ON THE QUESTION OF THE DRILLING FORMULA AND OF A RATIONAL FORM FOR PROCESSING EXPERIMENTAL DATA OBTAINED IN THE DESTRUCTION OF ROCKS

P. M. ALABUZHEV

1. In the work by Associate Professor A. I. Medvedko, “The Drilling Formula” [1], it is noted that “The technique of blasthole drilling progresses from year to year, but the theory of this process has essentially not yet been created” [1, 13].

The destruction of rocks has been studied in considerable detail by V. S. Fedorov [2], E. F. Epstein [3], L. A. Schreiner [4], P. A. Rebinder [5], B. K. Buchnev [6], I. S. Pokrovskii [7], and many others. Each investigator proposes his own point of view, his own approach to the question under consideration; however, as A. I. Ter-Grigoryan notes, “Not one of the proposed formulas determining the drillability of rocks can lay claim to completeness. The complex problem of establishing the drillability of rocks may be considered solved only when an equation is given that relates the mechanical drilling speed to the physical properties of the rock and the medium, the elements of the drilling regime, the parameters of the bit, and its variable wear” [8, 36].

On the basis of acquaintance with works on rock destruction [1–8], as well as with the physics of solids by Corresponding Member of the Academy of Sciences of the USSR V. D. Kuznetsov [9] and the mechanics of rocks by Prof. P. M. Tsimbarevich [10], the author of the present article attempts to consider the question of the drilling formula from the standpoint of dimensional theory.

Dimensional analysis, as a method for the preliminary study of natural phenomena, is often applied in the study of branches of physics, mechanics, hydrodynamics, and aerodynamics, and has firmly entered educational and scientific literature [11–22]. Without dwelling on the diversity of views concerning the doctrine of dimension and sufficiently fully discussed in the work of Prof. M. F. Malikov [18], we shall note that, on the basis of dimensional analysis, it is possible to establish the “structure of functional relationships between physical quantities” [15] even in the case when there is no differential equation describing the process; moreover, it is possible to establish relationships not only between the physical quantities characterizing the phenomenon, but also between their dimensionless combinations, i.e., as relationships between similarity criteria. Thus the formulation of the experiment, its orientation, and the processing and systematization of the experimental data obtained are facilitated.

Prof. K. M. Polivanov writes: “The methods of similarity theory and dimensional analysis are extremely close, and the difference between them may be discerned only in the fact that dimensional analysis does not consider the fundamental equations of precisely the given particular system, the processes in which may be determined only by a part of the general equations underlying the chosen system of units” [20, 233]. And somewhat further: “However, this distinction is also erased in those cases when, for greater definiteness of the solution, in ...”

in analysis one uses a system of units specially constructed so as to be most convenient for the analysis of the processes of precisely this system [20, 234].

In the twentieth century the fundamental theorem of similarity theory and dimensional theory was proved, concerning the possibility of expressing physical laws in the form of a dependence between dimensionless numbers (called similarity criteria) that characterize the phenomenon—the so-called \(\Pi\)-theorem, whose author is unjustly said to be the American physicist Buckingham [14; 15], who published a proof in 1914. “In reality the \(\Pi\)-theorem first found practical applications in Russia before Buckingham’s article appeared. In 1911 there appeared a paper by A. Federman, a lecturer at the St. Petersburg Polytechnic Institute, who proved a more general theorem from which the \(\Pi\)-theorem is derived as a consequence. It was recently established that the \(\Pi\)-theorem had already been used in 1909 in the work of the Kuchino Aerodynamic Institute, organized and operating under the direction of N. E. Zhukovsky. A proof of this theorem was published in the proceedings of the Kuchino Institute in 1912” [21, 99].

Let us add that borrowing of the results with reference to these works is contained in the book by L. Civita and U. Amaldi [12, 370] and is known in the literature under the name “method of zero dimensions” [12; 13; 17].

1. The method of zero dimensions is based on the theorem of homogeneity of formulas for physical quantities.

If there exists a functional relation between a number of dimensional \((u_i)\) and dimensionless \((c_i)\) quantities of the form: \(f(u_1, u_2, u_n \ldots c_1, c_2 \ldots c_m)=0\). From this equation any quantity may be singled out and represented as a function of the remaining quantities, for example:

\[ u_1 = f_1(u_2, u_3 \ldots u_n, c_1, c_2 \ldots c_m). \tag{1} \]

Choose from among the arguments any three quantities independent of one another, for example, \(u_2, u_3, u_4\), and take them as the fundamental units of our system\(^1\); then the remaining quantities \(u_1, u_5 \ldots u_n\) will be derived quantities in this new system of units and may be expressed through the fundamental units \(u_2, u_3, u_4\). The dimension of \(u_1\) in the new system will be \([u_1]=u_2^p\cdot u_3^q\cdot u_4^r\); equation (1) must possess triple homogeneity with respect to \(u_2, u_3, u_4\). Replacing in (1) the values \(u_2, u_3, u_4 \ldots u_n\) by the ratios of these quantities to their dimensions in the new system \((u_2, u_3, u_4)\), we obtain:

\[ \frac{u_1}{u_2^p \cdot u_3^q u_4^r} = \Phi\left( 1, 1, 1, \frac{u_5}{[u_5]_{2,3,4}}, \ldots, \frac{u_n}{[u_n]_{2,3,4}}; c_1, c_2 \ldots c_m \right). \]

Here

\[ \frac{u_i}{[u_i]_{2,3,4}} \]

denotes a dimensionless quantity—\(\Pi_i\) (similarity criterion), equal to the ratio of the quantity \(u_i\) to its dimension in the system of units \((u_2, u_3, u_4)\), and, obviously,

\[ \frac{u_2}{[u_2]_{2,3,4}}=1;\quad \frac{u_3}{[u_3]_{2,3,4}}=1;\quad \frac{u_4}{[u_4]_{2,3,4}}=1. \]

Between the dimensionless expressions there exists the dependence

\[ \Pi_1=\Phi(\Pi_5,\Pi_6\ldots\Pi_n) \]

(this is the \(\Pi\)-theorem)

\(^1\) The quantities \(u_2, u_3, u_4\) are mutually independent if the determinant composed of the exponents of their dimensions is not equal to zero, i.e.

\[ \Delta= \begin{vmatrix} p_1 & q_1 & r_1\\ p_2 & q_2 & r_2\\ p_3 & q_3 & r_3 \end{vmatrix} \ne 0. \]

or

\[ u_1=u_2^p\cdot u_3^q\cdot u_4^r\cdot \Phi(\Pi_5,\Pi_6\ldots \Pi_n;c_1c_2\ldots c^m). \]

If the number of physical parameters is equal to \(n\), and the number of fundamental units is equal to \(k\) (usually \(k=3\)), then the number of independent dimensionless expressions will be equal to \(n-k\).

Obviously, the smaller the difference \(n-k\), the easier and more reliable it is to find the form of the function \(\Phi\) that relates the dimensionless quantities.

As an example of the application of the method of zero dimensions, let us find the equation of the thermodynamic state of an ideal gas, if it is known that it must contain four dimensional quantities

\[ \Phi(p,v,R,T)=0 \]

or

\[ v=f(p,R,T). \]

Here

\[ n=4,\qquad k=3,\qquad n-k=1 \]

therefore, there must be one dimensionless expression. Let us find the dimensions of the quantities in the system \((m-l-t)\):

\[ \text{pressure, kg/cm}^2\quad [p]=\frac{mlt^{-2}}{l^2}=ml^{-1}\cdot t^{-2}, \tag{1} \]

\[ \text{specific volume, cm}^3/\text{kg}\quad [v]=\frac{l^3}{mlt^{-2}}=m^{-1}\cdot l^2\cdot t^2, \tag{2} \]

\[ \text{temperature, kg}\cdot\text{cm}\quad [T]=[mv^2]=ml^2\cdot t^{-2}, \tag{3} \]

\[ \text{gas constant, kg}\cdot\text{m}/\text{kg}\cdot\text{degree}\quad [R]=[l\cdot T^{-1}]=m^{-1}\cdot l^{-1}\cdot t^2. \tag{4} \]

Let us take \((p,R,T)\) as the fundamental system of units. The quantities \(p\), \(v\), \(T\) are mutually independent, since the determinant composed of the exponents of their dimensions

\[ \Delta= \begin{vmatrix} 1,&-1,&-2\\ -1,&-1,&2\\ -1,&2,&-2 \end{vmatrix} \ne 0. \]

Let us find the dimensions of all quantities in this new system of units \((p,R,T)\).

\[ [p]=1;\quad [R]=1;\quad [T]=1;\quad [v]=m^{-1}\cdot l^2\cdot t^2 \]

from (4)

\[ m^{-1}=\frac{R}{l^{-1}\cdot t^2};\quad \text{from (3) } l^2=\frac{T^2}{mt^{-2}};\quad \text{from (1) } t^2=\frac{ml^{-1}}{p}, \]

then

\[ [v]=m^{-1}\cdot l^2\cdot t^2 =\frac{R}{l^{-1}\cdot t^2}\cdot \frac{T}{mt^{-2}}\cdot \frac{m\cdot l^{-1}}{p} =\frac{RT}{p}. \]

Replacing \(v,p,R,T\) by the ratios of these quantities to their dimensions in the new system \((p,R,T)\), we shall have:

\[ \frac{v}{\dfrac{RT}{p}}=\Phi(1,1,1)=\mathrm{const}=c. \]

From experiment we determine \(c=1\). Consequently, \(pv=RT\).

As is known, the equation of state of an ideal gas can be represented graphically in the form of a thermodynamic surface in the coordinates: \(p, v, T\).

1. Let us see what indications dimensional theory gives with respect to the study of rock destruction. Let us consider the given question from the point of view of penetration rate and consumed power.

According to the assertion of Prof. L. A. Schreiner that the drilling rate is inversely proportional to hardness [4], it follows that for hard rocks the degree of their comminution (dispersity) \(\beta\) must also increase. Further, if one takes into account the considerations of Prof. P. M. Tsimbarevich [10] according to the theory of mineability, then the drilling rate is (in the first approximation) a function of the following quantities:

\[ v=\Phi(N,D,d,\beta), \]

where \(v\) is the drilling rate, \(\dfrac{\mathrm{cm}}{\mathrm{sec}}\);

\[ N \]

is power, \(\dfrac{\mathrm{kg}\cdot \mathrm{m}}{\mathrm{sec}}\);

\[ D \]

is mineability, \(\dfrac{\mathrm{cm}^{2}}{\mathrm{kg}\cdot \mathrm{m}}\);

\[ d \]

is the bit diameter, \(\mathrm{cm}\);

\[ \beta \]

is dispersity, \(1/\mathrm{mm}\).

If as the basic independent units we take \(N, D, d\) (it is not difficult to show that \(\Delta \ne 0\)), then

\[ \frac{v}{DNd^{-1}}=\Phi\left(1,1,1,\frac{\beta}{d^{-1}}\right)=\Phi(\beta d). \]

The form of the function \(\Phi\) is determined from experiment.

If the drilling rate is inversely proportional to the hardness of the rock, and consequently also to the dispersity, then one may put \(\Phi(\beta d)=\dfrac{c}{\beta d}\),

where \(c=\mathrm{const}\). Then

\[ v=\frac{DN}{d}\cdot\frac{c}{\beta d}=k\frac{N}{d^{2}} \]

(the formula of A. I. Medvedko).

Here the coefficient \(k=\dfrac{cD}{\beta}\) must be found from experiment. Under the stated assumptions, the drilling rate is proportional to the power and inversely proportional to the square of the diameter.

Incidentally, let us note that the work of destruction of brittle bodies is inversely proportional to the dispersity of the material being destroyed (Rittinger’s law), and according to the law of similarity (V. L. Kirpichev) is proportional to the volumes of the bodies being deformed. As L. A. Schreiner notes, with a decrease in the linear dimensions of a body its strength increases: “The law of similarity with a correction for the scale factor must be in agreement with practical data. Nothing else can be expected, since physically it is quite well substantiated, as is Rittinger’s law” [4, 16].

The drilling rate \(v\) depends on the quality of the tool, the drilling regime, the properties of the medium being destroyed, the system of development, and the organization of work.

We shall consider the physico-technical side of the question; then the drilling rate

\[ v=\Phi\bigl( \underbrace{Q_{\delta},d,l,\alpha,e}_{\substack{\text{bit}\\ \text{parameters}}}, \underbrace{\Theta}_{\text{wear}}, \underbrace{R_z,\mu,\lambda}_{\substack{\text{properties}\\ \text{of the medium}}}, \underbrace{N,M_{\mathrm{cr}},A_{\delta},P,\omega,v_{\delta},z}_{\text{drilling regime}} \bigr), \]

where: \(Q_d\) is the weight of the bit, kg;

\(d\) is the bit diameter, cm;

\(\alpha\) is the bit point angle, \([\alpha]=1\);

\(e\) is the number of bit blades, \([e]=1\);

\(Q\) is the wear of the bit due to friction forces, kg·cm\(^2\);

\(R_z\) is the absolute hardness of the rock, kg/cm\(^2\);

\(\mu\) is the coefficient of friction between the medium and the bit, \((\mu)=1\);

\(\lambda\) is the bottom-hole cleaning coefficient, \([\lambda]=1\);

\(N\) is the power consumed by the machine, h.p. (kW);

\(M_{kr}\) is the torque on the bit, kg/m;

\(A_b\) is the work of one blow on the striking bar in percussion-rotary drilling, kg·m;

\(P\) is the feed force, kg;

\(\omega\) is the angular velocity of rotation of the bit, \(1/\text{s}\);

\(v_b\) is the velocity of the striking bar at the instant of impact, cm/s;

\(z\) is the number of blows of the striking bar per min., blows/min.

Here we have thirteen dimensional quantities and four dimensionless ones. Let us take as the basic independent quantities: \(P, d, \omega\); in this case \((\Delta \ne 0)\). \(n-k=13-3=10\). Thus, with this formulation of the problem we have ten dimensionless expressions

\[ \frac{v}{d\omega} = \Phi\left( \frac{Q_d}{P},\, \frac{l}{d},\, \frac{Q}{P d^{-2}},\, \frac{R_z}{P d^{-2}},\, \frac{N}{P d\omega},\, \frac{M_{kr}}{P d},\, \frac{A_b}{P\cdot d};\, \frac{v_b}{d\omega},\, \frac{z}{\omega} \right). \]

united with one another by the function \(\Phi\). In the general case, finding the function \(\Phi\) is very difficult. Judging from the dimensions of the arguments, the idea arises of a connection between:

1) bit wear and rock hardness, i.e. \(\theta=\Phi_1(R_z)\),

2) the work on the striking bar and the torque: \(M_{kr}=\Phi_2(A_b)\),

3) the power of the machine, the velocity of the striking bar, and the feed force, i.e.
\(N=\Phi_3(P,v)\).

In order to identify the basic connections and relations, we introduce some simplifying assumptions.

In rotary drilling the role of \(A_b\) is not significant; then, with the same tool parameters and the same operating regime of the drilling machine, neglecting wear, we have:

\[ v=\omega\cdot d\cdot \Phi\left(\frac{R_z}{P\cdot d^{-2}}\right) \]

the form of the function \(\Phi\) is found from experience.

If we assume

\[ \Phi\left(\frac{R_z}{P\cdot d^{-2}}\right) = \frac{P\cdot d^{-2}}{R_z}, \]

then the guess, or guiding idea, is the obvious condition that the rate of advance decreases with increasing rock hardness; then the drilling rate will be inversely proportional to the rock hardness, i.e.

\[ v=\omega d\cdot \frac{P}{R_z d^2} = \frac{\omega P}{R_z\cdot d} = k\cdot \frac{P}{R_z} \quad ^1) \]

\(^1\) According to data of Prof. L. A. Schreiner, the volumetric rate of destruction is

\[ v=a\cdot \frac{\omega}{p}\cdot P = k'\frac{P}{p}. \]

Here \(a\) is a dimensional coefficient, \(\omega\) is the speed of displacement, \(p=R_z\) is the hardness to indentation, \(P\) is the load, and \(k'\) is a dimensional coefficient [4, 149–151].

According to L. A. Schreiner, in developing rational drilling regimes one should proceed from the hardness of rocks.

If, for one and the same rock hardness, we are interested in the penetration rate as a function of power, then it is not difficult to obtain the structure of the formula:

\[ v=\omega d\cdot \Phi\left(\frac{N}{P\cdot d\omega}\right). \]

If we set

\[ \Phi\left(\frac{N}{Pd\omega}\right)=k\cdot \frac{N}{Pd\omega}, \]

where \(k=\mathrm{const}\), determined from experiment, then

\[ v=k\cdot \frac{N}{P}. \]

At one and the same power, the drilling rate is inversely proportional to the feed force (with excessive pressure, the friction forces between the bit and the rock are large). Under the optimum regime, “the magnitude of the axial pressure must correspond to the power of the drilling machine” [6].

At one and the same impact energy, for one and the same medium, within known limits, the drilling rate is proportional to the number of impacts:

\[ v=d\omega\cdot \Phi\left(\frac{z}{\omega}\right), \quad \text{if we set} \quad \Phi\left(\frac{z}{\omega}\right)=\frac{z}{\omega}\cdot k, \]

then

\[ v=kdz=k'\cdot z. \]

The preservation of linearity of the penetration rate with respect to the number of impacts (or revolutions) over broader limits is evidently accompanied by a simultaneous increase in the pressure and power of the drilling machine \(\left(v=k\cdot \frac{N}{P}=k'\cdot z\right)\).

*

4. Under the combined action of a number of factors on the drilling process, finding the function \(\Phi\), which characterizes the drilling rate \(v\), is difficult, and, evidently, the principal regularities will most likely be revealed experimentally with rational processing of the experimental data.

Proceeding from the structure of the general formula, the idea arises of setting up experiments to determine relations of the following form:

\[ v=\omega\cdot d\cdot \Phi_1\left(\frac{Q_d}{P},\ \frac{v_d}{\omega d},\ \frac{A_b}{Pd}\right), \tag{1} \]

\[ v=\omega d\cdot \Phi_2\left(\frac{N}{\omega dP},\ \frac{M_{kr}}{P\cdot d},\ \frac{z}{\omega}\right), \tag{2} \]

\[ v=\omega d\cdot \Phi_3\left(\frac{R_z}{P\cdot d^{-2}},\ \frac{\Theta}{Pd^{-2}},\ u,\ \lambda\right), \tag{3} \]

\[ v=\omega d\,\Phi_1\left(\frac{l}{d},\ z,\ e,\ \frac{Q_b}{P}\right). \tag{4} \]

According to formula (1), the drilling rate is found in percussion and percussion-rotary drilling (as a special case, when \(M_{kr}=0\), only in rotary drilling) as a function of the relation of the impacting masses, the impact energy, and the deformation rate; according to formula (2), the drilling rate is determined as a function of the relation between dimensionless quantities that include the power, torque, and number of revolutions; according to formula (3), the rate of destruction is determined as a function of tool wear.

and the properties of the medium; formula (4) gives an idea of the drilling rate as a function of the bit parameters and the axial pressure; finally, to link the formulas with one another. On the basis of the available experiments one may additionally construct a nomogram:

\[ v=\omega d \cdot \Phi_5\left(\frac{Q_b}{P},\ \frac{l}{d}\cdot \frac{N}{\omega dP},\ \frac{R_z}{P d^{-2}}\right) \tag{5} \]

and others.

On the basis of the experimental material obtained, it is possible to construct a series of tables, graphs, and nomograms which, in a first approximation, in parts and from various sides, reflect the complex process of drilling. For example, one may reveal the influence on the magnitude of the power of the machine performing the drilling of the following criteria:

\[ N=Pd\omega \cdot f_1\left(\frac{R_z}{P d^{-2}},\ \frac{M_{\mathrm{cr}}}{Pd},\ \frac{A_b}{Pd}\right), \]

\[ N=Pd\omega \cdot f_2\left(\frac{z}{\omega},\ \mu,\ \lambda\right), \]

\[ N=P\cdot d\omega \cdot f_3\left(\frac{l}{d},\ \frac{Q_b}{P},\ \alpha,\ e\right). \]

Undoubtedly, it would be more desirable to have a single formula relating all the quantities indicated above; but, as noted in the literature, there is as yet no satisfactory formula, and at the present stage in the development of the science of the theory of drilling it is hardly possible to obtain in the near future a single universal formula comprehensively encompassing the complex (many-sided) process of drilling—a formula sufficiently simple and convenient for practical applications.

It is not useless to recall the statement of the major Russian theoretician, practical engineer, and founder of mining mechanics I. A. Time, that “...the magnitude of errors is proportional to the number of coefficients, and therefore simpler (approximate) formulas, corrected by a smaller number of coefficients, may often prove in practice more accurate than more complicated ones” [23, V].

When drilling with pneumatic drills, the rate of advance

\[ v=\Phi\left( \underbrace{Q_b,\ d,\ l,\ \alpha,\ e}_{\text{drill parameters}},\ \underbrace{\Theta}_{\text{wear}},\ \underbrace{R_z,\ \mu,\ \lambda}_{\text{properties of the medium}},\ \underbrace{S,\ Q_y,\ Q_m,\ D}_{\text{drill parameters}},\ \underbrace{p,\ H,\ M,\ N,\ A_b,\ P,\ n,\ z,\ h}_{\text{drilling regime}} \right); \]

Here the following are denoted (in addition to those considered above):

\(S\)—stroke of the striker, cm;
\(D\)—diameter of the hammer cylinder, cm;
\(Q_y\)—weight of the striker, kg;
\(Q_m\)—weight of the hammer;
\(p\)—air pressure in the hammer cylinder, kg/cm\(^2\);
\(H\)—air consumption, m\(^3\)/min;
\(h\)—feed clearance, mm.

Omitting the determination of the general form of the formula as a function of all dimensionless parameters, let us dwell on certain particular dependencies.

When comparing different designs of pneumatic drills, under approximately identical operating conditions, the drilling rate of advance \(v\) may be assumed to depend on the air pressure \(p\), the number of blows \(z\), the stroke length of the striker \(S\), the weight of the striker \(Q_y\), the cylinder diameter \(D\), and the feed force \(P\),

air consumption \(H\), power \(N\), number of revolutions of the drill \(n\), magnitude of the torque \(M_k\), and rock strength \(\mathrm{R}_z\), i.e.

\[ v=\Phi(p,z,S,Q_y,D,P,H,N,n,M_k,\mathrm{R}_z). \]

Choosing as the fundamental units \(p, S, z;\ (\Delta \ne 0)\), we obtain:

\[ \frac{v}{zS} = \Phi\left( 1,1,1, \frac{Q_y}{pD^2}, \frac{D}{S}, \frac{P}{pD^2}, \frac{H}{D^2Sz}, \frac{N}{D^2Spz}, \frac{n}{z}, \frac{M_k}{pD^2S}, \frac{\mathrm{R}_z}{p} \right), \]

or

\[ v = zS\cdot \Phi\left( \frac{D}{S}, \frac{n}{z}, \frac{Q_y}{P}, \frac{H}{D^2Sz}, \frac{N}{D^2Spz}, \frac{M_k}{pD^2S}, \frac{\mathrm{R}_z}{p} \right), \]

obviously, one can construct nomograms:

\[ v=zS\cdot \Phi_1\left(\frac{D}{S},\frac{n}{z},\frac{Q_y}{P}\right); \]

\[ v=z\cdot S\cdot \Phi_2\left( \frac{H}{D^2Sz}, \frac{N}{D^2Spz}, \frac{M_k}{pD^2S} \right); \]

\[ v=zS\cdot \Phi_3\left(\frac{\mathrm{R}_z}{p}\right) \]

or, for the last formula, \(v=z\cdot S\cdot \xi\), where ultimately \(\xi=f(p)\).

At the same time one can construct the graphs:

\[ H=f_1\left(\frac{N}{p}\right) \quad \text{and} \quad M_k=f_2\left(\frac{N}{z}\right). \]

Suppose that we are interested in the values of the torque on the drill \(M_k\) as a function of the hammer weight \(Q_y\), the drill weight \(Q_b\), the number of blows \(z\), the number of revolutions of the drill \(n\), and the air consumption \(H\), i.e.

\[ M_k=f(Q_y,Q_b,z,n,H). \]

Choosing as the fundamental units the quantities \(Q_y,z,H,\ (\Delta \ne 0)\), we obtain:

\[ M_k = Q_b \sqrt[3]{\frac{H}{z}}\cdot \Phi\left(\frac{Q_b}{Q_y},\frac{n}{z}\right). \]

This dependence can be represented in the form of a graph in dimensionless coordinates:

\[ \frac{M_k}{Q_b\cdot \sqrt[3]{H/z}} \quad \text{and} \quad \frac{n}{z} \quad \text{with parameter} \quad \frac{Q_b}{Q_y}. \]

Similarly, if it is necessary to find the dependence of the feed force \(P\) on the pressure \(p\), the number of blows \(z\), the air consumption \(H\), and the number of revolutions of the drill \(n\), i.e. \(P=f(p,z,H,n)\), then, taking as the fundamental quantities \(p,z,H;\ (\Delta \ne 0)\), we obtain:

\[ P = p\sqrt[3]{\left(\frac{H}{z}\right)^2} \cdot \Phi\left(\frac{n}{z}\right). \]

Finally, let us also consider the question of the energy of the blow on the bit

\[ A_b=\Phi(z,H,p,Q_b,N,n,M_k,P). \]

Taking as the fundamental quantities \(z,H,p;\ (\Delta \ne 0)\),

we obtain:

\[ A_{\sigma} = \frac{pH}{z}\cdot \Phi\left(\frac{n}{z},\ \frac{P}{Q},\ \frac{N}{pH},\ \frac{M_{\mathrm{kr}}}{\frac{pH}{z}}\right). \]

The basis of the theory of dimensions and the theory of similarity is the socio-historical productive practice of humankind as a whole; the relations established on the basis of the theory of dimensions are only an approximation (often a preliminary stage of research), and at the same time the relations established on the basis of the theory of dimensions, in the form of generalized regularities, make it possible to identify the most general and essential: the general structure of formulas, indicating the direction of experimental investigations. The formulation and assignment of an experiment on the basis of the theory of dimensions and similarity is simplified and facilitated when there is a functional relation between entire complexes of quantities that determine the phenomenon, since in a number of cases there is no need to study the influence on the process of each factor separately and, in addition, it becomes possible to extend, within known limits, the results of experiments to a series of similar systems.

1. The task of science is the study not only of the relations between objects of the real world, but chiefly the study of the properties of the objects themselves. Along with establishing the generalized relations given by the theory of dimension (which are neither final nor unique), it is necessary to study more deeply the specificity of each phenomenon, each process, under concrete conditions—otherwise form may become detached from content (and the danger of idealism may arise). Experiment is one of the cornerstones of knowledge, a method of research by means of which the validity of hypotheses, conjectures, and models is tested and the values of a number of coefficients and indices are established.

Having the correct initial premises, it is important to know all the quantities characterizing the process of rock destruction under definite conditions, as well as the values of a number of parameters and coefficients (the role of mass, strain rate, operating regime of the given machine, properties of the medium being destroyed: hardness, coefficient of friction, etc., which for rocks vary over a very wide range).

Along with the task of creating a reliable and more effective tool for drilling, it is necessary to investigate new drilling regimes (the creation of more advanced drilling machines) and the application of a number of factors that promote the destruction of rocks (hardness reducers, removal of drilling fines from the hole, etc.).

A more subtle experiment is required in studying the mechanism of rock destruction, which cannot be represented simplistically, statically.

It is obvious that the problem of rock destruction cannot be solved in isolation from other fields of science and technology.

Only a combination of theory and practice will make it possible to eliminate uncertainty in work and that naked empiricism which, unfortunately, still has a place in the field of the theory of rock destruction.

References

1. A. I. Medvedko. The Drilling Formula. Mining Journal, Nos. 11–12, 1946.
2. V. S. Fedorov. Bits for Oil Drilling. Azgosnefteizdat, Baku, 1941.
3. E. F. Epstein. The Theory of Drilling-Cutting of Rocks by Hard Alloys. GONTI, 1939.
4. L. A. Schreiner. Physical Foundations of the Mechanics of Rocks. Gostoptekhizdat, Moscow–Leningrad, 1950.

1. P. A. Rebinder, L. A. Schreiner, K. F. Zhigach. Hardness reducers in drilling. Publishing House of the Academy of Sciences of the USSR, Moscow, 1944.

2. V. K. Buchnev. Drilling and blasting operations. Ugletekhizdat, Moscow, 1950.

3. I. S. Pokrovskii. Theory of percussion drilling. Mining Journal, No. 12, 1949.

4. A. I. Ger-Grigoryan. Bits for rotary drilling. Aznefteizdat, Baku—Leningrad, 1950.

5. V. D. Kuznetsov. Physics of the Solid State, vol. V, Tomsk, 1949.

6. P. M. Tsimbarevich. Mechanics of Rocks. Ugletekhizdat, Moscow, 1948.

7. V. L. Karpichev. Conversations on Mechanics. GITTL, Moscow, 1950.

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