Drill Collar Length is a Major Factor in Vibration Control Don W. Dareing, * SPE

Summary Drill collar length directly affects the overall vibration response of drillstrings. Drill collar length is partly responsible for severe vibrations in hard rock drilling but can also be the solution to vibration control. This paper gives a new interpretation to the cause and control of drillstring vibrations and presents the results in terms of formulas that can be directly applied by the drilling engineer.

Introduction It is standard practice to design the length of drill collars or bottomhole assemblies (BRA's) so that the neutral point is located in the collars. The neutral point is the point where compressive stress is equal to local hydrostatic pressure. The calculation is based on static forces only, including buoyant weight of the collars and static bit weight. One formula commonly used to calculate drill collar length is

WB L= ...... " ...... , .......... (1)

0.85 Wa FB

According to this formula, the distance to the neutral point is 85 % of the total drill collar length, allowing for a margin of bit weight overload. Righer bit weights re­quire longer BRA's.

Natural frequencies and inertia loading in BRA's are not considered in this calculation. As a result, the present practice of calculating drill collar length often leads to natural tuning of the collars with bit displacement fre­quencies. This means that many drill collars are uninten­tionally designed to vibrate, and collar length selection, based on statics alone, may account for rough running.

Drill collars can vibrate in three modes: (1) axial or longitudinal, (2) torsional, and (3) transverse or lateral. Because the collars are confined by the wellbore, lateral vibrations are not usually a major source of stress and are not covered in this paper. Drill collars are free to move axially and torsionally, and these two modes of vibration can become severe. Kelly bounce and whipping of the drawworks cables indicate axial vibrations in drill­strings. Torsional vibrations are normally not seen from the rig floor because the rotary table drive tends to "fix" the vibrational angular motion at the surface. Nonetheless, large dynamic torque can be generated at the rotary table.

As in any mechanical system, severe vibrations in • Now with Norton Christensen Drilling Products.

0149-2136/84/0041-1228$00.25 Copyright 1984 Society of Petroleum Engineers of AIME

APRIL 1984

drillstrings are the result of resonance or frequency tun­ing. Resonance exists when the frequency of the applied force is equal to a natural free vibration frequency. The drill collar section, however, controls the overall vibra­tion response because its cross-sectional area is several times the cross-sectional area of drillpipe. The collars act as receivers and amplifiers of vibration energy from the drill bit. In one sense the drill collar section is the dog wagging the tail, which in this case is the drillpipe sec­tion. This observation, supported by calculations and field data, is explained further in the paper.

Assumptions made in the analysis are as follows. 1. The BRA is a constant-OD and -ID drill collar

section. 2. The drill bit is a roller cone rock bit. 3. The formation is medium to hard. 4. The natural frequency of damped free vibration of

the BRA is not significantly different from its natural frequency of undamped free vibration.

5. Axial and torsional stiffness of stabilizers do not significantly alter the natural modes and vibration.

6. Role inclination and curvature do not affect natural frequency of BRA's.

One goal of the paper is to give alternative vibration control techniques for alleviating rough running. Shock absorbers are proved alternatives. Rough' running can also be alleviated by adjustments in BRA design. A third alternative is rotary speed selection, based on techniques given in the paper.

Natural Frequency of Drill Collar Assembly BRA's are often made up of different sizes of drill col­lars, stabilizers, and downhole tools. In general, critical rotary speed should be based on the natural frequency of the composite BRA. For simplicity, the following discussion assumes the drill collar section has a uniform cross section from the bit to the collar/drillpipe interface and contains no downhole tools. It can be shown that the natural frequencies of BRA's made up of different collar sizes can be reasonably approximated by assuming uniform drill collars. An exception to this simplification is heavy drillpipe in tandem with drill collars. The natural frequencies of nonuniform BRA's, however, can be calculated from classical vibration equations.

In calculating natural frequencies for both axial and torsional modes, the drill collars are assumed fixed at the drillbit and free at the collar/drillpipe interface. The free constraint at the top of the collar section is based on relatively low dynamic force (or torque) applied to the top of the collars by drillpipe. This low dynamic force is the result of a small area ratio between drill pipe and drill

637

DYNAMIC AXIAL DISPLACEMENT

4212 f"'=-L- CPS

Fig. 1-Drill collar vibration mode.

collars. The shape of the fundamental vibration mode is shown in Fig. I along with corresponding dynamic force distribution. Torsional and longitudinal modes have the same shape.

The derivation of natural frequency equations for vibrating bars and shafts is well documented and is not repeated here. I The solution to equations of motion for a vibrating bar gives the following equation for natural fre­quencies of longitudinal modes.

irE Ina=---..j-, i=l, 3,5 .................... (2)

4L p

or

iVa Ina = 4L' ................................ (3)

Since the speed of a compression (tension) wave in steel is 16,850 ft/ sec [5136 m/ s], the natural frequency of the fundamental drill collar longitudinal mode is

4,212 Ina =-L- cycles/sec ...................... (4)

Similarly, the natural frequency of the fundamental tor­sional mode is

Ino = :L ~ ............................. (5)

638

1.4

1.2

§ .... 1.0 )(

t: I .8

....I

~ .6 ....I

5 o :::l .4

~ .2

\ \ \ \ \ \ \ AXIAL MODE

\ \

/\ TORSIONAL "

MODE , , " " ........ ,

.......... --°O~--~2----~4----~6----~8-----1~O---­

NATURAL FREQUENCY, CPS

Fig. 2-Natural frequency VS. drill collar length.

or

Vo Ino= 4L' ................................ (6)

Since the speed of a shear wave in steel is 10,650 ft/sec [3246 m/s], the natural frequency of the fundamental drill collar torsional mode is

2,662 Ino = -L- cycles/sec. . .................... (7)

Note that both Ina and Ino depend only on drill collar length. Cross-sectional area is not a factor. According to Eq. 4 the natural longitudinal frequency of 700 ft [213 m] of drill collars is 6 cycles/sec [6 Hz]. According to Eq. 7 the natural torsional frequency of700 ft [213 m] of drill collars is 3.8 cycles/sec [3.8 Hz]. Eqs. 4 and 7 are plotted in Fig. 2 to show how the longitudinal and tor­sional modes vary with drill collar length.

Source of Excitation Possible sources or means of exciting the axial and tor­sional drill collar natural modes include time-varying longitudinal forces and torsional loads, which can be ap­plied to the drill collars from various sources, such as pump pressure, sidewall friction, and drill-bit/formation interaction. More studies are needed to identify operating conditions where each may dominate. In addi­tion, the various types of drill bits (roller cone, diamond, polycrystalline diamond compact) generate different loading conditions to the bottom end of the drill collars, and more field data and analysis are needed to determine

JOURNAL OF PETROLEUM TECHNOLOGY

Fig. 3-Three-lobed bottom hole core taken from a hard dolomitic limestone formation (courtesy of A. Scovil Murray).

how interactive dynamic forces between bit and forma­tion are generated.

Measurements of drillstring vibrations over the past 25 years show that drill-bit displacement frequencies are three cycles per bit revolution for three-cone bits. 2,3

This frequency has been consistently measured at the drill bit and at the kelly. The three-cycle-per-revolution frequency can sometimes be verified by counting the number of kelly bounces over a given period of time. This can be done especially at low rotary speeds such as 60 rev/min. The vertical vibration of the kelly is the result of a three-cone bit rolling over high and low spots in the formation. Cores taken from hard rock formations often show a three-lobed pattern associated with rough­running areas (Fig. 3). This type of three-lobed pattern will induce axial and torsional bit displacement frequen­cies of three cycles per revolution, which can be related to rotary speed by

3N N f= - = - cycles/sec. . .................... (8)

60 20

Total vertical distance from the low point to the high point is around 0.25 to 0.5 in. [0.6 to 1.0 cm]. 4

Laboratory drilling tests and bottomhole patterns made with large-diameter drill bits have sometimes generated multiple three-lobed patterns-for example, six lobes. How these bottomhole lobes are formed is not complete­ly understood, but they seem to be hammered out by natural vibrations of the drill collar section. Once they are formed, they develop even larger vibrations in the drill collar section and are self-sustaining.

Field measurements taken with a downhole recorder show the transition from a smooth-running drill bit to the rough-running or resonant-type drilling (Fig. 4). The measured bit weight record shows a random variation in bit weight magnitude up to a point where bit force became periodic and built to the most extreme dynamic level. This transition occurred over an interval of about 15 seconds. Bending in the collar section also became periodic with one cycle for each rotation of the collar section. These two measurements, bit weight and collar

APRIL 1984

l-:z: 0 W ;= l-iii

A

150

100

50

'0

..: .... 0· Z III 2

~~ 0 W;:) ·2 1110

A :z: l-

Fig. 4-Bit force transition from smooth to rough running (rotary speed is 90 rev/min). 2

bending, suggest that the collar section took a fixed posi­tion in the well bore and rotated about its own axis in­stead of wobbling about the axis of the wellbore. Under this condition the bit would tend to cut into the formation over a small area, cutting out a low spot and eventually developing three low spots and three high spots in the formation as the bit penetrated and vibrated into the formation.

During this particular field test, the natural longitudinal frequency of the drill collar section hap­pened to be tuned to 3 cycles/rev, and energy was fed in­to the collar section with the slightest amount of bit displacement, encouraging further dynamic bit forces that possibly created the three low spots. In other words, the excitation frequency, as determined by Eq. 8, was tuned to the natural fundamental frequency, Eq. 4, ofthe drill collar section. Resonance developed, accompanied by severe dynamic bit forces.

Drill Collar Resonance and Critical Speeds Drill collar resonance occurs when the frequency of the source of excitation is tuned to a natural frequency of either longitudinal or torsional modes. Axial and tor­sional modes can be excited separately or simultaneous­ly. The development of the three-lobed pattern can be at­tributed more to axial bit forces than to torsional bit loads, and for this reason drill collar resonance is proba­bly a result of axial mode tuning;

According to Eq. 4 the natural frequency (axial mode) of 700 ft [213 m] of collars is 6 cycles/sec [6 Hz]. A three-cone bit rolling over a three-lobed pattern can generate a 6-cycle/sec [6-Hz] exciting force frequency at a rotary speed of 120 rev/min (Eq. 8). In addition, 800 ft [244 m] of collars can be excited at a rotary speed of 105 rev/min. Critical rotary speeds for various collar lengths can be determined by combining Eqs. 4 and 8, giving

N 4,212 ---- .............................. (9) 20 L

or 84,240

Ncr = -L-- rev/min. . ................... (10)

639

iii 5 150

~ ~ 100 l­I-a 50 w !: !:: 0 co TIME

...: 101 SEC .1 u.

a:i ..J

iii => ~ 5 I-

Fig. 5-Dynamic bit forces and torque caused by drill collar resonance (rotational speed is 110 rev/min). 2

This equation shows that drill collar lengths commonly used in the field have critical rotary speeds which are also commonly selected for drilling. For example, ac­cording to Eq. 1, 765 ft [233 m] of 61h-in. [16.5-cm]-OD (95.9-lbf/ft [426.6-N/m]) drill collars would be carried in 13-lbm/gal [5.9 kg/m3] mud to generate 50,000 lbf [222 411N] bit weight. Ifthe drilling program calls for a rotary speed of 110 rev/min, the drill collars and entire drillstring have an excellent chance of running rough. An example of drill collar resonance is shown in Fig. 5.

Critical rotary speeds should be based on drill collar length and not total drill string length. Either Eq. 10 or Fig. 6 should be used to determine critical speeds for various drill collar lengths. This recommended method differs from the critical rotary speed formulas in API RP 7G, Sec. 9. 5

The first critical speed formula in API RP 7G is based on whirling-shaft technology. In this case, a joint of drill pipe is modeled as a whirling shaft, pinned at each end. The critical speed is determined by mUltiplying the natural frequency of lateral vibration by 60 to convert cycles per second to cycles per minute. Three basic assumptions are made in applying the whirling-shaft solution to drillpipe.

1. Tension is ignored. 2. Hole confinement and hole curvature are ignored. 3. There is no bending moment in either of the adja­

cent tool joints. The whirling-shaft model does not seem to represent drillpipe in a wellbore.

640

1.4

1.2

§ ... 1.0 )(

t-Il.

~ .S CJ Z w .... a:: .6 cc .... .... 0 ()

~ .4 ~ Q

.2

°0~--~2-----4~--~6~--~S----~10~---I NA~URAL F~EQUE~Y, CPS I

o 40 SO 120 160 200 CRITICAL ROTARY SPEED, RPM

Fig. 6-Critical rotary speeds.

The second equation for determining critical speeds, as given in API RP 7G, is based on treating the entire drill string as a pinned-free bar with the bottom end pinned and the top of the drill string completely free and unconstrained in the longitudinal direction. The model does not account for the drill collar section and assumes that the source of excitation is 1 cycle/rev of the drill­string. Measurements of rough-running drillstrings show that the frequency of the source of excitation is 3 cycles/rev instead of 1 cycle/rev. The second API RP 7G critical-speed formula ignores the importance of the BHA and in addition is off by a factor of three.

Drillstring Response To support further the importance of drill collar length in the critical speed calculation, consider a 6,800-ft [2073-m] drillstring containing 800 ft [244 m] of drill collars. The longitudinal vibration calculation for the en­tire drill string corresponds to the critical speed of 105 rev/min based on drill collar length (Eq. 10). Vibration response predictions are based on forced vibration response equations similar to the ones given in Ref. 4. Input conditions for this example calculation are given in Table 1.

Calculated displacements, stresses, and forces are given in Table 2. When axial movement of the drill bit is 0.25 in. [0.64 cm] (or 0.5-in. [1.27-cm] peak-to-peak total movement), kelly bounce or vertical movement is 1.1 in. [2.8 cm] (or 2.2 in. [5.6 cm] total vertical move­ment). The largest dynamic forces occur in the drill col­lar section, as expected, with the maximum dynamic

JOURNAL OF PETROLEUM TECHNOLOGY

TABLE 1-INPUT CONDITIONS FOR EXAMPLE CALCULATION

Drill collar length, ft (6.5 in., 95.9 Ibflft)

Drillpipe length, ft (4.5 in., 16.6 Ibf/ft)

Rotary speed, rev/min Bit displacement amplitude, in. Weight of swivel and traveling block, Ibm Spring constant of drawworks, Ibflin. Damping in drill collar,

Ibf/in.lsec/in. Damping in drillpipe section,

Ibf/in.lseclin.

800

6,000

105 0.25

20,000 50,000

0.1

0.01

force at the drill bit. The dynamic force magnitude at the top of the collars is 14,000 lbf [62 275 N] or less than 10% of the 183,200-lbf [814 914-N] dynamic force magnitude at the bit. Drillpipe constraint to the top of the collars is relatively small, which supports the simplified drill collar model of Fig. 1. Note also, the drill collar section has developed an approximate quarter wave mode displacement similar to the one shown in Fig. 1. Therefore, under a critical speed condition, the drill col­lar section essentially maintains its identity as a pinned­free bar. The drill collar section thus becomes the primary receiver of vibration energy from the drill bit.

Frequency domain diagrams (Fig. 7) also show how drill collars dominate overall drill string vibrations. The vertical axis in Fig. 7 is the ratio of the maximum displacement amplitude in the drillstring to bit displace­ment amplitude. The multiple peaks indicate large displacement in the drillpipe section or resonance of the overall drillstring. Note that at typical rotary speeds, high-order vibration modes are excited.

The envelope under this frequency response curve is significant and indicates drill collar response. The peak of the envelope occurs at a frequency equal to the natural frequency of the drill collars. This frequency can be determined independently from Eq. 4. Also important are the following.

1. The frequency band width for resonance of each drillstring vibration mode is very narrow. This requires careful rotary speed control and maintenance.

2. Resonance modes are not established instantaneous­ly. Time is required to feed energy into the entire drill­string to establish a vibration mode. Work done by cyclic bit forces must be converted into kinetic energy throughout the drillstring.

3. The envelope or frequency response of the collar section has a wide frequency band width. The drill collar section, therefore, is receptive to bit force frequencies over a wide range of rotary speeds.

4. Drill collars are relatively short; therefore, their vibration modes can be established quickly.

5. Drillpipe acts primarily as a very soft spring at the top of drill collars prior to and during vibration buildup. During this period, the top of the collar is essentially un­constrained by the drill pipe section.

To summarize, field measurements and computer calculations indicate critical speeds, and vibration con­trol techniques should be based completely on the fre­quency response of the drill collar section.

APRIL 1984

TABLE 2-DYNAMIC RESPONSE OF DRILLSTRING ROTATING AT CRITICAL

SPEED

Axial Distance Displacement Stress Force

(ft) (in.) (psi) (Ibf)

6,800 1.10 1,586 7,000 6,600 0.89 3,739 16,500 6,400 0.52 5,268 23,200 6,200 0.09 5,916 26,000 6,000 0.39 5,575 24,600 5,800 0.79 4,309 19,000 5,600 1.06 2,354 10,400 5,400 1.15 665 3,000 5,200 1.05 2,574 11,300 5,000 0.78 4,496 19,800 4,800 0.40 5,709 25,200 4,600 0.23 5,994 26,400 4,400 0.60 5,311 23,400 4,200 0.94 3,816 16,800 4,000 1.14 1,978 8,700 3,800 1.16 1,812 8,000 3,600 1.00 3,644 16,000 3,400 0.70 5,262 23,200 3,200 0.39 6,123 27,000 3,000 0.47 6,052 26,700 2,800 0.81 5,091 22,400 2,600 1.09 3,538 15,600 2,400 1.22 2,309 10,200 2,200 1.17 3,048 13,400 2,000 0.97 4,709 20,700 1,800 0.68 6,005 26,500 1,600 0.52 6,511 28,700 1,400 0.71 6,123 27,000 1,200 1.01 4,990 22,000 1,000 1.23 3,626 16,000

Drill Collars

800 1.29 500 14,000 600 1.19 2,590 73,200 400 0.92 4,627 130,800 200 0.52 5,994 169,500

0 0.25 6,480 183,200

Vibration Control Vibration control techniques can be used either to in­crease or to decrease vibration forces in drillstrings. However, vibration elimination is the main concern here because large dynamic forces are normally associated with drill string failures. As more is understood about op­timal dynamic force levels, it may be desirable to create specific dynamic bit forces to increase penetration rate without damaging drillstring components.

Four basic techniques for reducing the magnitude of mechanical vibrations are: (1) change natural frequency, (2) change forcing frequency, (3) increase or apply mechanical damping, and (4) eliminate source of excita­tion. This section gives operational alternatives, based on the control techniques, for alleviating rough running. The drill string given in the previous example (Table 2) will be used as a reference to evaluate the different alter­natives. Note that the numbers in Table 2 correspond to a critical rotary speed of 105 rev/min based on 800 ft [243 m] of drill collars. Dynamic bit force amplitude predicted for this condition is 183,200 lbf [824 400 N]. The dynamic force distribution along the 6,800-ft [2073-m] drill string is also listed in Table 3 under the Case 1 column.

641

10.0

0 j:

• ell:

"" Q ~ ~ ::; ~

~ • ~ ~

~ >< • ~

9.0

8.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

DRILL PIPE

RESPONSE

I \ I \ I \ I \ I \ I

I \ I \

/ , I ,

I ,

DRILL :+.-___ COLLAR

~ RESPONSE

o~-~-~-~-~~--~--~--~ o 1.0 2.0 3.0 4.0 5.0 6.0 7.0

BIT DISPLACEMENT FREQUENCY - CPS I I

o 20 40 60 80 100 120 140 ROTARY SPEED - RPM

Fig. 7-Frequency response of an 8,000-ft [2438-m] drillstring containing 800 ft [243 m] of collars. 4

The frequency response diagram in Fig. 8 shows how 800 ft [243 m] of drill collars responds to different axial drill bit displacement frequencies. Stripping out the multiple spikes representing drillpipe response reduces a complex model to a simpler one that is adequate for ex­plaining vibration control techniques. For rotary speeds up to about 60 rev/min, the drill collar section moves up and down as a lump mass with little relative movement between the bottom and top of the collar section. Be­tween 80 and 130 rev/min, the 800-ft [243-m] collar sec­tion responds with a large amount of stretching with the maximum amount of dynamic stretching occurring at the critical rotary speed of 105 rev/min. Beyond about 160 rev/min, collar stretching is relatively low until 320 rev/min, where the second axial mode of vibration becomes active. The first mode usually falls within the range of most rotary speeds. Large dynamic bit forces can develop between rotary speeds of 90 and 120 rev/min or speeds that excite the first mode and around 320 rev/min or speeds that excite the second mode.

Shock Absorber. One way to detune the natural fre­quency of the BHA from a drill bit excitation frequency is by using a shock absorber directly above the drill bit. A shock absorber lowers the natural frequency of the BHA and shifts the resonant peak to the left of drill bit excitation frequency. Fig_ 9 shows how the resonance peak is shifted away from the desired rotary speed range

642

• .0 '2 14 II FREQUENCY, CPS

I [ !

eo .20 .eo 200 210 320 ROTARY SPEED. RPM

Fig. 8-Frequency response of 800 ft [243 m] of drill collars.

I

I I H I I I

I FREaUE:CY. CPS

I I

'0 •• II

I I I I

40 eo .20 .eo 200 240 210 320

ROTARY SPEED.RPM

Fig. 9-Frequency response of 800 ft [243 m] of drill collars with shock absorber.

of90 to 120 rev/min. When a shock absorber is used, the drill collar vibrates (first mode) as a lumped mass on top of the spring (shock absorber). The natural frequency of the BHA in this case can be determined from

fna=-l g ........................... (11) 211" mt

For example, when a SO,OOO-lbf/in. [222 411-N/cm] shock absorber is placed directly above the bit in Case 1, the natural frequency of the BHA, according to Eq. 11, is f=2.52 cycles/sec [2.52 Hz]. The shock absorber reduces natural frequency from 5.25 cycles/sec [5.25 Hz] to 2.52 cycles/sec [2.52 Hz] and detunes the BHA from the 5.2S-cycles/sec [5.2S-Hz] excitation frequency.

The effect of shifting the resonance peak to the left of the 105 rev/min rotary speed is shown in Table 3. Case 1 refers to the 800-ft [243-m] drill collar length rotating at a critical speed of 105 rev/min. Case 2 refers to the 800-ft [243-m] drill collar section including a 50,OOO-lbf/in. [222 411-N/cm] shock absorber directly above the bit (Fig. 10); rotary speed is still 105 rev/min. Dynamic force levels throughout the drill string are con­siderably reduced as a result of frequency detuning with the shock absorber. Dynamic bit force amplitudes are reduced from 183,000 lbf [814024 N] (Case 1) to 12,300 lbf [54 713 N] (Case 2).

JOURNAL OF PETROLEUM TECHNOLOGY

TABLE 3-COMPARISON OF VIBRATION CONTROL METHODS (Ibf)

Distance (ft) Case 1 Case 2 Case 3 Case 4 -- -- --6,800 7,000 500 2,900 2,300 6,600 16,500 1,000 7,000 5,400 6,400 23,200 1,600 9,900 7,600 6,200 26,000 1,800 11,100 8,500 6,000 24,600 1,600 10,500 8,000 5,800 19,000 1,300 8,100 6,200 5,600 10,400 700 4,400 3,400 5,400 3,000 200 1,200 960 5,200 11,300 760 4,800 3,700 5,000 19,800 1,300 8,400 6,500 4,800 25,200 1,700 10,700 8,200 4,600 26,400 1,800 11,300 8,600 4,400 23,400 1,600 10,000 7,700 4,200 16,800 1,100 7,200 5,500 4,000 8,700 600 3,700 2,900 3,800 8,000 540 3,400 2,600 3,600 16,000 1,100 6,800 5,300 3,400 23,200 1,500 9,900 7,600 3,200 27,000 1,800 11,500 8,800 3,000 26,700 1,800 11,300 8,700 2,800 22,400 1,500 9,600 7,300 2,600 15,600 1,050 6,700 5,100 2,400 10,200 700 4,300 3,300 2,200 13,400 910 5,700 4,400 2,000 20,700 1,400 8,900 6,800 1,800 26,500 1,800 11,300 8,700 1,600 28,700 1,900 12,200 9,400 1,400 27,000 1,800 17,400 10,700 1,200 22,000 1,500 24,900 12,000 1,000 16,000 1,100 30,500 13,000

800 14,000 1,000 32,700 13,600 600 73,200 5,000 31,200 13,900 400 130,800 8,800 26,700 14,300 200 169,500 11,400 21,500 15,100

0 183,200 12,300 19,700 19,600

Drill Collars. Another way to lower natural frequency ofBHA's is to increase drill collarlength (Case 3, Table 3 and Fig. 10); a shock absorber is not used in this case. Long drill collar sections have lower natural frequencies than short drill collar sections.

According to Eq. 4, the length of drill collars that has a natural frequency of 2.52 cycles/sec [2.52 Hz] is about 1,700 ft [518 m]. In other words, 1,700 ft [518 m] of drill collars has the same natural frequency as 800 ft [243 m] of drill collars on top of a 50,000-lbf/in. [222 411-N] shock absorber. One would expect the dynamic force amplitude at the drill bit to be about the same order of magnitude in both cases. Computer calculations (Table 3, Case 3) show that a drill collar length of 1,700 ft [518 m] in a 6,800-ft [20n-m] string reduces dynamic bit force amplitude from 183,200 lbf [814914 N] to 19,700 lbf [87 629 N]. This is the same order of reduction as achieved with a shock absorber.

It may not be necessary to reduce the natural frequency of the drill collar section to the level previously in­dicated. Further studies are needed to evaluate the effect of more practical collar lengths on drillstring response.

Heavy Drill Pipe. Carrying an excessive number of drill collars is not desirable because of cost, extra handling, and considerable added weight to the rig. However, computer calculations (Table 3, Case 4) show that a BHA made up of 400 ft [121 m] of drill collars and 1,300

APRIL 1984

DRILL PIPE

HEAVY DRILL PIPE

DRILL COLLARS

DRILL SHOCK BIT ABSORBER

.,/

CASE I CASE n CASEm CASEN

Fig. 10-Vibration control alternatives.

ft [396 m] of heavy driilpipe (Fig. 10, Case 4) will ac­complish about the same reduction in dynamic bit force. In this case, dynamic bit force amplitudes are reduced from 183,200 lbf [814914 N] to 19,600 lbf [87 185 N].

This force reduction is the result of natural frequency reduction accomplished by drill collars and heavy drill­pipe having a combined length of 1,700 ft [518 m]. The economics of this vibration control alternative need to be evaluated.

Phase Selection. Phase selection may be another way to alleviate vibrations and at the same time increase penetration rate. There is a phase change (depending on damping) when the excitation frequency passes through resonance. On the low side of resonance, drill collar displacements are in phase with bit displacements-i.e., when the cones roll over a high spot in the formation, each point in the drill collars reaches peak displacements at the same instant the drill bit reaches its peak displace­ment. This means that instantaneous maximum dynamic bit loads impact on the low spots of a three-lobed bot­tomhole pattern.

On the high side of resonance, points in drill collars move downward when the bit moves upward or drill col­lar displacements are approximately 180 0 out of phase with bit displacements. This means that the maximum dynamic bit force impacts the high spots of a three-lobed bottomhole pattern.

643

In the first case (rotary speed on the low side of critical speed) vibrations are sustained. In the second case (rotary speed on the high side of critical speed) vibra­tions should be eliminated because the instantaneous maximum bit force always seeks to drill off the high spots. Usually rotary speed is reduced when drill string vibrations are severe. Increasing rotary speed beyond critical may eliminate the source of vibration complete­ly. This concept needs to be tested in the laboratory.

A companion paper explains how to determine critical speeds for integrated BHA's. 6

Conclusions Severe drill string vibrations are an indication of drill col­lar or BHA resonance. In areas where drill strings run rough, it would be worthwhile to calculate the natural frequencies of both axial and torsional modes in the col­lars (or BHA) and compare those frequencies with the 3-cycle/rev excitation frequency, assuming the bit is a three-cone bit. Adjustments in the design of the BHA may help reduce the vibrations.

For engineering calculations, the natural frequency and critical rotary speed of BHA's can be approximated by assuming that the top end of the BHA is uncon­strained by drillpipe. Heavyweight drillpipe should be included as part of the BHA.

The critical speed of a given BHA is useful, because it is a reference speed for judging how much the rotary speed should be increased or decreased to reduce rough drilling.

This study reinforces the shock absorber as an effec­tive vibration control tool. Field data and economic studies are needed to evaluate heavy drill pipe as a vibra­tion control tool.

Nomenclature

644

E = modulus of elasticity, lbf/sq ft [N/m2] I = frequency of source of excitation,

cycles/sec [Hz] Ina' InO = natural longitudinal and torsional fre­

quencies of BHA, cycles/sec [Hz] F B = buoyancy factor

g = acceleration caused by gravity, 32.2 ft/sec 2 [m/s2]

G = shear modulus, lbf/sq ft [N/m2]

i = natural vibration mode, first, second, etc.

k = shock absorber spring constant, lbf/ft [N/m]

L = length of BHA, ft [m] m t = total mass of bottomhole assembly,

slugs N = rotary speed, rev/min

Va = speed of compression (tension) wave, 16,850 ft/sec [5136 m/s]

Vo = speed of shear wave, 10,650 ft/sec [3246 m/s]

Wa = weight in air per unit length, lbf/ft [N/m]

W B = weight on bit, lbf [N] p = mass density, slugs/ cu ft [slugs/m 3]

References 1. Timoshenko, S., Young, D.H., and Weaver, W. Jr.: Vibration

Problems in Engineering, fourth edition, John Wiley & Sons Inc., New York City (1974) 364.

2. Deily, F.H., Dareing, D.W., Paff, C.H., Ortloff, J.E., and Lynn, R.D.: "Downhole Measurements of Drill String Forces and Mo­tions," Trans., ASME (1968) 217-25.

3. Garrett, W.R.: "The Effect of a Downhole Shock Absorber on Drill Bit and Drill Stem Performance," paper ASME 62-Pet-21 presented at the AS ME 1962 Petroleum and Mechanical Engineer­ing Conference, Dallas, Sept. 23-26.

4. Dareing, D.W. and Livesay, BJ.: "Longitudinal and Angular Drill-String Vibrations With Damping," Trans., AS ME (1968) 1-9.

5. Recommended Practice for Drillstem Design and Operating Limits, tenth edition, API, Dallas (1981), 65-69.

6. Dareing, D.W.: "Guidelines for Controlling Drill String Vibra­tions," paper AS ME 83-Pet-9 presented at the 1983 ASME Energy Technology Conference and Exhibition, Houston, Jan. 30-Feb.3.

SI Metric Conversion Factors

cycles/sec X 1.0 ft X 3.048*

lbf X 4.448222

* Conversion factor is exact.

E+OO E-Ol E+OO

Hz m N

JPT

Original manuscript received in Society of Petroleum Engineers office Aug. 27, 1982. Paper accepted for publication March 21, 1983. Revised manuscript received July 18, 1983. Paper (SPE 11228) first presented at the 1982 SPE Annual Technical Con­ference and Exhibition held in New Orleans Sept. 26-29.

JOURNAL OF PETROLEUM TECHNOLOGY