Methodology for Selecting Optimal Drilling Conditions

V. I. Zvarygin, S. S. Sulakshin

(Presented by the scientific seminar of the Department of Exploration Technology)

Increasing the mechanical drilling speed of boreholes depends on a number of technological and geological factors, the principal ones of which, besides the properties of the rocks, are the axial pressure on the bit and the speed of its rotation.

Earlier, as a result of theoretical and experimental studies on the operation of a single cutter, we obtained a formula for cutting rock under bottom-hole conditions:

\[ h=\frac{P\cos\alpha}{4\sigma(b+l)}, \tag{1} \]

where

\[ \sigma=\frac{\sigma_{\text{comp}}}{6.8} \]

is the critical stress;

\(P\) — axial pressure on the cutter;
\(\alpha\) — angle of chipping of the rock element;
\(h\) — depth of penetration of the cutter;
\(b\) — width of the cutter;
\(l\) — a constant; for sandstones and siltstones \(l=4\).

The mechanical drilling speed in general form may be expressed by the dependence

\[ v_{\text{mech}}=f(h,m,n). \]

Here \(m\) is the number of cutters in the bit;
\(n\) is the number of revolutions of the bit per minute.

From experimental studies we have established that the mechanical drilling speed is inversely proportional to the square root of the number of cutters in the bit. In addition, when the rotational speed of the bit is increased from 1 to 200 rpm, the depth of penetration of the cutters decreases by a factor of 4. Moreover, the decrease in the depth of penetration of the cutters within these limits of bit revolutions is linear in character. Since at 200 rpm the decrease in penetration depth is \(3/4h\), then at \(n\) rpm (within the range 0–200 rpm) the depth of penetration of the cutters will be:

\[ h_n=h\left(1-\frac{3}{4}\cdot\frac{n}{200}\right) =h\left(1-\frac{n}{267}\right). \tag{2} \]

Or, substituting the value of \(h\), we obtain:

\[ h_n=\frac{P\cdot\cos\alpha}{4\sigma(b+l)}\left(1-\frac{n}{267}\right). \tag{3} \]

And then the mechanical drilling rate, taking into account the influence of the number of cutters for rotational speeds of the bits of \(1\text{--}200\ \mathrm{rev/min}\), will be

\[ v_{\mathrm{mech}}=\frac{P\cdot\cos\alpha\cdot n}{4\sigma(b+l)\sqrt[4]{m}}\left(1-\frac{n}{267}\right). \tag{4} \]

For \(n=200\):

\[ v_{\mathrm{mech}}=\frac{P\cdot\cos\alpha\cdot200}{4\sigma(b+l)4\sqrt{m}}. \tag{5} \]

As experimental studies show, at bit speeds above \(200\ \mathrm{rev/min}\) for poorly abrasive sandstones and aleurolites, the mechanical drilling rate is directly proportional to the square root of the number of revolutions.

\[ v_{\mathrm{mech}}=\frac{P\cos\alpha\sqrt{200n}}{4\sigma(b+l)4\sqrt{m}} =\frac{0.875P\cos\alpha\sqrt{n}}{\sigma(b+l)\sqrt{m}}. \tag{6} \]

In previous works it was established that, when cutting rock under bottom-hole conditions, \(F=0.5p\). Substituting this value into formula (6), we obtain:

\[ v_{\mathrm{mech}}=\frac{1.75F\cos\alpha\sqrt{n}}{\sigma(b+l)\sqrt{m}}. \tag{7} \]

Thus, the mechanical drilling rate is directly proportional to the axial pressure and to the square root of the number of revolutions of the bit. But increasing the axial pressure and the number of revolutions of the bit has a number of limitations.

First, an increase in the technological parameters \((P,n)\) causes increased energy consumption. The transfer of a large amount of energy directly to the bottom of the borehole is associated with certain difficulties, since there are large energy losses in useless work (idle rotation of the rods, friction of the drill string against the borehole walls, etc.), insufficiently high strength of the drill string, and others. Let us denote the power going directly to the destruction of the rock by \(N\). According to the formula of elementary physics:

\[ N=F\cdot v, \tag{8} \]

where \(F\) is the force acting on the body (in our case, the circumferential force);

\(v\) is the velocity of motion of the body (the rotational speed of the bit).

Substituting into this formula the values of \(v\) and \(F\),

\[ v=\frac{\pi d n}{60};\quad F=0.5P; \]

here \(d\) is the bit diameter in m,

we obtain

\[ N=\frac{0.5\pi Pdk}{60}=\frac{Pdn}{40}. \tag{9} \]

Comparing formulas (6) and (9), one may note that, from the standpoint of energy consumption, increasing the axial pressure is considerably more advantageous than increasing the revolutions of the bit. Thus, when the axial pressure is increased by a factor of 4, the mechanical drilling rate increases by a factor of 4, by 4 times

increases, as does energy consumption; whereas when the rotational speed is increased by a factor of 4, the mechanical drilling rate increases only by a factor of 2, while energy consumption increases by a factor of 4.

When selecting optimal technical drilling conditions, it is also necessary to take into account cutter wear, which leads to a reduction in the mechanical drilling rate. V. S. Fedorov [1] established that, with an increase in axial pressure during volumetric destruction of the rock, the specific wear of the tool decreases, while during surface abrasion it remains constant. The English researcher R. Shepherd [2] also established that, as the axial pressure on the bit increases, cutter wear decreases. “Minimum cutter wear,” concludes R. Shepherd, “occurs at a constant maximum load. Considerably greater cutter wear occurs under a load that gradually increases from zero to the maximum than under a load that decreases from the maximum.”

As the rotational speed of the tool increases, according to V. S. Fedorov, up to a certain value the wear is proportional to the number of revolutions; when the number of revolutions is increased above this value, the increase in wear proceeds faster than the increase in the number of revolutions. E. F. Epshtein, as a result of experiments he conducted [3], found that when the rotational speed of the bit increases, wear increases and then, after a certain decrease, the wear intensity again rises sharply.

Thus, from the standpoint of cutter wear as well, preference should be given to increasing the axial pressure.

On the other hand, an increase in axial pressure leads to an increase in circumferential force, and the latter is limited by the strength of the cutters in the bit. Consequently, the axial pressure must not exceed the critical value at which cutter breakage may occur.

The strength condition for the cutters can be written as the inequality:

\[ F < \frac{\sigma_{\text{iz}} \cdot a \cdot b \cdot m}{h_p} \tag{10} \]

or

\[ P < \frac{2\sigma_{\text{iz}} \cdot a \cdot b \cdot m}{h_p}, \tag{10} \]

where
\(\sigma_{\text{iz}}\) is the ultimate bending strength of the cutter;
\(b\) is the cutter width;
\(a\) is the cutter thickness;
\(m\) is the number of cutters;
\(h_p\) is the projection of the cutters from the body of the bit.

It is obvious that an increase in the strength of the cutters of the bit can be achieved in the following way:

1. Improve the quality of the hard alloy (increase bending strength).
2. Increase the cross section of the cutters: their width and thickness.
3. Increase the number of cutters in the bit.

However, increasing the width and number of cutters leads to a reduction in the mechanical drilling rate.

From a comparison of formulas (10) and (6), it is evident that the increase in cutter strength when \(m\) and \(b\) are increased occurs faster than the decrease in the mechanical drilling rate; and when the strength is increased by a factor of \(k\), the axial pressure can also be increased by a factor of \(k\), which accordingly leads to an increase in the mechanical drilling rate by a factor of \(k\). Consequently, in the case of a sufficiently strong drill-rod column (new equipment set), with which considerable axial loads can be applied, it is rational to use bits with a large number of cutters or bits with wide cut-

Thus, for example, if we increase the number of cutters in the bit by a factor of 4 and correspondingly increase the axial pressure, the mechanical drilling rate will increase due to the increase in \(P\) by a factor of 4 and will decrease due to the increase in the number of cutters by a factor of 2, i.e., in total the mechanical drilling rate increases by a factor of 2.

If the cutter width is increased from 2 to 8 mm, the mechanical drilling rate will increase owing to the possible fourfold increase in axial pressure and will decrease owing to the twofold increase in cutter width; in total, the mechanical drilling rate will increase by a factor of 2.

Thus, at shallow depths and with a small borehole diameter, and also when working with a strong drill string or a string with protectors, when significant axial loads can be applied, bits should be selected with a large number of cutters or with cutters of increased width. In cases where the conditions listed above are absent and the most dangerous factor is the strength of the drill-rod string, and axial pressures are produced only in small magnitudes, bits should be selected with a smaller number of cutters and with cutters of small width.

When the width and number of cutters are increased, their wear will decrease and, consequently, the drop in the mechanical drilling rate with time will be correspondingly smaller, and the length of the run will increase. All this will ultimately lead to an increase in drilling productivity. The wear factor should also be taken into account when selecting the minimum value of the number of cutters in the bit.

A reduction in the protrusion of the cutters from the body of the bit may also affect the mechanical drilling rate, since the drilled cuttings will be in confined conditions and therefore part of the energy will be expended on secondary crushing of the rock. Therefore, simultaneously with decreasing the protrusion of the cutter from the bit body, the front face of the cutter should be exposed to a greater degree by means of a special cutout.

Such a cutout, as we have already considered in previous articles, will eliminate the possibility of the formation of a buildup of compacted rock arising during drilling as a result of the limited space in front of the cutter between the end face of the bit and the bottom of the borehole. In addition, the presence of cutouts in front of the cutter is favorable from the standpoint of removing drilled particles. Indeed, the flow of flushing fluid will be directed into these cutouts ahead of the cutters and will carry the sludge directly out from under the cutters.

Increasing the rotational speed of the bit, as we saw from the experimental results, does not lead to an increase in the circumferential force. Therefore, from the standpoint of cutter strength there are no restrictions on increasing the revolutions.

Thus, the axial pressure should preferably be increased up to \(P_{\text{cr}}\), above which cutter breakage is possible. Thereafter, to increase the mechanical drilling rate, the rotational speed of the bit should be increased until the full capacity of the rig is used (the power that can be delivered to the bottom of the borehole).

But when increasing the technological drilling parameters, one should also take into account the strength and working conditions of the drill string.

Increasing the axial pressure leads to greater bending of the tool: the force pressing the drill string against the walls of the borehole increases; consequently, the friction and wear of the rods increase. When the number of revolutions is increased, the centrifugal forces pressing the tool against the walls of the borehole increase, which also leads to wear of the rods, large energy losses, and a reduction in the axial pressure on the bit; the latter leads to a reduction in the mechanical drilling rate.

Finally, at high rotational speeds of the drill string, owing to the presence of a long drive shaft at relatively large borehole diameters, strong vibration of the rig and of the core barrel arises. Strong vibration of the rig may lead to its failure; therefore operation at high speeds under the existing drilling conditions is practically impossible. Vibration of the core barrel leads to chipping of the cutter edges in the bit and to destruction of the drilled core column. As a result of the presence of crushed rock and of the flow of flushing fluid directed into the clearance between the bit and the core column, self-jamming of the core occurs. In this case, the greater the axial pressure, the stronger the wedging of the core, and the lower the mechanical drilling rate. To reduce self-jamming of the core, it is evidently necessary to eliminate vibration of the core barrel and to use reverse flushing, or to install a pump that draws the flushing fluid, together with the sludge and large pieces of rock, out of the core barrel.

The result of bending of the drill string, its friction against the walls of the borehole, vibration of the core barrel, which contributes to chipping of the cutters and to self-jamming of the core at elevated technological drilling parameters, etc., is, as observations by researchers [4] show, the fact that in core drilling, with an increase in the axial pressure and rotational speed of the string after the mechanical rate has increased up to \(v_{\max}\), it then decreases. This fact substantially limits the increase in the mechanical drilling rate; therefore it should also be taken into account when selecting optimal technological drilling regimes.

Thus, a major obstacle to increasing the mechanical drilling rate is the presence of a long drive shaft; therefore one should strive either to reduce the energy losses due to friction of the string against the borehole walls, or to abandon the rod method altogether and switch to a rodless method of drilling.

One of the ways to reduce friction of the drill-pipe string is to use protectors of a diameter corresponding to the borehole diameter, mounted on the rods on bearings.

In the course of drilling, as drilling time passes, the cutters wear, and as a result the depth of penetration of the cutters decreases. The circumferential force also decreases; therefore, to increase the mechanical drilling rate, the axial pressure should be increased up to \(F_{\mathrm{cr}}\), which is favored by a reduced circumferential force.

It is necessary that, throughout the entire drilling process, the circumferential force not exceed \(F_{\mathrm{cr}}\) (the circumferential force above which cutter failure is possible); on the other hand, in order to obtain the maximum value of the mechanical drilling rate, the circumferential force should be maximal, which depends on the strength of the bit cutters \((F_{\mathrm{cr}})\), i.e., it is necessary that \(F_{\max} = F_{\mathrm{cr}}\) during drilling be constant and at the maximum permissible value. Maintaining a constant value of \(F\) during drilling is an impossible task for the shift driller; therefore it is necessary to develop the design of an automatic regulator capable of responding to changes in \(F\) due to blunting of the cutters or to any other causes.

Apparently, one must proceed from the constancy, for the given conditions, of the rotational speed of the string and of the power expended on the drilling process, i.e.,

\[ F = \frac{N_{\mathrm{const}}}{v_{\mathrm{const}}} = \mathrm{const}. \]

If one has a continuously variable transmission, which is achieved by using oil to transmit energy, then, at constant values, in the course of drilling, as the cutters wear, \(F\) will decrease; but since the oil pressure in the hydraulic system will remain constant, owing to the reduction in peripheral forces, the greater part of the oil will be supplied to create axial pressure. The pressure will increase until \(F\) assumes its initial value, i.e., until the equality is restored:

\[ N = Fv. \]

Thus, by means of a continuously variable transmission, automatic regulation of the technological drilling conditions can be achieved.

From the foregoing, the following conclusions may be drawn:

1. To increase the mechanical drilling rate, one should strive to raise, up to a certain limit, the parameters \(P\) and \(n\), with preferential increase of \(P\).

2. Given a sufficiently strong drilling assembly, in order to make it possible to increase the strength of the cutters and thereby raise the axial pressures, bits of increased strength should be used (with a larger number of cutters or with cutters of greater width).

3. To eliminate the factor limiting the increase in drilling rate and to reduce useless work, it is necessary either to eliminate bending of the rods and their friction against the borehole walls by means of special protectors, or to abandon the rod method of drilling altogether.

4. To reduce self-wedging of the core in the core barrel during drilling, either reverse circulation should be used, or a suction pump should be installed above the core barrel.

5. For automatic regulation of drilling conditions and continuously variable transmission of energy to the bit, a drilling machine with a hydraulic drive should be developed.

REFERENCES

1. V. S. Fedorov. Scientific Foundations of Drilling Conditions. Gostoptekhizdat, 1951.

2. R. Scheffer. Physical Properties and Drillability of Rocks. Ugletekhizdat, 1956.

3. E. F. Epstein. Wear of Hard Alloys in Friction against Rocks during Core Drilling. Gostoptekhizdat, 1952.

4. A. S. Stanishevsky, A. P. Rudenko, Yu. V. Baklanov, G. M. Zakharov, O. V. Ivanov, I. S. Lyubovich, A. N. Yagodin. Core Drilling Conditions. ONTI, VITR, issue 27, 1960.