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# Drill n Vibration Control

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## Content

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.
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

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\
\

1.2

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§

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.... 1.0

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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
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

......:

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

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