Lateral Vibration of Oilwell Drillstring During Backreaming Operation

Cristiano E. Agostini1, Rodrigo Nicoletti11Department of Mechanical Engineering, Sao Carlos School of Engineering, University of Sao Paulo

Trabalhador Sao-Carlense 400, 13566-590 Sao Carlos, Brazilcr.agostini@gmail.com, rnicolet@sc.usp.br

ABSTRACT: Oilwell drillstring vibrations are renowned by their high damaging effects in the drillstring elements. In the case ofbackreaming operations, i.e. when the drillstring is pulled out of the wellbore, abnormal lateral vibrations have been reported byoperators in the field. Such lateral vibrations of the drillstring can cause several problems during wellbore construction, e.g. BHAelectronic equipment failure, falling rocks into the well, drillstring blockage, which result in high financial losses when they occur.This work focuses on the study of the dynamics of oilwell drillstrings during the withdrawing operation (backreaming). For that,a non-linear mathematical model is used for representing the drillstring vibrations considering fluid pumping and string rotationsimultaneously. The proposed model focuses on the effects of lateral vibrations on the lower portion of the drill string, commonlyknown as Bottom Hole Assembly (BHA). The modelling approach is based on analytical, nonlinear and lumped parameters, whichconsiders the effects of drilling fluid damping, stabilizer and drill collar contact with the borehole wall. The numerical results arecompared to experimental results obtained in a scaled test rig.

KEY WORDS: Rotor dynamics; Drillstring dynamics; Backreaming operation; Mechanical vibration.

1 INTRODUCTION

During the borehole drilling of oilwells, when the final depthis reached, there comes the task of pulling out the drillstring.There are two possible ways of pulling out the drillstring: dryoperation (no rotation and no pumping of fluid) and backream-ing operation (rotation and pumping of fluid). The dry opera-tion represents a faster operation, but the friction between thedrillstring and the borehole becomes so high that it usually jeop-ardizes the pulling-out operation. The backreaming operationis more commonly adopted, but their optimum conditions arestill controversial in literature. In some applications, the back-reaming operation represents a much better friction conditionfor pulling out the drillstring but, in other applications, the se-vere lateral vibration of the drillstring can cause serious damageto the Bottom Hole Assembly (BHA), and even borehole col-lapse with drillstring imprisonment [1].

The challenge of modelling the drillstring during backream-ing operation is the correct representation of the coupling be-tween the torcional and lateral movements of the drillstring andits longitudinal movement in contact with the borehole. Thefriction between the drillstring and the borehole during longitu-dinal movement, together with occasional impacts due to lateralvibration, lead the system to a strong nonlinear behaviour.

In literature, one can find few studies on the subject, mainlyfocused on axial vibration. In [2,3], axial vibration of the drill-string is studied to improve the dynamic behaviour of the sys-tem during stick-slip phenomenon and to optimize the designof the stabilizers. Friction between the drillstring and the bore-hole is studied in [4], based on a low order mathematical modeland experimental correlation to results shown in [5]. The resultsshow that friction plays an important role in the dynamics of the

system and in the impact level of the drillstring against the bore-hole walls. The control of the rotating speed of the drillstringhas also been investigated to minimize lateral vibrations. In [6],the adopted mathematical model took into account the longitu-dinal movement of the drillstring and results showed that vi-bration is self-excited. Hence, in some cases, vibration was re-duced by increasing the rotating speed of the drillstring. How-ever, in other cases, the lateral vibration observed was severe.

Despite the advances in the study of lateral vibrations of drill-strings, there are no related studies on the backreaming opera-tion, considering drillstring rotation and fluid pumping. In thiswork, one develops a mathematical model for the drillstringduring backreaming. The modelling approach is based on ana-lytical, nonlinear and lumped parameters, which considers theeffects of drilling fluid damping, stabilizer and drill collar con-tact with the borehole wall. The results of numerical simula-tions show the occurrence of abnormal lateral vibrations duringthe drillstring withdrawal, but this effect decreases with higherfriction coefficients, and for rotating speeds near the natural fre-quency of the system. A comparison between numerical andexperimental results obtained in scaled test rig showed goodagreement for conditions of low friction coefficients.

2 MATHEMATICAL MODEL

The following mathematical model was based on the twodegree-of-freedom model presented by Jansen [7,8], whichtakes into account the pumping fluid, contact of the stabiliz-ers with the borehole walls, contact of the drillstring with theborehole walls, and excitation due to unbalance. The model isthen complemented with additional degrees-of-freedom of tor-sion and longitudinal movement. The basic hypotheses of the

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

893

model are:

• Pumping Fluid: drag in the annular gap between the drill-string and the borehole is proportional to the square of therotating speed. There fluid also adds inertia to the system,as showed in [9];

• Stabilizers: the hydrodynamic effects in the gap betweenthe stabilizers and the borehole walls are neglected;

• Borehole: the cross section of the borehole is circular, andcontact between the drillstring and the borehole obey theCoulomb law;

• Drillstring Vibration: the adopted rotating speeds areclose to the first natural frequency of the drillstring associ-ated to the first bending mode. Hence, lateral vibration ofthe drillstring will be limited to that of the bending modeof a simply supported beam, which means that both stabi-lizers will be in contact at the same time;

• Longitudinal Movement: Coulomb friction is consideredin the longitudinal direction of movement. The weight ofthe drillstring, and its effects, will not be considered in themodel.

The mathematical model is composed of a cylindrical beamsimply supported, represented by lumped parameters. Thebeam represents the BHA and the supports are the stabilizers(Fig. 1). Vibration is analyzed in the mid section of the beam(section A-A of Figs. 1 and 2). The model has two degrees-of-freedom to represent the lateral movements of the BHA indirections x1 and x2, one degree-of-freedom to represent thetorsion of the drillstring, and one degree-of-freedom to repre-sent the longitudinal motion in x3 direction. The origin of thecoordinate system lies in the centre of the borehole. The unbal-ance of the BHA is represented by an eccentricity, given by thedifference between the centre of mass of the BHA and its thegeometric centre (Fig. 2).

Figure 1: Extremity of the drillstring and main components.

The inertia forces due to acceleration of the BHA and of thefluid can be described by:

Fm,x1 = −(m+mf ) x1 +mΩ2eo cos(ϕ− Ωt)

Fm,x2 = −(m+mf ) x2 +mΩ2eo sin(ϕ− Ωt)(1)

where m is the equivalent mass of the BHA, mf is the equiv-alent mass of the fluid in the gap between the BHA and the

Figure 2: Section A-A with bending of the BHA and contact ofthe stabilizers with the borehole wall [8].

borehole, x1 and x2 are the coordinates of the geometric centreof the BHA, Ω is the rotating speed of the drillstring, t is time,eo is the eccentricity of the center of mass, and ϕ is the angularposition of the BHA.

The resultant damping force of the fluid is given by [8]:

Fd,x1 = −cf v2x1√x21 + x22

= −cf x1√x21 + x22

Fd,x2 = −cf v2x2√x21 + x22

= −cf x2√x21 + x22

(2)

where cf is the equivalent damping coefficient of the fluid, andv =

√x21 + x22 is the amplitude of the lateral velocity of the

BHA.When the stabilizer hits the borehole wall (q > S0 – Fig. 3),

the restoring force can be decomposed into radial and tangentialcomponents, as follows:

Fk,rad = −k p cosβ

Fk,tan = −k p sinβ(3)

where q is the radial deflection of the BHA geometric centre,S0 = 1

2 (Dh − DS) is the gap between the stabilizer and theborehole (Dh is the borehole diameter, DS is the stabilizer di-ameter), and p = q cosβ + S0 cos γ = q cosβ − S0 cosφ isthe distance between the geometric centre of the BHA and thegeometric centre of the stabilizer. It is important to note that,if q < S0, there will be no restoring force in the system (nocontact between the stabilizer and the borehole wall – Fig. 3).

The equations of the restoring force during contact can besimplified by a first order Taylor series of the angular terms ofthe equations, as follows:

Fk,rad = −k (q − S0)

Fk,tan = −k φ(S0 −

S20

q

) (4)

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Figure 3: Contact force between the stabilizer and the boreholewall, and restoring force on the BHA [8].

This is justified by the low friction values observed in prac-tice during operation. Transforming to the Cartesian system ofcoordinates:

Fk,1 = −Fkradx1q

+ Fk,tanx2q

Fk,2 = −Fk,radx2q

− Fk,tanx1q

(5)

where q =√x21 + x22.

The restoring force due to the contact of the BHA with theborehole wall is given by spring-damper model:

Fw,rad = −kw(q − co)− cw q

Fw,tan = −S µc Fw,rad

(6)

where µc is the friction coefficient between the BHA and theborehole wall, kw and cw are the equivalent stiffness and damp-ing coefficient of the borehole wall, respectively, and S =sign(vg − vtan) (Fig. 4). The BHA velocity and the tangen-tial velocity in the contact point are given by:

vg =√x21 + x22 (7)

vtan =Dc

2Ω (8)

During the longitudinal movement of the drillstring, the con-tact force against the borehole wall is given by [10]:

Fc,axial = sign(x3)(Fk,rad µ+ Fw,rad µc) tanψ (9)

where tanψ = x3/ΩRc. Such contact force is only consideredin the model when there is contact between the BHA and theborehole wall, or between the stabilizer and the borehole wall.

Figure 4: Representation of the relative velocities in the BHAand in the borehole wall.

By adopting the following adimensional parameters:

β =m+mf

mξ =

cf com

ζ =So

co(10)

η =Ω

ωρ =

kwk

τ = ωt ν =cwmω

(11)

ω =

√k

mco =

1

2(Dh −Dc) yi =

xico

(12)

a =√x21 + x22 b =

√x21 + x22 c = x21 + x22 (13)

one can write the equations of motion of the drillstring for thebackreaming operation, as follows:

βy1 + ξby1 + αFk,rad y1 − αFk,tan y2 + γFw,rad y1

− γFw,tan y2 = ϵ cos(ϕ− ητ)(ϕ− Ω)2

ω2

− ϵ

ω2sin(ϕ− ητ)ϕ = 0

βy2 + ξb y2 + αFk,rad y2 − αFk,tan y1 + γFw,rad y2

+ γFw,tan y1 = −ϵ sin(ϕ− ητ)(ϕ− Ω)2

ω2

− ϵ

ω2cos(ϕ− ητ)ϕ = 0

ϕ+ 2ξtwsϕ+ w2s

(ϕ− ητ +

Tokt

)+ chby1 sin(ϕ− ητ)

− chby2 cos(ϕ− ητ) =F (αFk,tan + γFw,tan)[R− ϵ cos(ϕ− ητ)]

− F (αFk,rad + γFw,rad)ϵ sin(ϕ− ητ)

ma

KJy3 +

caKJω

y3 +ka

KJω2y3 − Fc,axial = 0

(14)where:

Fk,rad =

(1− ζ

a

)(15)

Fk,tan = φ

(ζ

a− ζ2

c

)(16)

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Fw,rad = ρ (a− 1) + ν b (17)

Fw,tan = −S µc Fw,rad (18)

and:

α = 1 and γ = 0 , if a > ζ and a ≤ 1(impact of stabilizer)

α = 1 and γ = 1 , if a > 1(impact of stabilizer and BHA)

α = 0 and γ = 0 , if a ≤ ζ (no impact)(19)

The equations of motion are integrated in time for differentoperating conditions of rotating speed of the drillstring, lon-gitudinal velocity of the drillstring, and friction coefficient ofthe borehole wall. The adopted parameter values are those ofthe scaled test rig. Some of the parameters are uncertain (e.g.friction coefficients and damping coefficients) and they must beadjusted accordingly to the experimental results.

3 TEST RIG

The test rig was built in scale according to the characteristicsof a real 16” oilwell. The drillstring is represented by a steelshaft with diameter of 12 mm, mounted in the vertical direc-tion, simply supported by a self aligning ball bearing (Fig. 5).In the extremity of the shaft above the bearing, it is mounteda inertia cylinder to represent the inertia of the drillpipe andBHA. The cylinder+shaft is driven by an electric motor througha flexible coupling, which represents the torsional stiffness ofthe drillpipe. The rotating speed of the shaft is measured byan encoder mounted in between the ball bearing and the inertiacylinder.

The electric motor and the ball bearing are rigidly mountedon the structure of the test rig. Hence, the relative longitudi-nal motion between the BHA (shaft) and the oilwell is repro-duced by the movement of an acrylic tube in which the shaftis mounted and it is free to rotate and translate (Fig. 5). Thistube is mounted on a linear guide driven by an electric servomotor. The rotating speed of the electric motor of the shaft iscontrolled by a frequency inverter, whereas the velocity of theservo motor of the linear guide is controlled by a servo drive.Analogue inductive proximity sensors, mounted in a positionequidistant to the stabilizers mounted in the shaft, are responsi-ble for measuring the behaviour of the shaft inside the tube.

The dimensions of the shaft, the inertia cylinder, and the flex-ibility of the coupling were determined according to param-eters observed in the oil field (offshore platforms). For ex-ample, the ratios ω1/Ω (first bending natural frequency of theBHA/rotating speed) and ωt/Ω (first torsion natural frequencyof the drillpipe/rotating speed) were the same as those observedin practice. The dimensions of the stabilizers mounted in theshaft also obey the proportions observed in practice. The ve-locity of the longitudinal movement of the tube is chosen ac-cording to the ratio LBHA/t (BHA length/time) observed in

Figure 5: Schematic view of the test rig.

Table 1: Characteristics of the test rig.

description value unitshaft length 1.43 mshaft diameter 12.0 mmdistance between stabilizers 0.73 mstabilizer diameter 19.0 mmcylinder mass 1.85 kgtube inner diameter 21.0 mmtube outer diameter 25.0 mmcoupling torsion stiffness (nominal) 0.17 Nm/radmaximum rotating speed 53.3 Hzmaximum longitudinal speed 0.53 m/smaximum tube stroke 0.40 m

practice. Table 1 presents the geometric and operational char-acteristics of the test rig.

4 RESULTS

Experimental tests were performed for different rotatingspeeds of the shaft and different longitudinal velocities of thetube. The tube was filled with drilling fluid with density of 1070kg/m3 and dynamic viscosity of 0.01 Pa.s. The lateral vibra-tion of the shaft was measured during the tests and the resultantRMS acceleration is presented in Fig. 6 in adimensional form,i.e. the adimensional acceleration is given by RMS(x1)/ω

2co.As one can see in Fig. 6, good agreement is observed between

numerical and experimental results, especially in the case whenthe tube has vertical movement, thus showing that the devel-oped model for the system is suitable for describing the sys-tem dynamics. It is also observed that above 2400 rpm thereis a sudden reduction in the lateral acceleration of the shaft,irrespective of the vertical velocity of the tube. Considering

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800 1200 1600 2000 2400 28000

0.090.12

0.16

0

0.5

1

Vert.

Veloc

ity (m

/s)

Rotating Speed (rpm)

RM

S A

dim

. Acc

eler

atio

n

exp.model

Figure 6: RMS lateral adimensional acceleration of the shaft asa function of the rotating speed and the longitudinal velocity ofthe tube: comparison between model and experiment.

that the natural frequency of the shaft (associated to the firstbending mode) is 39.9 Hz (equivalent to 2396 rpm), this reduc-tion in lateral acceleration of the shaft occurs after crossing thecritical velocity of the system. A possible explanation for thisphenomenon is the inversion of phase of the unbalance (self-centring effect).

The vertical velocity of the tube does not affect significantlythe results. One can observe that there is a slight increase ofthe lateral acceleration as one increases the vertical velocity ofthe tube. Such low effect of the vertical motion on system dy-namics can be explained by the low friction coefficient betweenthe stabilizers and the tube (estimated during model correlationas µ = 0.013). Considering the model correlated to the exper-imental results, one simulated numerically the system for thesame operating conditions but with a higher friction coefficientbetween the stabilizers and the tube: µ = 0.2. The results arepresented in Fig. 7.

800 1200 1600 2000 2400 28000

0.090.12

0.16

0

0.5

1

Vert.

Veloc

ity (m

/s)

Rotating Speed (rpm)

RM

S A

dim

. Acc

eler

atio

n

= 0.2µ = 0.013µ

Figure 7: RMS lateral adimensional acceleration of the shaft asa function of the rotating speed and the longitudinal velocity ofthe tube: variation of friction coefficient (numerical).

As one can see in Fig. 7, the lateral acceleration of the shaftincreased in all tested conditions with higher friction coefficientbetween the stabilizers and the tube (around 30% higher). Inaddition, the effect of the vertical velocity of the tube in thesystem dynamics (increase of lateral acceleration) is more evi-dent, and the decrease of lateral acceleration above the criticalspeed is still noticeable.

In summary, the results show that supercritical operation con-dition is interesting (irrespective of the longitudinal velocity)because this condition presents lower lateral accelerations ofthe shaft, and the longitudinal velocity tends to increase the lat-eral vibration of the shaft.

4.1 Simulations with Impacts of the Shaft against the Tube

In all the tested operation conditions, one observed that theshaft did not shock against the tube (a common occurrence dur-ing practical backreaming operation). This no-impact operationwas probably caused by the small unbalance of the shaft in thetest rig (basically the residual unbalance of the cylinder shaft).In practice, the unbalance is higher due to the irregular massdistribution in the BHA caused by instrumentation. Hence, themodel adopted previously was numerically solved with higherunbalance (100 times higher), and one observed that the shaft(BHA) began to touch the inner side of the tube (borehole) –Fig. 8.

Figure 8: Adimensional lateral deflection of the shaft as a func-tion of time: (a) 1630 rpm (low unbalance), (b) 1830 rpm (highunbalance).

In Fig. 8, one presents the deflection of the shaft duringtime in adimensional form (ratio between the displacement qand distance co). That means, if q/co equals 1, then the shafttouches the inner surface of the tube. One can clearly see that,in the conditions tested in the experiments (Fig. 8(a)), the shaftnever touches the tube, whereas adopting a higher unbalancevalue (Fig. 8(b)), the shaft continuously touches the tube.

In order to evaluate the level of impact of the shaft againstthe tube in each operation condition, one adopted the followingdefinition of total impulse:

I =

∫ τ

0

(Fk,rad + Fw,rad) dt (20)

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0.05 0.1 0.15 0.2 0.25 0.3 0.35200

400

600

800

1000

1200

1400

1600

1800

2000

2200

Vertical Velocity (m/s)

Adi

men

sion

al Im

puls

e

815 rpm1220 rpm1830 rpm3940 rpm3050 rpm

Figure 9: Adimensional impulse as a function of the longitudi-nal velocity and the rotating speed of the shaft (numerical).

By calculating the total impulse for different operation con-ditions under the same unbalance, one obtained the results pre-sented in Fig. 9. As one can see, as the rotating speed increases,the total impulse of the shaft against the tube increases signif-icantly. After crossing the critical speed of 2396 rpm, thereis a sudden reduction of the total impulse (similarly to whathappened to the lateral acceleration of the shaft in the exper-iments). However, for high rotating speeds (much above thecritical speed), the total impulse further increased.

Such results show that the supercritical operation is still inter-esting when shocks occur between the BHA and the borehole.However, it is clear that one cannot increase the rotating speedindefinitely.

Considering the vertical velocity of the tube, again no sig-nificant effect on the results is observed, except in the case of3940 rpm (rotating speed above the critical speed). In this case,a low longitudinal velocity of the tube results in a total impulse50% lower than that observed in higher longitudinal velocities.This is an indication that, in case of an operation with impacts,the longitudinal velocity of the BHA must be as small as possi-ble to reduce the total impulse against the borehole walls.

5 CONCLUSION

This work presents a simplified mathematical model for theoilwell drillstring during backreaming operation. A compari-son between numerical and experimental results showed thatthe developed model is suitable to describe the dynamics of thesystem under such operating conditions (backreaming). Theobtained results also showed that:

• a supercritical operation is an interesting operating condi-tion due to the lower lateral accelerations (and total im-pulse) presented by the shaft, irrespective of the longitudi-nal velocity adopted in the backreaming operation. How-ever, the rotating speed cannot increase indefinitely, other-wise the total impulse, and eventually the lateral accelera-tion of the shaft, will further increase;

• the longitudinal velocity of the shaft during the adoptedbackreaming conditions does not affect significantly theresults. When there is no impact against the boreholewalls, an increase of the longitudinal velocity slightlyincreases the lateral acceleration presented by the shaft.When there is impact against the borehole walls, a sig-nificant reduction of total impulse is only observed whenthe longitudinal velocity is small and the rotating speed isabove the critical speed.

Such are promising results in terms of optimum backream-ing operating conditions. However, due to the highly nonlinearcharacteristics of the system, other metrics shall be adopted toevaluate system response and better understand the system dy-namics.

ACKNOWLEDGEMENTS

This project was supported by the Brazilian research founda-tions FAPESP (Fundacao de Amparo a Pesquisa do Estado deSao Paulo) and CNPq (Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico).

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[2] Forster, I., Asymmetric vibration damping tool: small scale rigtesting and full scale field testing, SPE/IADC Drilling Confer-ence, Louisiana, USA, n.128458, 2010.

[3] Forster, I., Axial excitation as a means stick-slip mitigation:small scale rig testing and full scale field testing, SPE/IADCDrilling Conference, Amsterdam, Netherlands, n.139830, 2011.

[4] Liao, C.M., Balachandran, B., Karkoub, M., Abdel-Magid,Y.L., Drill-string dynamics: reduced-order models and ex-perimental studies, J. Vibration and Acoustics, v.133, n.4,pp.041008, 2011.

[5] Melakhessou, H., Berlioz, A., Ferraris, G., A Nonlinear Well-Drillstring Interaction Model. J. Vibration and Acoustics, v.125,n.1, pp.46-52, 2003.

[6] Yigit, A.S., Christoforou, A.P., Coupled torsional and bend-ing vibrations of drillstrings subject to impact with friction, J.Sound and Vibration, v.215, n.1, pp.167-181, 1998.

[7] Jansen, J.D., Non-linear rotor dynamics as applied to oil-well drillstring vibrations, J. Sound and Vibration, v.147, n.1,p.115135, 1991.

[8] Jansen, J.D., Non linear dynamics of oilwell drillstrings, DelftUniversity, Delft, 1993 (Master Dissertation).

[9] Wambsganss, M.W., Chen, S.S., Jendrzejczyk, J.A. Addedmass and damping of vibrating rod in confined viscous fluid,Argonne National Laboratory, Argonne, ANL-CT-75-08, 1974.

[10] Aadnoy, B.S., Fazaelizadeh, M., Hareland, G., A 3D analyticalmodel for wellbore friction, J. Canadian Petroleum Technology,v.49, n.10, pp.25-36, 2010.

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