SOME ISSUES IN THE THEORY OF VIBRATORY-ROTARY DRILLING WITH AN ANNULAR FACE

V. M. MATROSOV

(Presented to the scientific seminar of the Departments of Exploration Technology and Mining Machines and Mine Transport)

The great effectiveness of vibratory mechanisms used for the destruction of rocks is attracting ever greater attention to them. A number of research and production organizations are carrying out extensive work on introducing the vibration method into mining and drilling practice.

However, further improvement of the vibration method is being hindered by the absence of sufficiently coherent theories on the fundamentals of rock destruction by this method. Despite long experience in the operation of vibratory mechanisms, there is still no theory for calculating the principal design parameters of vibratory installations; nor have the optimal relationships been determined among the parameters of the vibratory-rotary drilling regime (disturbing force, oscillation frequency, axial load, and peripheral speed of the bit), which would ensure the best performance indicators for the use of vibratory mechanisms from the technical and economic standpoint.

The present article is devoted to solving one particular problem in the theory of the vibration method, relating to the mechanism of rock destruction in vibratory-rotary drilling with an annular face. On the basis of the adopted scheme of rock destruction at the face and of experimental investigations carried out, the optimal relationships are established among the principal parameters of the vibratory-rotary drilling regime with an annular face in hard rocks.

The experimental part was carried out in the drilling laboratory of the “Exploration Technology” Department of the Tomsk Polytechnic Institute on a vibratory-rotary installation specially assembled for this purpose.

On the mechanism of rock destruction in vibratory-rotary drilling

The high efficiency of mechanisms to which vibrations are imparted is due to the following. The action of vibratory mechanisms on an object is impulsive in character, and the working process is obtained as the result of the cumulative effect of a large number of individual cycles following one after another. Although in one cycle a very small amount of work is performed, on the whole a high production—

... a substantial effect owing to the high frequency of impulses per unit time.

It should also be noted that the action of the drill bit on the bottomhole during each impulse, with comparatively small displacements, occurs at considerable velocity and with large accelerations, which is of great importance in the destruction of rock by impact.

The phenomena that occur in the interaction of a vibrating drill bit with rock are very complex and have so far been studied only in their most general features. The phenomena arising when drill bits are sunk by the vibratory method into soft soils have been investigated comparatively more fully [2].

The mechanism of destruction in vibratory-rotary drilling in hard rocks has at present been studied far from sufficiently. One can only state with confidence that it has much in common with bottomhole processes occurring when wells are drilled with a solid bottomhole by the rotary-percussive method and with roller bits. In all these cases, destruction of the rock at the bottomhole occurs under the action of dynamic loads.

A complete analogy, however, cannot be drawn when comparing the action on the bottomhole by the indicated methods with vibratory-rotary action, since in the former case the drilling tool is constantly pressed against the bottomhole under the action of the axial load or feed force, whereas in vibratory-rotary drilling the drilling tool delivers blows while separating from the bottomhole. Separation of the bit or crown from the bottomhole is a characteristic feature of the vibratory-rotary drilling method, exerting a decisive influence on the nature of the interaction of the drill bit with the rock.

In rotary-percussive drilling, the principal motion of the tool, which provides the maximum work of bottomhole destruction, is the rotary motion [1].

In percussion-rotary drilling, destruction of the rock at the bottomhole occurs under the action of an impact load; in this case the cutter penetrates into the rock vertically, at an angle of 90 degrees to the bottomhole surface.

The mechanism of rock destruction at the bottomhole in vibratory-rotary drilling with an annular bottomhole may be represented as follows (Fig. 1). As is clear from the diagram, crown 1, together with the cutter, simultaneously participates in two motions: a rotary motion with constant circumferential velocity and an axial motion with a vertical velocity that varies in magnitude and sign. As a result of this, the motion of the cutter takes place along a complex trajectory 2, which in the simplest case approaches a sinusoid. The presence of the natural vibrations of the drilling tool, as well as the elastic properties of the rock, distort the harmonic motion of the cutter to one degree or another. Nevertheless, when the cutter meets rock 3, the phenomenon of oblique impact takes place: depending on the ratio of the circumferential and vertical velocities, the cutter

Fig. 1. Diagram of rock destruction at the bottomhole. 1—crown, 2—cutter trajectory, 3—rock, \(w\)—circumferential velocity, \(v\)—vertical velocity, \(A\)—amplitude of oscillations.

penetrates into the rock at one or another angle to the surface of the bottom hole.

As was established earlier [3], the volume of rock chipped off in one blow depends on the angle at which the cutter meets the rock and reaches a maximum at the optimal angle of encounter “$\rho_0$,” which is complementary to the angle of friction between the cutter and the rock, “$\varphi$,” i.e. $\rho_0 + \varphi = 90^\circ$.

From an analysis of the adopted scheme of rock destruction in vibratory-rotary drilling with an annular bottom hole, the following conclusions follow:

1) at the optimal ratio of the circumferential and vertical velocities of the cutter, the drilling rate should be at a maximum;

2) the power taken from the shaft of the rotator should be less in vibratory-rotary drilling than in static drilling, which is due to the phenomenon of separation of the core bit from the bottom hole and to the reduction of total friction.

The stated propositions can be verified only experimentally, for which special experiments were conducted.

Methodology of the Experimental Investigations

Experiments on vibratory-rotary drilling in hard rocks with an annular bottom hole were carried out on a special installation having the following arrangement (Fig. 2).

Fig. 2. Vibratory-rotary installation for experimental drilling. 1 — drilling tool, 2 — rotor, 3 — oil seal, 4 — vibrator, 5 — load, 6 — gearbox, 7 — winch, 8 — flexible shaft, 9 — frame, 10 — alternating-current electric motor, 11 — direct-current electric motor.

From electric motor 10, through gearbox 6 and rotor 2, rotation is transmitted to drilling tool 1, consisting of a driving rod, core barrel, and bit. At the upper end of the driving rod is screwed—

a stuffing box 3 is screwed on, which is connected by means of a rubber hose to the flushing pump. On the stuffing box, by means of conical roller bearings, vibrator 4 is mounted; it remains stationary while the tool rotates. On the upper cover of the vibrator are placed weights 5, rigidly connected to the vibrator by bolts. The vibrator is set in motion through flexible shaft 8 from direct-current electric motor 11.

The rotor of the installation had three rotational speeds: 38, 75, and 148 rpm. Vibrational oscillations were transmitted to the drill string by a two-block, directed-action vibrator of P. F. Pal’yanov’s design, with an eccentric moment of \(25\ \text{kg}\cdot\text{cm}\). The number of revolutions of the vibrator was regulated by means of a rheostat connected into the circuit of the direct-current electric motor. The limits of regulation lay from 500–700 to 1300–1400 revolutions per minute.

Experimental drilling was carried out in blocks of granodiorite, belonging to category IX in drillability by the core method, granite—category X, and marble—category V. The granite and granodiorite were drilled by the shot method; the marble, by means of bits equipped with octagonal cutters made of hard alloy. In shot drilling, steel shot—chaff—was used. The diameters of the bits were 65 mm (outer) and 52 mm (inner).

Flushing was carried out by a “45/15” pump at a capacity of 6–15 l/min; the flushing fluid was water.

The power consumed by the alternating-current electric motor from the mains was measured by means of two wattmeters connected into the circuit according to Aron’s scheme. The power taken from the shaft of the rotator was determined from the wattmeter readings, with allowance for the load characteristic of the motor.

Experimental data

The first series of experiments was conducted in order to determine the nature of the change in drilling speed as a function of the vibration frequency of the vibrator. The results of the experimental observations are given in Table 1.

Table 1

On the basis of the data obtained, graphs were constructed showing the influence of vibrations on the drilling speed at different peripheral speeds of the bit (Fig. 3). The peripheral speeds of the bit corresponding to the rotor speeds were: \(W_1=0.12\ \text{m/sec}\), \(W_2=0.23\ \text{m/sec}\), and \(W_3=0.45\ \text{m/sec}\).

It follows from the graph that the maxima of drilling speed are observed when drilling granite and granodiorite at rotor speed I, and marble at speed II.

In order to establish the dependence of drilling speed on the angle of incidence of the cutters with the face surface, the graph given below should be plotted in new coordinates: “penetration rate—angle of incidence.” The angle of incidence was calculated by the formula

\[ \tan \alpha = \frac{v}{W}, \tag{1} \]

where \(v\) is the vertical velocity of the bit at the moment of incidence with the face surface.

Fig. 3. Graphs of the effect of vibrations on drilling speed.
1—drilling in granite, 2—drilling in granodiorite, 3—drilling in marble.
a) rotor 38 rpm, b) rotor 75 rpm, c) rotor 148 rpm; \(U\)—drilling speed without vibrations, \(U_v\)—with vibrations.

The vertical velocity of the bit was calculated by the formulas given in the next chapter, and was also determined from the recorded vibrograms of the oscillations of the drilling assembly.

From the graph in Fig. 4 it follows that the maximum penetration rate corresponds to the following optimal angles of incidence: for granite—84°; for granodiorite—83°; for marble—78°. After the angle of incidence exceeds the optimum, the drilling speed begins to fall to a certain limit. With a subsequent increase in the impulse along the abso-

…absolute value, and also with an increase in the number of impulses per unit time, the drilling rate begins to increase again. At high rotational speeds, for the given parameters of the vibrator, the angle of engagement does not reach the optimum.

Fig. 4. Graph of the dependence of the drilling rate on the angle of engagement.

The next series of experiments pursued the aim of determining the power consumption when vibrations are imposed on the drilling tool. Measurements of the power taken from the motor shaft during static drilling are given in Table 2.

Graphs of the variation of power consumption as a function of the frequency of oscillations of the drilling assembly when drilling in various rocks are pre—

Table 2

shown in Fig. 5. As follows from these graphs, the power taken from the motor shaft decreases when vibrations are imposed on the drilling tool, and more intensively so in hard rocks.

Fig. 5. Graphs of the dependence of power consumption on oscillation frequency: a) drilling in marble, \(Q = 240\) kg; b) drilling in granite, \(Q = 120\) kg; c) drilling in granite, \(Q = 240\) kg. 1—38 rpm of the rotor, 2—75 rpm of the rotor, 3—148 rpm of the rotor. \(N\) — power consumption with vibrations. \(N'\) — power consumption without vibrations.

Formula Relating the Main Parameters of the Optimal Regime of Vibratory-Rotary Drilling with an Annular Face

In light of the above conception of the mechanism of rock destruction in drilling by the vibratory-rotary method with an annular face in hard rocks, it is possible to derive a general formula relating the main parameters of the drilling technology. These parameters are as follows:

\(P_{\max}\) — maximum exciting force of the vibrator, kg;

\(Q\) — axial load on the face, kg;

\(n\) — oscillation frequency of the vibrator, rpm;

\(W\) — peripheral speed of the bit, m/sec.

The manifestation of the vibration effect begins from the moment when the maximum exciting force of the vibrator \(P_{\max}\) exceeds the value of the axial load \(Q\). Under the action of the exciting force, the drilling assembly enters into oscillations with separation from the face and with transmission of the dynamic impulse to the face by the entire oscillating mass. In this case, from some-

As a first approximation, we assume that the bit is rigidly connected with the vibrator and performs oscillations of the same amplitude as the vibrator. In the case of drilling shallow wells with surface vibrators, and also when drilling deep wells with downhole vibrators, the above assumption will not be of substantial significance.

Fig. 6. Diagram of the acting forces.

The diagram of the acting forces will be represented by a sinusoid shifted downward by the magnitude of the axial load \(Q\) (Fig. 6).

To calculate the dynamic impulse transmitted by the bit to the bottomhole, we shall use the equation of this sinusoid and, by integration, determine the area bounded by the sinusoid and the time axis (shaded in Fig. 6). The impulse will be numerically equal to this area:

\[ y = P_{\max}\cdot \sin(\omega\cdot t) - Q. \]

\[ P\,t = P_{\max}\int_{t_2}^{t_3}\sin(\omega t)\,dt - Q\int_{t_2}^{t_3}dt \]

\[ = -\frac{P_{\max}}{\omega}\cos(\omega t_3) + \frac{P_{\max}}{\omega}\cos(\omega t_2) - Q(t_3-t_2). \]

Here

\[ \omega=\frac{\pi n}{30} \]

is the angular velocity of the eccentric, \(1/\mathrm{sec}\).

To determine the limits of integration, we set the sinusoid equation equal to zero and find the values \(t_2\) and \(t_3\):

\[ \sin(\omega t)=\frac{Q}{P_{\max}}; \]

\[ \omega t=\arcsin\frac{Q}{P_{\max}}; \]

\[ t_1=\frac{30}{\pi n}\arcsin\frac{Q}{P_{\max}}; \]

\[ t_2=\frac{30}{n}-t_1; \]

\[ t_3=\frac{60}{n}-t_1. \]

The values of \(t\) are substituted into the formula for determining the impulse

\[ P \cdot t = - \frac{30 \cdot P_{\max}}{\pi \cdot n} \cdot \cos \frac{\pi \cdot n}{60} \left( \frac{60}{n} + \frac{30}{\pi \cdot n}\arcsin \frac{Q}{P_{\max}} \right) + \]

\[ + \frac{30 \cdot P_{\max}}{\pi \cdot n} \cdot \cos \frac{\pi \cdot n}{30} \left( \frac{30}{n} - \frac{30}{\pi \cdot n}\arcsin \frac{Q}{P_{\max}} \right) - \]

\[ - Q \cdot \left( \frac{60}{n} + \frac{30}{\pi \cdot n}\arcsin \frac{Q}{P_{\max}} \cdot \frac{30}{n} + \frac{30}{\pi \cdot n}\arcsin \frac{Q}{P_{\max}} \right). \]

After the corresponding transformations, the impulse in absolute magnitude will be equal to

\[ P \cdot t = \frac{60 P_{\max}}{\pi n} \sqrt{1 - \frac{Q^2}{P_{\max}^2}} - \frac{30 \cdot Q}{n} \left( 1 + \frac{2}{\pi}\arcsin \frac{Q}{P_{\max}} \right) \tag{2} \]

Let us denote

\[ \frac{1}{\pi}\arcsin \frac{Q}{P_{\max}} = K. \]

Then we obtain

\[ P \cdot t = \frac{30}{n} \left[ 0.64 \sqrt{P_{\max}^2 - Q^2} - Q \cdot (1 + 2K) \right]. \tag{3} \]

Under the optimum regime of vibratory-rotary drilling, equality (3) must be satisfied

\[ P \cdot t = \frac{Q}{g} W \cdot \tg \rho_0. \tag{4} \]

Whence it follows that

\[ W = \frac{P \cdot t \cdot g}{Q \cdot \tg \rho_0}; \]

Substituting into this expression the value of the impulse from formula 3,

\[ W = \frac{294.3}{n \cdot \tg \rho_0} \left[ 0.64 \sqrt{ \left(\frac{P_{\max}}{Q}\right)^2 - 1 } - (1 + 2K) \right]. \tag{5} \]

Using this formula, one can select and calculate the values of the leading parameters for optimum operation of the vibrating installation, depending on the specific drilling conditions.

For example, let us calculate what the circumferential speed of the bit was during experimental drilling in granite (Fig. 4). The parameters of the drilling regime corresponding to the maximum penetration rate were as follows: \(P_{\max} = 610\) kg, \(n = 1050\) rpm, \(Q = 120\) kg, \(\rho_0 = 84^\circ\) (granite)

\[ K = \frac{1}{3.14}\arcsin \frac{120}{610} = 0.06. \]

\[ W = \frac{294.3}{1050 \cdot \tg 84^\circ} \left[ 0.64 \sqrt{ \left(\frac{610}{120}\right)^2 - 1 } + (1 + 2 \cdot 0.06) \right] = \]

\[ = 0.0284(3.20 + 1.12) = 0.121 \text{ m/sec}. \]

In actual fact, drilling was carried out at a peripheral speed of the bit of 0.12 m/sec. Consequently, the difference between the actual and calculated speeds is insignificant.

On the basis of the results of the studies conducted, the following conclusions may be drawn.

1. The mechanism of rock destruction in vibratory-rotary drilling with an annular face has its own special features, distinguishing it from other methods of drilling. These differences are associated with the phenomenon of oblique impact of the bit cutters on the face.

2. The maxima of the rate of penetration observed in vibratory-rotary drilling are due to a favorable ratio of velocities (peripheral and vertical), which leads to an optimal angle of encounter between the cutters and the surface of the face.

3. Vibrations of the drilling tool reduce the power consumption taken from the shaft of the rotator, as compared with static drilling.

4. The formula proposed by the author can be used in designing drilling rigs of vibratory-rotary action and in selecting the optimal drilling regime.

REFERENCES

1. Alimov O. D., Lyapichev I. G., Serov Ya. A. Investigation of rotary-impact drilling. Izvestiya TPI, vol. 106, 1958.

2. Barkan D. D. Vibrations in Construction. Gosstroiizdat, 1959.

3. Molchanov V. I., Matrosov V. M. On the selection of initial parameters of drilling machines of vibratory-rotary action. Journal Razvedka i okhrana nedr, No. 2, 1959.