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Stick-slip Oscillations in Oillwell Drilstrings:

Distributed Parameter andNeutral Type Time Delay Model

Time Delay Systems

M. Belem Saldivar [email protected]

CINVESTAV Mexico /IRCCYN France.

Sabine Mondi [email protected]

Departamento de Control Automtico delCentro de Investigacin y Estudios Avanzados

del IPN, Mxico.

15-02-12

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Contents

1 Introduction

2 Drilling system modelDistributed parameter model

Mechanical system model

Neutral type time delay model

DAlembert transformationFrequency domain analysisTorque on the bit model

3 Reduction of stick-slip phenomenonPractical strategiesManipulation of the weight on the bitManipulation of the damping at the bottom extremity

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IntroductionOilwell drilling system

Oilwell drillstrings are mechanisms that play a key role in the petroleum extraction

industry.Failures in drillstrings can be signicant in the total cost of the perforation process.

Fig. 1. Vertical oilwell drilling system.

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IntroductionNon-desired oscillations

The drillstring interaction with the borehole gives rise to a wide variety ofnon-desired oscillations which are classied depending on the direction theyappear.

Three main types of vibrations can be distinguished:

torsional (stick-slip oscillations),axial (bit bouncing phenomenon) and

lateral (whirl motion due to the out-of-balance of the drillstring).

The stick-slip phenomenon can originate problems such as: drill pipe fatigueproblems, drillstring components failures, wellbore instability.

Torsional drillstring vibrations are a source of failures which reduce

penetration rates and increase drilling operation costs.

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IntroductionStick-slip phenomena

Torsional drillstring vibrations appears due to downhole conditions, such assignicant drag, tight hole, and formation characteristics.

The stick-slip phenomena can cause the bit to stall in the formation while therotary table continues to rotate.

When the trapped torsional energy (similar to a wound-up spring) reaches a levelthat the bit can no longer resist, the bit suddenly comes loose, rotating andwhipping at very high speeds.

This stick-slip behavior can generate a torsional wave that travels up the drillstringto the rotary top system. Because of the high inertia of the rotary table, it actslike a xed end to the drillstring and reects the torsional wave back down thedrillstring to the bit. The bit may stall again, and the torsional wave cycle repeats.

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Distributed parameter modelDrilling system

The mechanical system is described by the following partial dierential equation

(Challamel, 2000):

GJ2v

2(, t) I

2v

t2(, t) v

t(, t) = 0, 2 (0, L), 0

with boundary conditions:

v(0, t) = t, GJv

(L, t) + IB2v

t2(L, t) = T

vt

(L, t)

where:

v(, t) is the angle of rotation,

is the angular velocity at the surface,T is the torque on the bit, L is the length of the rod,

IB is a lumped inertia (it represents the assembly at the bottom hole)

0 is the damping (viscous and structural)I is the inertia, G is the shear modulus and J is the geometrical moment of

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Neutral type time delay modelDAlembert transformation

Let the damping = 0. The distributed parameter model reduces to the

unidimensional wave equation: 2

v2 (, t) = p2

2

vt2 (, t), 2 (0, L), wherep =

pI/GJ. The general solution of the wave equation can be written as

v(, t) = () + (),

where = t + p, = t p, , are continuously dierentiable real-valuedfunctions. Observe that

v

t=

v

+

v

,

v

= p

v

p v

.

Then, the boundary conditions can be written as:

v(

0,t) =

(t) +

(t) =

(t),

v

(L, t) = p

(t + ) p

(t ) = IB

GJ

2

2(t + ) +

2

2(t )

1

GJ

T(t), = pL.

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Let

w(t) = v(L, t) = (t +

) + (t ).considering and as functions of t, we obtain:

(t) + (t) = (t), (1)

(t + ) + (t ) = w(t), (2)p(t + ) p(t ) = IB

G Jw(t) 1

GJT(t). (3)

Equation (2) implies(t + ) = (t ) + w(t). (4)

Substituting (4) into (1) and (3) we get:(t) (t 2) + w(t ) = (t) (5)

and

2p(t

) + pw(t) =

IB

GJ

w(t)

1

G J

T(t).

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Then, we have

(t) = 12

w(t + ) + IB2p

IGJw(t + ) + 1

2p

IGJT(t + ). (6)

Substituting (6) into (5), we get:

12

w(t + ) + IB2p

IGJw(t + ) + 1

2p

IGJT(t + ) 1

2w(t )

IB2p

IGJw(t ) 1

2p

IG JT(t ) + w(t ) = (t).

Finally we obtain:

w(t) w(t 2) +p

IGJ

IBw(t) +

pIG J

IBw(t 2) = 1

IBT(t) +

1

IBT(t 2)

+2p

IG J

IB(t ). (7)

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Neutral type time delay modelFrequency domain analysis

By denition,

(v(, t)) = v(, s) =

Z0

estv(, t)dt.

where (v(, t)) is the Laplace transform of v(, t).

2

v(, t)t2

= s2 (v(, t)) sv(, 0) vt(, 0) = s2v(, s)

sv(, 0) vt(, 0),

v(, t)

t = s (v(, t)) v(, 0) = sv(, s) v(, 0),

2v(, t)

2

=

2(v(, t))

2=

2v(, s)

2,

v(, t)

=

(v(, t))

=

v(, s)

.

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Considering null initial conditions, the wave equation can be written as:

GJ2v(, s)

2= Is2v(, s) (8)

The general solution of (8) is:

v(, s) = v1(s)eqIGJs

+ v2(s)eqIGJs

(9)

where v1(s) and v2(s) are particular solutions to be determined. From theboundary conditions we have that:

v(0, t) = (t).

then, the boundary conditions are:

(s) = sv1(s) + sv2(s) (10)

GJv

(L, s) + IBs

2v(L, s) = T(s). (11)

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The derivative of (9) is:

dv(, s)

d= v1(s)

sI

GJse

qI

GJs v2(s)s

I

GJseq

IGJs, (12)

substituting (12) into (11),

GJ

v1(s)s

IGJ

seq

IGJsL v2(s)

sI

GJseqIGJsL!

+IBs2

v1(s)e

qI

GJsL + v2(s)eq

IGJsL

= T(s)

or, in matrix form:s s

es(p

IGJs + IBs2) es

pIGJs + IBs2

v1(s)v2(s)

=

(s)T(s)

where

=qI

GJL.

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Then, the particular solutions v1(s) and v2(s) are:v1(s)v2(s)

= 1

D(s)

es

pIGJs + IBs2 ses(pIGJs + IBs2) s

(s)T(s)

where

D(s) = sh

esp

IGJs + IBs2 es(

pIGJs + IBs

2)i,

then,

v1(s) =es

pIGJs + IBs2(s) + sT(s)s

espIGJs + IBs2 es(pIGJs + IBs2)

=e2s

pIGJs + IBs2

(s) + sesT(s)

s e2s p

IGJs + IB

s2 (p

IGJs + IB

s2) ,

v2(s) =es(pIGJs + IBs2)(s) sT(s)

s

espIGJs + IBs2 es(pIGJs + IBs2)

=(pIGJs + IBs2)(s) sesT(s)

s

e2

spIGJs + IBs2 (pIGJs + IBs2) .

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Let w(s) = v(L, s) = v1(s)es + v2(s)e

s. Substituting v1 and v2, yields

w(s) =2pIGJes(s) + T(s) e2sT(s)

e2spIGJs + IBs2 (pIGJs + IBs2) .

equivalently

w(s)h

e2sp

IGJs + IBs2 (

pIGJs + IBs

2)i

= 2p

IGJes(s)

+T(s) e2s T(s)

s2w(s) e2ss2w(s) + pIGJIB

sw(s) + pIGJIB

e2ssw(s) = (13)

1IB

T(s) +1

IBe2sT(s) + 2

pIGJ

IBes(s)

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Finally, the neutral type time delay equation(7) is obtained through the inverseLaplace transform of (13).

1

s2w(s)

= w(t),

1 e2ss2w(s) = w(t 2),1 (sw(s)) = w(t),

1

e2ssw(s)

= w(t 2),1 (T(s)) = T(t),

1

e2sT(s)

= T(t 2),1

es(s)

= (t ).

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Neutral type time delay modelTorque on the bit

The following switched equation allows to approximate the physical phenomenonat the bottom hole (Navarro & Corts, 2007):

T(w(t)) = cbw(t) + Wob Rbb (w(t)) sgn (w(t))

where Rb > 0 is the bit radius and Wob > 0 the weight on the bit.

The bit dry friction coecient b(w(t)) is modeled as

b (w(t)) = cb + (sbcb )e

bvf jw(t)j

where sb , cb 2 (0, 1) are the static and Coulomb friction coecients and0 < b < 1 is a constant dening the velocity decrease rate.

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Stick-slip phenomenonSimulations of the neutral type model

Neutral type time-delay model (7) coupled to the model for the torque on the bitdescribes the drillstring behavior and the occurrence of stick-slip oscillations.

The model parameters used for the simulations presented in the sequel are:

I =0.095Kg m L =1172m cb =0.5

G =79.3x109N/m2 Rb =0.155575m sb =0.8

J =1.19x105m4 cb=0.03Nms/rad b=0.9

vf= 1 ca = 2000Nms

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Stick-slip phenomenon

A simulation of the neutral type model of the drilling system (7) coupled with themodel for the torque on the bit with Wob = 97347N and = 10rad/seg is

shown.

Fig. 2. Stick-slip phenomenon. a) Velocity atthe bottom extremity. b) Torque on the bit.

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Reduction of stick-slip phenomenonPractical strategies

According to drillersexperience, two practical strategies to avoid the stick-slip

phenomenon are: the reduction of the weight on the bit Wob and the increase of thevelocity at the surface . Simulations results of this model show an important reductionof the stick-slip oscillations by decreasing Wob from 97347N to 31649N (Fig. 3) and byincreasing from 10rad/s to 20rad/s (Fig. 4).

Fig. 3. Reduction of stick-slip phenomenon by decreasing Wob . Fig. 4. Reduction of stick-slip phenomenon by increasing .

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Reduction of stick-slip phenomenonManipulation of the weight on the bit

From eld data experience and from simulations of the model studied, it isconcluded that the manipulation of the weight on the bit can be a solution forstick-slip oscillations even for low velocities at the surface .

The variation of the weight on the bit is proposed in (Navarro & Surez, 2004) as

follows:Wob (w) = Kw jw(t)j + Wob0

with Wob0 > 0 and Wob > Wob0.

This expression captures the main characteristics of the weight on the bit. When

w(t) decreases, Wob decreases. As too low values ofWob would make drilling stop,the weight on the bit must be maintained at a minimal value Wob0 to insure adesirable rate of penetration.

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Reduction of stick-slip phenomenonManipulation of the weight on the bit

Wob variation law can be substituted into model (7) for a drilling parameterscombination for which stick-slip oscillations are presented. This is the case of considering

= 10rad/s and Wob0= 97347N as it is shown on Fig. 5.

Fig. 5. Reduction of stick-slip oscillations by means of considering the Wob

variation law.

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Reduction of stick slip phenomenon
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Reduction of stick-slip phenomenonManipulation of the damping at the bottom extremity

Another strategy for reducing stick-slip oscillations is by increasing the damping at the

down end of the drillstring. This can be done modifying the drilling uid characteristics

(it could be approximated by means of increasing the damping coecient cb). Thebehavior of the velocity in the bottom extremity of the system (7) with the model for the

torque on the bit for dierent values of cb is shown on Fig. 6.

Fig. 6. Velocity at the bottom extremity for dierent values

of cb (Nms/rad): a)0.8, b)15, c)65, d)150.(DCA - CINVESTAV) 15-02-12 22 / 23

References
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References

Challamel N., Rock destruction eect on the stability of a drilling structure,

Journal of sound and vibration, 233 (2), 235-254, 2000.Fridman E., Mondi S. and Saldivar M.B., Bounds on the Response of aDrilling Pipe Model. IMA J. of Mathematical Control and Information, 27,(4), 513-526, 2010.

Navarro E. and Surez R, Practical approach to modelling and controllingstick-slip oscillations in oilwell drillstrings, Proceedings of the 2004 IEEEInternational Conference on Control Applications, 1454-1460, Taipei, Taiwan,2004.

Navarro E., Corts D., Sliding-mode of a multi-DOF oilwell drillstring withstick-slip oscillations, Proceedings of the 2007 American Control Conference,3837-3842, New York City, USA, 2007.

Perreau P.J., Rey Fabret I.F., Gomel M.M. and Mabile C.M., New results inreal time vibrations prediction, 8th International Petroleum Exhibition andConference, Abu Dhabi, U.A.F., 190-199, 1998.

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