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

L=

WB

...... " ...... , .......... (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

1.4

\

\

\

\

\

1.2

\

§

\

.... 1.0

\

t:

I

.8

....I

DYNAMIC AXIAL

DISPLACEMENT

4212

f"'=-L- CPS

~

.6

....I

/\

TORSIONAL

MODE

5o

:::l

~

AXIAL MODE

\

)(

"

,

,

" " ........,

..........

.4

.2

--

°O~--~2----~4----~6----~8-----1~O---

NATURAL FREQUENCY, 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

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

638

~

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

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

4L p

Ino = :L

Fig. 2-Natural frequency

............................. (5)

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

A

l-

150

0

100

:z:

W

;=

50

l-

'0

iii

......:

0·

Z III 2

~~ ·20

W;:)

1110

A

:z:

l-

Fig. 3-Three-lobed bottom hole core taken from a hard

dolomitic limestone formation (courtesy of A. Scovil

Murray).

Fig. 4-Bit force transition from smooth to rough running

(rotary speed is 90 rev/min). 2

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

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.

3N N

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

60 20

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

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

N

4,212

20

L

- - - - .............................. (9)

or

84,240

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

639

1.4

1.2

§...

iii

)(

5 150

~

t-

Il.

~

~ 100

CJ

Z

w

lI-

a

a::

!:

0

.S

....

50

w

!::

co

1.0

TIME

cc

....

....

0

.6

~

.4

()

~

Q

...: 1 0 1SEC

u.

.1

.2

a:i

..J

iii

=>

~

5

I-

°0~--~2-----4~--~6~--~S----~10~--I

o

40

NA~URAL F~EQUE~Y, CPS

SO

120

160

I

200

CRITICAL ROTARY SPEED, RPM

Fig. 5-Dynamic bit forces and torque caused by drill collar

resonance (rotational speed is 110 rev/min). 2

Fig. 6-Critical rotary speeds.

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 adjacent tool joints.

The whirling-shaft model does not seem to represent

drillpipe in a wellbore.

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.

640

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.

TABLE 2-DYNAMIC RESPONSE OF

DRILLSTRING ROTATING AT CRITICAL

SPEED

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

Axial

Displacement

(in.)

Stress

(psi)

Force

(Ibf)

6,800

6,600

6,400

6,200

6,000

5,800

5,600

5,400

5,200

5,000

4,800

4,600

4,400

4,200

4,000

3,800

3,600

3,400

3,200

3,000

2,800

2,600

2,400

2,200

2,000

1,800

1,600

1,400

1,200

1,000

Drill Collars

1.10

0.89

0.52

0.09

0.39

0.79

1.06

1.15

1.05

0.78

0.40

0.23

0.60

0.94

1.14

1.16

1.00

0.70

0.39

0.47

0.81

1.09

1.22

1.17

0.97

0.68

0.52

0.71

1.01

1.23

1,586

3,739

5,268

5,916

5,575

4,309

2,354

665

2,574

4,496

5,709

5,994

5,311

3,816

1,978

1,812

3,644

5,262

6,123

6,052

5,091

3,538

2,309

3,048

4,709

6,005

6,511

6,123

4,990

3,626

7,000

16,500

23,200

26,000

24,600

19,000

10,400

3,000

11,300

19,800

25,200

26,400

23,400

16,800

8,700

8,000

16,000

23,200

27,000

26,700

22,400

15,600

10,200

13,400

20,700

26,500

28,700

27,000

22,000

16,000

800

600

400

200

0

1.29

1.19

0.92

0.52

0.25

500

2,590

4,627

5,994

6,480

14,000

73,200

130,800

169,500

183,200

Distance

(ft)

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

9.0

8.0

0

•

7.0

""Q

6.0

j:

DRILL

PIPE

RESPONSE

ell:

I

I

I

I

I

~

~

::;

~

~

5.0

• 4.0

I \

\

•

.0

[

!

FREQUENCY, CPS

I

.20

.eo

200

ROTARY SPEED. RPM

eo

\

\

\

'2

14

II

210

320

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

~

~

I

~

/

>< 3.0

•

I

~

2.0

I

\

\

I

,

,

DRILL

:+.-_ _ _ COLLAR

~ RESPONSE

1.0

I

I

I

I

I

H

,

o~-~-~-~-~~--~--~--~

o

1.0 2.0 3.0 4.0 5.0 6.0 7.0

BIT DISPLACEMENT FREQUENCY - CPS

I

o

20

I

40

60

80

100

ROTARY SPEED - RPM

••

'0

FREaUE:CY. CPS

120 140

II

I

I

I

I

I

I

I

I

40

eo

.20

.eo

200

240

210

320

ROTARY SPEED.RPM

Fig. 7-Frequency response of an 8,000-ft [2438-m] drillstring

containing 800 ft [243 m] of collars. 4

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

with shock absorber.

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.

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

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

fna=-l

211"

g ...........................

mt

(11)

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

6,800

6,600

6,400

6,200

6,000

5,800

5,600

5,400

5,200

5,000

4,800

4,600

4,400

4,200

4,000

3,800

3,600

3,400

3,200

3,000

2,800

2,600

2,400

2,200

2,000

1,800

1,600

1,400

1,200

1,000

800

600

400

200

0

7,000

16,500

23,200

26,000

24,600

19,000

10,400

3,000

11,300

19,800

25,200

26,400

23,400

16,800

8,700

8,000

16,000

23,200

27,000

26,700

22,400

15,600

10,200

13,400

20,700

26,500

28,700

27,000

22,000

16,000

14,000

73,200

130,800

169,500

183,200

Case 2

500

1,000

1,600

1,800

1,600

1,300

700

200

760

1,300

1,700

1,800

1,600

1,100

600

540

1,100

1,500

1,800

1,800

1,500

1,050

700

910

1,400

1,800

1,900

1,800

1,500

1,100

1,000

5,000

8,800

11,400

12,300

Case 3

-2,900

7,000

9,900

11,100

10,500

8,100

4,400

1,200

4,800

8,400

10,700

11,300

10,000

7,200

3,700

3,400

6,800

9,900

11,500

11,300

9,600

6,700

4,300

5,700

8,900

11,300

12,200

17,400

24,900

30,500

32,700

31,200

26,700

21,500

19,700

Case 4

2,300

5,400

7,600

8,500

8,000

6,200

3,400

960

3,700

6,500

8,200

8,600

7,700

5,500

2,900

2,600

5,300

7,600

8,800

8,700

7,300

5,100

3,300

4,400

6,800

8,700

9,400

10,700

12,000

13,000

13,600

13,900

14,300

15,100

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

BIT

CASE I

SHOCK

ABSORBER

.,/

CASE

n

CASEN

CASEm

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 out of phase

with bit displacements. This means that the maximum

dynamic bit force impacts the high spots of a three-lobed

bottomhole pattern.

0

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

=

I =

E

Ina' InO =

g

=

=

G

=

FB

644

modulus of elasticity, lbf/sq ft [N/m 2]

frequency of source of excitation,

cycles/sec [Hz]

natural longitudinal and torsional frequencies of BHA, cycles/sec [Hz]

buoyancy factor

acceleration caused by gravity, 32.2

ft/sec 2 [m/s2]

shear modulus, lbf/sq ft [N/m 2]

i

= natural vibration mode,

k =

L

mt

=

=

N =

Va

=

Vo

=

Wa

=

WB

=

=

p

first, second,

etc.

shock absorber spring constant, lbf/ft

[N/m]

length of BHA, ft [m]

total mass of bottomhole assembly,

slugs

rotary speed, rev/min

speed of compression (tension) wave,

16,850 ft/sec [5136 m/s]

speed of shear wave, 10,650 ft/sec

[3246 m/s]

weight in air per unit length, lbf/ft

[N/m]

weight on bit, lbf [N]

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

Hz

E-Ol

m

N

E+OO

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

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

L=

WB

...... " ...... , .......... (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

1.4

\

\

\

\

\

1.2

\

§

\

.... 1.0

\

t:

I

.8

....I

DYNAMIC AXIAL

DISPLACEMENT

4212

f"'=-L- CPS

~

.6

....I

/\

TORSIONAL

MODE

5o

:::l

~

AXIAL MODE

\

)(

"

,

,

" " ........,

..........

.4

.2

--

°O~--~2----~4----~6----~8-----1~O---

NATURAL FREQUENCY, 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

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

638

~

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

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

4L p

Ino = :L

Fig. 2-Natural frequency

............................. (5)

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

A

l-

150

0

100

:z:

W

;=

50

l-

'0

iii

......:

0·

Z III 2

~~ ·20

W;:)

1110

A

:z:

l-

Fig. 3-Three-lobed bottom hole core taken from a hard

dolomitic limestone formation (courtesy of A. Scovil

Murray).

Fig. 4-Bit force transition from smooth to rough running

(rotary speed is 90 rev/min). 2

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

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.

3N N

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

60 20

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

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

N

4,212

20

L

- - - - .............................. (9)

or

84,240

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

639

1.4

1.2

§...

iii

)(

5 150

~

t-

Il.

~

~ 100

CJ

Z

w

lI-

a

a::

!:

0

.S

....

50

w

!::

co

1.0

TIME

cc

....

....

0

.6

~

.4

()

~

Q

...: 1 0 1SEC

u.

.1

.2

a:i

..J

iii

=>

~

5

I-

°0~--~2-----4~--~6~--~S----~10~--I

o

40

NA~URAL F~EQUE~Y, CPS

SO

120

160

I

200

CRITICAL ROTARY SPEED, RPM

Fig. 5-Dynamic bit forces and torque caused by drill collar

resonance (rotational speed is 110 rev/min). 2

Fig. 6-Critical rotary speeds.

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 adjacent tool joints.

The whirling-shaft model does not seem to represent

drillpipe in a wellbore.

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.

640

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.

TABLE 2-DYNAMIC RESPONSE OF

DRILLSTRING ROTATING AT CRITICAL

SPEED

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

Axial

Displacement

(in.)

Stress

(psi)

Force

(Ibf)

6,800

6,600

6,400

6,200

6,000

5,800

5,600

5,400

5,200

5,000

4,800

4,600

4,400

4,200

4,000

3,800

3,600

3,400

3,200

3,000

2,800

2,600

2,400

2,200

2,000

1,800

1,600

1,400

1,200

1,000

Drill Collars

1.10

0.89

0.52

0.09

0.39

0.79

1.06

1.15

1.05

0.78

0.40

0.23

0.60

0.94

1.14

1.16

1.00

0.70

0.39

0.47

0.81

1.09

1.22

1.17

0.97

0.68

0.52

0.71

1.01

1.23

1,586

3,739

5,268

5,916

5,575

4,309

2,354

665

2,574

4,496

5,709

5,994

5,311

3,816

1,978

1,812

3,644

5,262

6,123

6,052

5,091

3,538

2,309

3,048

4,709

6,005

6,511

6,123

4,990

3,626

7,000

16,500

23,200

26,000

24,600

19,000

10,400

3,000

11,300

19,800

25,200

26,400

23,400

16,800

8,700

8,000

16,000

23,200

27,000

26,700

22,400

15,600

10,200

13,400

20,700

26,500

28,700

27,000

22,000

16,000

800

600

400

200

0

1.29

1.19

0.92

0.52

0.25

500

2,590

4,627

5,994

6,480

14,000

73,200

130,800

169,500

183,200

Distance

(ft)

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

9.0

8.0

0

•

7.0

""Q

6.0

j:

DRILL

PIPE

RESPONSE

ell:

I

I

I

I

I

~

~

::;

~

~

5.0

• 4.0

I \

\

•

.0

[

!

FREQUENCY, CPS

I

.20

.eo

200

ROTARY SPEED. RPM

eo

\

\

\

'2

14

II

210

320

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

~

~

I

~

/

>< 3.0

•

I

~

2.0

I

\

\

I

,

,

DRILL

:+.-_ _ _ COLLAR

~ RESPONSE

1.0

I

I

I

I

I

H

,

o~-~-~-~-~~--~--~--~

o

1.0 2.0 3.0 4.0 5.0 6.0 7.0

BIT DISPLACEMENT FREQUENCY - CPS

I

o

20

I

40

60

80

100

ROTARY SPEED - RPM

••

'0

FREaUE:CY. CPS

120 140

II

I

I

I

I

I

I

I

I

40

eo

.20

.eo

200

240

210

320

ROTARY SPEED.RPM

Fig. 7-Frequency response of an 8,000-ft [2438-m] drillstring

containing 800 ft [243 m] of collars. 4

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

with shock absorber.

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.

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

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

fna=-l

211"

g ...........................

mt

(11)

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

6,800

6,600

6,400

6,200

6,000

5,800

5,600

5,400

5,200

5,000

4,800

4,600

4,400

4,200

4,000

3,800

3,600

3,400

3,200

3,000

2,800

2,600

2,400

2,200

2,000

1,800

1,600

1,400

1,200

1,000

800

600

400

200

0

7,000

16,500

23,200

26,000

24,600

19,000

10,400

3,000

11,300

19,800

25,200

26,400

23,400

16,800

8,700

8,000

16,000

23,200

27,000

26,700

22,400

15,600

10,200

13,400

20,700

26,500

28,700

27,000

22,000

16,000

14,000

73,200

130,800

169,500

183,200

Case 2

500

1,000

1,600

1,800

1,600

1,300

700

200

760

1,300

1,700

1,800

1,600

1,100

600

540

1,100

1,500

1,800

1,800

1,500

1,050

700

910

1,400

1,800

1,900

1,800

1,500

1,100

1,000

5,000

8,800

11,400

12,300

Case 3

-2,900

7,000

9,900

11,100

10,500

8,100

4,400

1,200

4,800

8,400

10,700

11,300

10,000

7,200

3,700

3,400

6,800

9,900

11,500

11,300

9,600

6,700

4,300

5,700

8,900

11,300

12,200

17,400

24,900

30,500

32,700

31,200

26,700

21,500

19,700

Case 4

2,300

5,400

7,600

8,500

8,000

6,200

3,400

960

3,700

6,500

8,200

8,600

7,700

5,500

2,900

2,600

5,300

7,600

8,800

8,700

7,300

5,100

3,300

4,400

6,800

8,700

9,400

10,700

12,000

13,000

13,600

13,900

14,300

15,100

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

BIT

CASE I

SHOCK

ABSORBER

.,/

CASE

n

CASEN

CASEm

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 out of phase

with bit displacements. This means that the maximum

dynamic bit force impacts the high spots of a three-lobed

bottomhole pattern.

0

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

=

I =

E

Ina' InO =

g

=

=

G

=

FB

644

modulus of elasticity, lbf/sq ft [N/m 2]

frequency of source of excitation,

cycles/sec [Hz]

natural longitudinal and torsional frequencies of BHA, cycles/sec [Hz]

buoyancy factor

acceleration caused by gravity, 32.2

ft/sec 2 [m/s2]

shear modulus, lbf/sq ft [N/m 2]

i

= natural vibration mode,

k =

L

mt

=

=

N =

Va

=

Vo

=

Wa

=

WB

=

=

p

first, second,

etc.

shock absorber spring constant, lbf/ft

[N/m]

length of BHA, ft [m]

total mass of bottomhole assembly,

slugs

rotary speed, rev/min

speed of compression (tension) wave,

16,850 ft/sec [5136 m/s]

speed of shear wave, 10,650 ft/sec

[3246 m/s]

weight in air per unit length, lbf/ft

[N/m]

weight on bit, lbf [N]

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

Hz

E-Ol

m

N

E+OO

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