigma_{\mathrm{cж}}U^k b+fC\;(\mathrm{kg}) \tag{7} \]

The active force developed by the drill-rig motor and applied to the bit must be equal to the sum of the reactive forces.

\[ P_{\mathrm{rot}} = P_{\mathrm{tot}} = P_{p} + F_{\mathrm{fr}}, \quad \text{and} \quad P_{\mathrm{rot}} = \frac{M_{\mathrm{tor}}}{r_{\mathrm{av}}}\eta, \tag{8} \]

where \(M_{\mathrm{tor}}\) is the torque on the spindle of the drilling rig \((\mathrm{kg \cdot cm})\),

\(\eta\) is the efficiency of the drill-rod column \((M_{\mathrm{tor}}\eta\) is the torque applied to the bit),

\(r_{\mathrm{av}}\) is the mean radius of the bit.

Before equating the rotational force to the sum of the reactive forces, it is necessary to introduce one more coefficient. The work of the drilling tool can be evaluated by the actually performed work of destruction, if the efficiency of the tool is known. Let us introduce the efficiency of comminution \((\eta_{\rho})\) and equate the active force to the sum of the reactive forces:

\[ \frac{M_{\mathrm{tor}}}{r_{\mathrm{av}}}\eta = \frac{60\,\sigma_{\mathrm{comp}}\,U^{k} b}{n\,\eta_{\rho}} + fC \tag{9} \]

or

\[ U^{k} = \frac{n\eta_{\rho}}{60\cdot\sigma_{\mathrm{comp}}\cdot b} \cdot \left( \frac{M_{\mathrm{tor}}}{r_{\mathrm{av}}}\eta - fC \right). \tag{9a} \]

Let us analyze the formula obtained. The rate of penetration is proportional to the number of revolutions, but the exponent \(k\), which depends on the properties of the rock, substantially changes the character of the dependence, and only in the special case \((k = 1)\) is the rate of penetration directly proportional to the number of revolutions. For viscous rocks \((k < 1)\), an increase in the number of revolutions is extremely effective; for brittle rocks, in which chiefly fractional drilling occurs, an increase in the number of revolutions increases the rate of penetration, but not in direct proportion.

The rate of penetration is greater the higher the efficiency of the bottom-hole tool, that is, the coarser the separated rock particles and the less work is spent on repeated crushing and dispersion of the material.

The rate of penetration is smaller the stronger the rock.

The rate of penetration decreases with increasing width of the annular bottom hole.

The expression in parentheses determines the adhesion of the tool to the rock. The law of its variation is unknown, but numerous checks by calculations carried out on the most varied data show that, at the optimum pressure \(^{1)}\) on the bit, the following relationship exists:

\[ \frac{M_{\mathrm{tor}}}{r_{\mathrm{av}}}\eta = 2fC_{\mathrm{opt}}. \]

This means that the adhesion of the cuttings to the rock at the optimum pressure is numerically equal to the friction force. The total force braking the bit at the optimum pressure is equal to twice the friction force measured in the complete absence of borehole deepening. Studying the character of the variation of the adhesion of the tool to the rock is our immediate task.

Determination of the Polytropic Exponent and Other Examples of the Use of the Equations

Using the results of experimental drilling, we calculated the polytropic exponent for granite. Under laboratory conditions, when setting up the experimental conditions, formula (9a) can be simplified by assuming that:

\[ \frac{\eta_{\rho}}{60\,\sigma_{\mathrm{comp}}\,b} = B = \mathrm{const}. \]

\(^{1)}\) The optimum pressure on the bit is that pressure at which the most intensive deepening of the borehole is observed.

We solve the system of equations:

\[ U_1^k = n_1 B\left(\frac{M_{kr1}}{r_{cr}} - f C_1\right); \]

\[ U_2^k = n_2 B\left(\frac{M_{kr2}}{r_{cr}} - f C_2\right); \]

\[ U_3^k = n_3 B\left(\frac{M_{kr3}}{r_{cr}} - f C_3\right). \]

The data required for solving these equations are taken from the experiment (Fig. 7).

\[ \begin{aligned} U_1 &= 0.0183\ \text{cm/sec},\\ U_2 &= 0.0160\ \text{cm/sec},\\ U_3 &= 0.0105\ \text{cm/sec},\\ f &= 0.130,\\ n_1 &= 380\ \text{rpm},\\ n_2 &= 240\ \text{rpm},\\ n_3 &= 96\ \text{rpm}, \end{aligned} \qquad \begin{aligned} \frac{M_{kr1}}{r_{cr}} &= 31.5\ \text{kg},\\[4pt] \frac{M_{kr2}}{r_{cr}} &= 36.0\ \text{kg},\\[4pt] \frac{M_{kr3}}{r_{cr}} &= 46.2\ \text{kg}, \end{aligned} \qquad \begin{aligned} C_1 &= 124\ \text{kg},\\ C_2 &= 142\ \text{kg},\\ C_3 &= 180\ \text{kg}. \end{aligned} \]

As a result of the calculations we obtain the following values:

\[ k_1 = 1.82,\qquad k_2 = 2.1,\qquad k_3 = 1.68. \]

Their average is \(k_{cr} = 1.87\).

It should again be noted that, in the relation between the friction force and the total rotating force applied to the end of the bit, in all cases the friction force (without accounting for adhesion) is equal to one half of the total force, or the adhesion is equal to the friction force.

Verification in experimental boreholes of various depths confirmed this relation: in all cases \(f \cdot C_{opt}\) amounts to one half of the rotating force.

Consequently, the optimum pressure on the bit can be expressed in two ways:

\[ C_{opt} = \frac{60\,\delta_{csh} U^k b}{n f \eta_p}, \tag{10} \]

\[ C_{opt} = \frac{M_{kr}\eta}{2 r_{kr} f}. \tag{11} \]

After completion of the studies it will be possible to use equation (10) to determine the optimum parameters of the regime, and equation (11) to determine the drive power of the drilling rig.

From the graph of the dependence of the rate of advance on the number of revolutions, taking into account the technical capabilities of the drilling equipment, we find the optimum number of revolutions of the bit \((n_{opt})\) and the corresponding rate of advance \((U)\) in rocks of the given category (for specified \(\delta_{csh}\) and \(k\)).

We substitute the values \(n_{opt}\) and \(U\) into equation (10) and find the optimum pressure on the bit when drilling with cast-iron or steel shot (depending on the quality of the shot, the coefficient of friction will change).

Using equation (11), we determine the torque at the end of the bit, and after multiplying by the efficiency of the drill-rod string, we find the torque at the spindle of the drilling rig.

Conclusions

1. Concepts of the nature of the operation of shot—as a cutting tool or as a crushing tool—do not explain many facts known from drilling practice and do not provide a basis for the scientific development of shot-drilling regimes.

2. Studies based on the new hypothesis of the rock’s resistance to drilling, and the identification of the principal parameters of the regime affecting the rate of penetration, make it possible, as a first approximation, to solve the question of optimum regimes of shot drilling and open up prospects for its further improvement. At the present stage of the research, the following may be asserted:

1) the optimum pressure on the bit does not correspond to the strength limit of the shot; it depends on the number of revolutions of the bit and varies with changes in the size of the shot particles, their shape, and also the material of the bit and of the shot—that is, it depends on the factors determining the coefficient of friction;

2) the dependence of the rate of penetration on the number of revolutions is expressed by the curve $U^k = Ln$, where the exponent $k$ depends on the rock and is determined as the degree of its plasticity;

3) the drillability of a rock—its resistance to drilling—is determined by its strength, expressed through the critical resistance to compression, and by the plasticity of the rock, expressed through the exponent of the degree of plasticity. The first shows how strongly the materials composing the rock are bound together; the second characterizes the rock’s capacity for plastic rearrangements—the capacity of the rock to flow.

1. The derived equation for the general dependence of the rate of penetration on the principal regime parameters, while to a certain extent empirical, nevertheless reflects the nature of the technological process and, after the necessary diagrams have been compiled, will make it possible to prescribe the drilling regime correctly under specific geological and technical conditions.

2. The classification of rocks by drillability, which at present is based on time observations and a subjective assessment of rock strength, can be carried out on the basis of an objective assessment of drillability.

Literature

1. E. V. Borovskii. Operation of shot as an abrading material. Journal “Razvedka nedr,” No. 13, 1934.

2. B. I. Vozdvizhenskii. Drilling Mechanics. Gosgeolizdat, 1949.

3. S. A. Volkov. Technology of Shot Drilling. Sverdlovsk, 1950.

4. Temporary Instruction for Shot Drilling. Gosgeolizdat, 1950.

5. A. V. Kuznetsov. Some calculations for core drilling. Journal “Razvedka nedr,” No. 14, 1933.

6. V. D. Kuznetsov. Physics of the Solid State, Vol. IV. Tomsk, 1947.

7. A. S. Korenyako and Kremenshtein. Theory of Mechanisms and Machines. Gostekhizdat, Kiev, 1952.

8. I. Kragelskii. On the friction of unlubricated surfaces. All-Union Conference on Friction and Wear in Machines. Proceedings, Vol. I, Publishing House of the Academy of Sciences of the USSR, 1939.

9. I. A. Ostroushko. Destruction of Rocks during Drilling. Gosgeolizdat, 1950.

10. P. M. Tsimbarevich. Mechanics of Rocks. Ugletekhizdat, 1948.

1. A more detailed discussion of these views is given in the article by I. S. Mityushkin and V. I. Molchanov (see the present collection, p. 78). ↩