research article mathematical modeling and analysis of drill string … · 2019. 7. 30. · drill...
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Research ArticleMathematical Modeling and Analysis of Drill StringLongitudinal Vibration with Lateral Inertia Effect
Jialin Tian12 Chunming Wu1 Lin Yang1 Zhi Yang1 Gang Liu1 and Changfu Yuan1
1School of Mechatronic Engineering Southwest Petroleum University Chengdu 610500 China2School of Mechanical Engineering Southwest Jiaotong University Chengdu 610031 China
Correspondence should be addressed to Jialin Tian tianjialin001gmailcom and Chunming Wu 1769752908qqcom
Received 16 July 2015 Revised 8 January 2016 Accepted 17 January 2016
Academic Editor Sergio De Rosa
Copyright copy 2016 Jialin Tian et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Comparative analysis whether considering the lateral inertia or not aiming at the longitudinal vibration of the drill string in drillingprogress is proposed In the light of the actual condition the mechanical model of the drill string about vibration is establishedon the basis of the theoretical analysis Longitudinal vibration equation of the drill string is derived from the Rayleigh-Love modeland one-dimensional viscoelastic model According to the Laplace transform method and the relationships among parameters ofthe model the solutions to complex impedance at the bottom of the drill string are obtained and then the comparison results areanalyzed which is the lateral inertia effect on longitudinal vibration characteristicsThe researches show that the smaller the lengthof the drill string the greater the cross-sectional area of the drill string the greater the damping coefficient of bottom hole on thebottom of the drill string and the more evident the effect on the dynamic stiffness of the drill string with lateral inertia effect ThePoisson ratio of the drill string only has some effects on it taking account of the lateral inertia effect and the influence is relativelysmall compared with the former three conditions
1 Introduction
In oil and gas drilling engineering the drill string is a slenderstructure connecting downhole tools and drilling rig Itsdynamic performance is one of the key factors affecting thedrilling efficiency and cost Also the results caused by drillstring vibration features include rock-breaking performanceof drill bit drilling ROP (rate of penetration) wellbore qual-ity and downhole tools fatigues [1ndash4] According to drillingcondition the drill string vibration can be divided into threeforms such as longitudinal lateral and torsional vibration inwhich the longitudinal vibration is the main factor of drillingefficiency and safety The longitudinal vibration makes thedrill bit move up and down which results in the constantchange of ROP the service life of drill bit cutting elementor drilling footage [5ndash7] To itself longitudinal vibration maycause the drill string fatigue fracture [8]
For the solutions of drill string vibration analysismethodswith complex influence factors and boundary conditionsthe related researches include taking the drill string as elas-tomer or obtaining vibration equations with dynamics FEM
(finite element method) or analyzing longitudinal vibrationconsidering the coupling effects of wellbore friction or thehydrodynamic interaction of drilling fluid [9ndash14] Howeverin light of the drilling actual condition and well structureas an elongated rod the lateral inertia has an importantinfluence on the longitudinal vibration of drill string Infact the drill longitudinal string vibration characteristicscannot be fully reflectedwith ignoring the lateral inertia effect[15] Moreover with the search results of reference articlesprevious researchworkwas not found about the lateral inertiaeffect on drill string longitudinal vibration Based on thisthe influence of the lateral inertia effect on the drill stringlongitudinal vibration is discussed in this paper Accordingto well structure and drill string working conditions thelongitudinal vibrationmodels whether considering the lateralinertia or not are established Moreover the solution methodof drill string key parameter impedance at the bottomhole isobtained
With the presented analysis models and solution equa-tions the influences of considering the lateral inertia effector not on drill string longitudinal vibration are discussed
Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 6281264 8 pageshttpdxdoiorg10115520166281264
2 Shock and Vibration
rPk
r998400Pk120591Pkminusi
120591Pkminuso
H
hk+1
hk
h1 120575Pk
kPkz
middot middot middot
middot middot middot
rPk
r998400Pk120591Pkminusi
120591Pkminuso
H
hk+1
hk
h1 120575Pk
kPkz
middot middot middot
middot middot middot
Figure 1 Longitudinal vibration model of drill string
Meanwhile by comparing different calculation results somekey parametersrsquo influences on vibration features are pre-sented such as the drill string length inside and outsideradius damping coefficient and Poissonrsquos ratio The researchmethod and model can provide reference for drill stringvibration research At the same time the research resultscan also present theoretical and actual value for optimizationof downhole tools design and safety evaluation of drillingoperations
2 Analysis Method and Mathematical Model
Based on the real condition of the drill string relatedassumptions are proposed to simplify the research for drillstring longitudinal vibration [16] The following assumptionsare made the cross-sectional surface of the drill string isequivalently circular the drill string axis coincides withborehole axis the drilling fluid is regarded as viscoelasticlayer which can be divided into several layers and thedrilling fluids inside and outside drill string have identicalcharacteristics the vibration velocity of drill string is muchlarger than the flow rate of drilling fluid
According to drilling condition the drill string vibrationanalysis model is presented as in Figure 1 wherein ℎ119896 denotesthe location layer of 119896th drill string 119867 is the total lengthof the drill string 119903119875119896 and 119903
1015840
119875119896 respectively denote the outer
and inner radius of 119896th drill string 120591119875119896minus119894 120591119875119896minus119900 respectivelydenote the shear stress of drilling fluid inside and outside119896th drill string 119896119875119896 is the stiffness coefficient and 120575119875119896 meansthe damping coefficient of the 119896th drill string Usually inthe absence of inner and outer mud pressure this vibrationreduces to the classical wave vibration for longitudinal wavesin a uniform rod
21 Model of Drill String without Lateral Inertia Effect Thedeformation produced in a uniform drill string while it
vibrates in a borehole is symmetric about its axis Forestablishing the longitudinal vibration model of drill stringwithout lateral inertia effect regarding the drill string as aone-dimensional sticky elastomer and by using the laws ofmass andmomentum conservation the vibration equation ofthe 119896th drill string segment can be obtained as
119864119875119896119860119896
1205972119908119896
1205971199112minus 2120587120591119875119896minus119900119903119875119896 minus 2120587120591119875119896minus119894119903
1015840
119875119896= 120588119875119896119860119896
1205972119908119896
1205971199052
(V119891 gt VV V119891 gt 0)
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587120591119875119896minus119900119903119875119896 minus 2120587120591119875119896minus119894119903
1015840
119875119896= 120588119875119896119860119896
1205972119908119896
1205971199052
(V119891 lt VV V119891 gt 0)
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587120591119875119896minus119900119903119875119896 + 2120587120591119875119896minus119894119903
1015840
119875119896= 120588119875119896119860119896
1205972119908119896
1205971199052
(V119891 gt VV V119891 lt 0)
119864119875119896119860119896
1205972119908119896
1205971199112minus 2120587120591119875119896minus119900119903119875119896 + 2120587120591119875119896minus119894119903
1015840
119875119896= 120588119875119896119860119896
1205972119908119896
1205971199052
(V119891 lt VV V119891 lt 0)
(1)
wherein the right-hand side of this equation denotes themudpressures which is substituted with axial displacements Theleft-hand side of it denotes inertia force and the shear stressof drilling fluid inside and outside the drill string119908119896 denotesthe longitudinal displacement of the 119896th drill string segmentwhich is function of time 119905 and coordinate 119911 119908119896 = 119908119896(119911 119905)VV V119891 represent the vibration velocity of drill string and theflow rate of drilling fluid respectively Moreover 119864119875119896 denotesthe elastic modulus of the 119896th drill string segment 120588119875119896 is
Shock and Vibration 3
the density119860119896 represents the cross-sectional area of 119896th drillstring segment and the expression is given by
119860119896 = 120587 (1199032
119875119896minus 11990310158402
119875119896) (2)
According to the real working condition of the drillingthe relationship between VV and V119891 is V119891 ≪ VV and themovement of the drill string is always to the right So thecalculation formula should be denoted as
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587120591119875119896minus119900119903119875119896 minus 2120587120591119875119896minus119894119903
1015840
119875119896
= 120588119875119896119860119896
1205972119908119896
1205971199052
(3)
To the shear stress of drilling fluid inside and outside 119896thdrill string 120591119875119896minus119894 120591119875119896minus119900 the expressions can be written as
120591119875119896minus119894 = 119870119896minus119894119908119896 (119911 119905)
120591119875119896minus119900 = 119870119896minus119900119908119896 (119911 119905)
(4)
where 119870119896minus119894 119870119896minus119900 are the longitudinal shear stiffness ofdrilling fluid at the position of inside and outside 119896th drillstring segment With applied Laplace transform of (1) andconsistent with the assumption conditions the followingequation can be obtained
119864119875119896119860119896
1205972119882119896
1205971199112
minus [1205881198751198961198601198961199042minus 2120587119870119896minus119900119903119875119896 + 2120587119870119896minus119894119903
1015840
119875119896]119882119896 = 0
(5)
in which the Laplace transform with respect to time of 119908119896 =119908119896(119911 119905) is represented by119882119896 = 119882119896(119911 119904) and the transformrelation is
119882119896 (119911 119904) = 119871 [119908119896 (119911 119905)] = int
+infin
0
119908119896 (119911 119905) 119890minus119904119905119889119905 (6)
The definition of symbol 120572 is as follows
120572 = radic120588119875119896119860119896119904
2minus 2120587 (119870119896minus119900119903119875119896 minus 119870119896minus119894119903
1015840
119875119896)
119864119875119896119860119896
(7)
Then the general solution of119882119896(119911 119904) can be gained as thefollowing form
119882119896 (119911 119904) = 119862119896119890120572119911+ 119863119896119890
minus120572119911 (8)
where 119862119896 119863119896 denote undetermined coefficients determinedby boundary conditions respectively
During oil and gas field drilling process the vibration ofdrill bit is caused partly because of the uneven bottomholeAssuming that the force of bottomhole to the drill bit is119891119896(119911 119905) according to the transitivity of force the force on thebottom of the drill string can be also denoted as 119891119896(119911 119905) Sothe bottom and top of the drill string can be described as
119864119875119896119860119896
120597119882119896 (119911 119904)
120597119911
10038161003816100381610038161003816100381610038161003816119911=0= 119891119896 (119911 119904) (9)
119864119875119896
120597119882119896 (119911 119904)
120597119911
10038161003816100381610038161003816100381610038161003816119911=119867= minus (119896119875119896 + 120575119875119896119904)119882119896 (119911 119904)
1003816100381610038161003816119911=119867 (10)
where the Laplace transform with respect to time of 119891119896(119911 119905)is represented by 119891119896(119911 119904) Furthermore with (8) and (9)substituted into (5) the following equations can be obtained
119862119896 minus 119863119896 =119891119896 (119911 119904)
120572119864119875119896119860119896
119862119896 =119864119875119896120572 minus (119896119875119896 + 120575119875119896119904)
119864119875119896120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867
119863119896
(11)
Here defining the symbol 120574 its expression is given by
120574 =119864119875119896120572 minus (119896119875119896 + 120575119875119896119904)
119864119875119896120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867
(12)
Then taking the results of 119862119896 119863119896 and definition of 120574into (7) the displacement of the drill string 119882119896(119911 119904) can beobtained as follows
119882119896 (119911 119904) =119891119896 (119911 119904)
(120574 minus 1) 120572119864119875119896119860119896
(120574119890120572119911+ 119890minus120572119911) (13)
Moreover assume that
119904 = 119894120596 (14)
Then the Laplace transform is equivalent to the unilateralFourier transform so the response of displacement frequencycan be expressed as119882119896(119911 119894119908)The complex impedance of thedrill string is derived as follows
119870119875119889 =120572119860119896 (120574 minus 1) 119864119875119896
120574 + 1 (15)
The complex impedance of the drill string bottom is equalto the complex stiffness which can be expressed in the pluralform as follows
119870119875119889 = 119870119903 + 119894119862119894 (16)
in which the plural real component 119870119903 denotes the realdynamic stiffness which reflects the drill string ability oflongitudinal deformation resistance In other words whenthe drill string is pressed by the vibration force the valueof the dynamic stiffness has close association with the defor-mation of the drill string On the other hand the imaginarycomponent 119862119894 denotes the dynamic damping which reflectsthe drill string energy dissipation
22 Model of Considering Lateral Inertia Effect Consideringthe influence of lateral inertia effect on drill string vibrationcharacteristics with actual situation the vibration of the 119896th
4 Shock and Vibration
drill string segment can be described by the theory ofRayleigh-Love rod as follows
119864119875119896119860119896
1205972119908119896
1205971199112minus 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 gt VV V119891 gt 0)
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 lt VV V119891 gt 0)
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 gt VV V119891 lt 0)
119864119875119896119860119896
1205972119908119896
1205971199112minus 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 lt VV V119891 lt 0)
(17)
where ]119875119896 denotes Poissonrsquos ratio of the 119896th drill stringsegment It is also worth noticing that both coupling termsare proportional to Poissonrsquos ratio This is reasonable sincethis ratio describes the radial contraction of a rod subjectedto axial strain According to the drilling field situation theanalysis condition can be obtained as V119891 minus VV ≪ 0 and V119891 gt 0and the above equation with (16) can be simplified to
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(18)
For obtaining the results of the equation similar to themethod which is used for the solution of model without con-sidering the influence of lateral inertia effect on drill stringvibration considering the initial and continuity conditionsthe vibration equation of the drill string is solved by Laplacetransform and the displacement expression of the drill stringwith considering the influence of lateral inertia effect can beobtained
Here defining symbols 120572 120573 and 120574 their expressions aregiven as
120572 = radic120588119875119896119860119896119904
2minus 2120587 (119870119896minus119900119903119875119896 minus 119870119896minus119894119903
1015840
119875119896)
119864119875119896119860119896 + 120588119875119896119860119896]2119875119896 (1199032
119875119896minus 11990310158402119875119896) 1199042
120573 = 120588119875119896]2
119875119896(1199032
119875119896minus 11990310158402
119875119896) 1199042
120574 =(119864119875119896 + 120573) 120572 minus (119896119875119896 + 120575119875119896119904)
(119864119875119896 + 120573) 120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867
(19)
Then the displacement expression of the drill string canbe obtained as
119882119896 (119911 119904) =119891119896 (119911 119904)
(120574 minus 1) 120572 (119864119875119896119860119896 + 119860119896120573)(120574119890120572119911+ 119890minus120572119911) (20)
Moreover the analytical solution of the complex impe-dance at the bottomhole of drill string can be expressed as
119870119875119889 =120572119860119896 (120574 minus 1) (119864119875119896 + 120573)
120574 + 1 (21)
3 Numerical Example and Analysis ofKey Parameters Influence
For contrastively analyzing the influence of design parame-ters on the characteristics of drill string longitudinal vibra-tion the complex impedance results are discussed by usingthe analysis models established above including the modelswith and without lateral inertia effect In the calculationresults figures the horizontal axis is the vibration frequencyand the longitudinal axis is the dynamic stiffness at thebottom of drill string First of all the material parametersof the drill string and drilling fluid including the densityelastic modulus stiffness coefficient and longitudinal shearstiffness are shown in Table 1
31 The Drill String Length The length of the drill string isalso on behalf of the well depth At the same time to theratio of drill string length and diameter the bigger ratio iscorresponding to the poorer vibration stability of drill stringThe influence of drill string length on its dynamic stiffnessat the bottom is analyzed as in Figure 2 including takingaccount of the lateral inertia effect or not The numericalexample parameters are shown in Table 2 including the drillstring Poissonrsquos ratio length radius and damping coefficientand the drill string length value includes 1800m and 2000m
In Figure 2 the symbol119867 denotes the length of drill stringmodel without considering the lateral inertia effect m Thesymbol ℎ denotes the length of drill string with consideringthe lateral inertia effect mThe results of drill string dynamicstiffness are presented corresponding to frequency of 20Hzto 30Hz including different drill string length and modelsof considering the lateral inertia effect or not The followingconclusions can be obtainedwith the increasing of frequencythe amplitude of drill string dynamic stiffness presentsincreasing trend including the results of different length and
Shock and Vibration 5
Table 1 Material parameters of the drilling string
Parameters name ResultsDensity of drill string 120588
119875119896(kgm3) 800000
Elastic modulus of drill string 119864119875119896 (MPa) 206 times 103
Stiffness coefficient 119896119875119896 (Nm) 100 times 105
Longitudinal shear stiffness of drilling fluid inside drill string119870119896minus119894 (Mpam) 03Longitudinal shear stiffness of drilling fluid outside drill string119870119896minus119900 (Mpam) 01
3
2
1
0
minus1
minus2
Kr
(Nm
)
times104
H1 = 1800
h1 = 1800
H2 = 2000
h2 = 2000
22 23 24 25 26 27 28 29 3021f (Hz)
Figure 2 Influence of the drill string length on dynamic stiffness
Table 2 Calculation parameters of the length of drill stringinfluence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string1198671 ℎ1 (m) 1800Length of the drill string1198672 ℎ2 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficient 120575119875119896 (Nsm) 10000
analysis models Meanwhile to the lateral inertia effect theinfluence on drill string dynamic stiffness can be almostnegligible when the frequency is low or zero However withthe increasing of frequency such that the frequency is greaterthan 25Hz the influence of lateral inertia effect becomesmore obvious To the results of models considering lateralinertia effect or not in the increasing period of dynamicstiffness the absolute value of considering lateral inertia effectis greater than that of ignoring lateral inertia effect To theresults of drill string different lengths the influence of lateralinertia effect is bigger when the drill string is shorter
32 The Inside and Outside Radius The inside and outsideradius of drill string are determined by the size specificationThe influence of radius change on the dynamic stiffness isanalyzed as in Figure 3 including the models of consideringthe lateral inertia effect or not As before the numericalexample parameters are shown in Table 3 including the drillstring different inside and outside radius value
543210
minus1
minus3minus2
minus4
Kr
(Nm
)
times104
22 23 24 25 26 27 28 29 302120f (Hz)
RPk1 = 01
rPk1 = 01
RPk2 = 011
rPk2 = 011
Figure 3 Influence of outside radius on dynamic stiffness
Table 3 Calculated parameters of the inside and outside radiusinfluence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string H (m) 2000Outside radii of the drill string 1198771198751198961 1199031198751198961 (m) 010Outside radii of the drill string 119877
1198751198962 1199031198751198962
(m) 011Inside radii of the drill string 1198771015840
1198751198961 1199031198751198961 (m) 008
Inside radii of the drill string 11987710158401198751198962
1199031198751198962 (m) 009Damping coefficient 120575119875119896 (Nsm) 10000
In Figures 3 and 4 the symbols 119877119875119896 and 1198771198751198961 denote theradius of drill string model without considering the lateralinertia effect m The symbols 119903119875119896 and 119903
1015840
119875119896denote the radius
of drill string model considering the lateral inertia effectm The results of drill string dynamic stiffness are presentedcorresponding to frequency of 20Hz to 30Hz as shown inFigures 3 and 4 including different inside and outside radiusandmodels with lateral inertia effect or not As before the fol-lowing conclusions can be got with the frequency increasingthe dynamic stiffness amplitude presents increasing trendMeanwhile when outside radius increases or inside radiusreduces equivalent to the area increasing the increasingtrend of the dynamic stiffness becomes more notable Alsoto the results of models with lateral inertia effect or not inthe increasing period of dynamic stiffness the value of modelwith lateral inertia effect is greater than that without lateralinertia effect Meanwhile with bigger difference between thedrill string outside and inside radius the lateral inertia effecthas a more important influence on the dynamic stiffness ofdrill string
6 Shock and Vibration
3
2
1
0
minus1
minus2
Kr
(Nm
)
times104
22 23 24 25 26 27 28 29 302120f (Hz)
R998400Pk1 = 008
r998400Pk1 = 008
R998400Pk2 = 009
r998400Pk2 = 009
Figure 4 Influence of inside radius on dynamic stiffness
Table 4 Calculation parameters of the damping coefficient influ-ence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficients 12057510158401198751198961
1205751198751198961 (Nsm) 8000Damping coefficients 1205751015840
1198751198962 1205751198751198962 (Nsm) 10000
33 The Damping Coefficient The damping coefficient ofthe bottomhole to drill string is influential to the drillstring longitudinal vibration and the lateral inertia effect isdirectly related to the damping coefficient According to thedrilling field conditions the calculation parameters analyzingdamping coefficient are shown in Table 4 taking the dampingcoefficient as 8000 and 10000Nsm
In Figure 5 the symbol 120575119875119896 denotes the damping coeffi-cient ofmodelwithout lateral inertia effectNsmThe symbol1205751015840
119875119896denotes the damping coefficient of model considering
lateral inertia effect Nsm Figure 5 shows the influence of thedamping coefficient on the drill string dynamic stiffness cor-responding to frequency of 30Hz to 40Hz It can be observedthat the increasing trend of the dynamic stiffness becomesmore notable when the damping coefficient decreased andthe cyclic frequency increased including models with lateralinertia effect or not In a single cycle when it is closer tothe crest or trough the difference of drill string dynamicstiffness is bigger When considering the lateral inertia effectthe absolute value of dynamic stiffness in increasing periodis bigger than that of the model without lateral inertia effectHowever the bigger damping coefficient does notmeanmoreobvious influence in the drill string dynamic stiffness whichindicates that the damping of bottomhole to drill string canweaken the influence of lateral inertia effect on the dynamicstiffness
34 Poissonrsquos Ratio Poissonrsquos ratio is the lateral deformationcoefficient of material which reflects the lateral deformationelastic constants of material In the static force condition
120575Pk1 = 8000
120575998400Pk1 = 8000
120575Pk2 = 10000
120575998400Pk2 = 10000
4
2
0
minus2
minus4
Kr
(Nm
)
32 33 34 35 36 37 38 39 403130f (Hz)
times104
Figure 5 Influence of the damping coefficient on dynamic stiffness
3
4
2
1
0
minus1
minus3
minus2
Kr
(Nm
)
32 33 34 35 36 37 38 39 403130f (Hz)
times104
1 = 01
9984001 = 01
2 = 03
9984002 = 03
Figure 6 Influence of Poissonrsquos ratio on dynamic stiffness
Table 5 Calculated parameters of Poissonrsquos ratio influence
Name of parameters ResultsPoissonrsquos ratios of the drill string ]1015840
1 ]1 010
Poissonrsquos ratios of the drill string ]10158402 ]2 030
Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficient 120575119875119896 (Nsm) 10000
Poissonrsquos ratio change of drill string is very small and it isgenerally taken as ] = 025 However in the dynamic loadcase the dynamic Poissonrsquos ratio is commonly taken as ] =01sim03 According to the analysis above the calculatedparameters analyzing Poissonrsquos ratio are shown in Table 5taking Poissonrsquos ratio as 010 and 030
In Figure 6 the symbol ] denotes Poissonrsquos ratio of modelwithout lateral inertia effect On the other hand the symbol]1015840 denotes Poissonrsquos ratio of model considering lateral inertiaeffect Figure 6 shows the influence of Poissonrsquos ratio on thedrill string dynamic stiffness in the frequency of 30Hz to40Hz It can be seen that the increasing trend of the drillstring dynamic stiffness becomes more obvious when Pois-sonrsquos ratio and cyclic frequency increase both of the modelswith lateral inertia effect or not
Shock and Vibration 7
Table 6 The free vibration natural frequency in different conditions
Boundary conditions Without lateral inertia Lateral inertiaThe first-order natural frequency (rads) 003 004The second-order natural frequency (rads) 005 006The third-order natural frequency (rads) 007 009
However different from other parameters results thechange of the drill string Poissonrsquos ratio has influence on theresults of model considering lateral inertia effect while ithas no influence on the results of model without consideringlateral inertia effect which indicates that the lateral inertiaeffect includes the effect of Poissonrsquos ratio Meanwhile whenconsidering the lateral inertia effect the absolute value ofdynamic stiffness is increasing with the drill string Poissonrsquosratio in the increasing period
35 Natural Frequency Assume that the length of the drillstring is 3500m and the boundary conditions are prescribedby the above as well as the structure parameters of the drillstring The longitudinal vibration model of the drill stringis analyzed with different conditions The first three-ordernatural frequencies of the drill string longitudinal vibrationare shown in Table 6 whether considering the lateral inertiaeffect or not
The calculation results illustrate that the natural fre-quency increases with the increase of frequency order Andwhen there is no inertia it is less than the situation withlateral inertia effect When considering the inertia effect thevibration is coupled with the longitudinal vibration whichwill cause the vibration difference According to adjusting thedrilling parameters the drill string resonance can be avoidedand make drilling optimization possible
4 Conclusion
According to the drilling field conditions the analysis modelof drill string longitudinal vibration is established with thelateral inertia effect and the Laplace transform is used forthe complex impedance solution of drill string bottom Byusing the established equations and analyzing the resultsof numerical examples the influence of design parameterson drill string dynamic stiffness is discussed including theresults of models considering the lateral inertia effect or notWithin the frequency range of drilling field conditions theseconclusions can be drawn
(1) The cyclical transformation amplitude of drill stringdynamic stiffness whether considering the radialinertia effect or not presents increasing with fre-quency and the length of the drill string Thereforefor resisting drill string longitudinal deformation andexcessive vibration the appropriate length of drillstring should be considered in the well structuredesign
(2) For the drill string dynamic stiffness increasing withthe cross-sectional area combining the relationshipof sectional area and inside and outside radius
the influence of sectional area on drill string dynamicstiffness can provide reference for the design of BHA(Bottomhole Assembly)
(3) With the damping coefficient increasing the ampli-tude of drill string dynamic stiffness is reducingwhich would lead to the vibration force increasingThe results are coincidingwith the drilling actual con-dition Moreover the consequences can provide basicdata for the design of downhole friction reductiontools
(4) To the influence of Poissonrsquos ratio on drill stringdynamic stiffness the results are some differencecomparing with other parameters Based on themodel without considering the lateral inertia effectthe change of Poissonrsquos ratio has no influence onthe results of drill string dynamic stiffness Howeverto the model including the lateral inertia effect theinfluence of Poissonrsquos ratio on drill string dynamicstiffness cannot be ignored According to the drillingfield condition the results are important for drillingdynamics analysis especially in deep and ultradeepdrilling or some new wells which is the basis forimproving the rock-breaking efficiency and ROP
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by Open Fund (OGE201403-05) ofKey Laboratory of Oil amp Gas Equipment Ministry of Edu-cation (Southwest Petroleum University) National NaturalScience Foundation of China (no 51074202 and no 11102173)Major Cultivation Foundation of Sichuan Education Depart-ment (12ZZ003 no 667) and Graduate Student InnovationFund of Southwest Petroleum University (CX2015SY06)
References
[1] J J Bailey and I Finnie ldquoAn analytical study of drill-stringvibrationrdquo Journal of Engineering for Industry vol 82 no 2 pp122ndash127 1960
[2] S A Al-Hiddabi B Samanta and A Seibi ldquoNon-linear controlof torsional and bending vibrations of oilwell drillstringsrdquoJournal of Sound andVibration vol 265 no 2 pp 401ndash415 2003
[3] Q Xue R Wang F Sun and Z Huang ldquoChaotic vibrationanalysis of the bottom rotating drill stringrdquo Shock andVibrationvol 2014 Article ID 429164 8 pages 2014
8 Shock and Vibration
[4] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010
[5] W Wang Q Hu H Liu andW Zhang ldquoAnalysis on longitudi-nal vibration of drill string and the corresponding applicationrdquoJournal of Vibration and Shock vol 6 pp 229ndash233 2011
[6] J Li T Yan X Sun and S Peng ldquoFinite element analysis ondrilling string axial vibration in a crooked holerdquo in Proceedingsof the International Conference on Pipelines and TrenchlessTechnology (ICPTT rsquo12) vol 10 pp 1328ndash1334 Wuhan ChinaOctober 2012
[7] T Yan XWang S Li X Bi andCHan ldquoFinite element analysisof longitudinal and lateral coupled vibration of drillstringrdquo OilField Equipment vol 3 p 12 2012
[8] X Yan J Sun M Zhang et al ldquoAnalysis of longitudinalvibration of drill strings using equivalent networkrdquo Oil FieldEquipment vol 4 p 5 2010
[9] S M Sahebkar M R Ghazavi S E Khadem and M HGhayesh ldquoNonlinear vibration analysis of an axially movingdrillstring system with time dependent axial load and axialvelocity in inclined wellrdquoMechanism and Machine Theory vol46 no 5 pp 743ndash760 2011
[10] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011
[11] T Ding andC Li ldquoAnalysis of coupling axial vibrations betweendrilling fluids and drillstringrdquo Chinese Journal of MechanicalEngineering vol 43 no 9 pp 215ndash219 2007
[12] V I Gulyayev and O I Borshch ldquoFree vibrations of drill stringsin hyper deep vertical bore-wellsrdquo Journal of Petroleum Scienceand Engineering vol 78 no 3-4 pp 759ndash764 2011
[13] C Bu X Li L Sun and B Xia ldquoArithmetic solution for the axialvibration of drill string coupling with a down-the-hole hammerin rock drillingrdquo Journal of Vibration and Control 2014
[14] Y A Khulief F A Al-Sulaiman and S Bashmal ldquoVibrationanalysis of drillstrings with self-excited stick-slip oscillationsrdquoJournal of Sound and Vibration vol 299 no 3 pp 540ndash5582007
[15] W-L Wang Q-A Hu H-W Liu and W Zhang ldquoAnalysison longitudinal vibration of drill string and the correspondingapplicationrdquo Journal of Vibration and Shock vol 30 no 6 pp229ndash233 2011
[16] A A Jafari R Kazemi and M F Mahyari ldquoThe effects ofdrilling mud and weight bit on stability and vibration of a drillstringrdquo Journal of Vibration and Acoustics Transactions of theASME vol 134 no 1 Article ID 011014 2012
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Shock and Vibration
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2 Shock and Vibration
rPk
r998400Pk120591Pkminusi
120591Pkminuso
H
hk+1
hk
h1 120575Pk
kPkz
middot middot middot
middot middot middot
rPk
r998400Pk120591Pkminusi
120591Pkminuso
H
hk+1
hk
h1 120575Pk
kPkz
middot middot middot
middot middot middot
Figure 1 Longitudinal vibration model of drill string
Meanwhile by comparing different calculation results somekey parametersrsquo influences on vibration features are pre-sented such as the drill string length inside and outsideradius damping coefficient and Poissonrsquos ratio The researchmethod and model can provide reference for drill stringvibration research At the same time the research resultscan also present theoretical and actual value for optimizationof downhole tools design and safety evaluation of drillingoperations
2 Analysis Method and Mathematical Model
Based on the real condition of the drill string relatedassumptions are proposed to simplify the research for drillstring longitudinal vibration [16] The following assumptionsare made the cross-sectional surface of the drill string isequivalently circular the drill string axis coincides withborehole axis the drilling fluid is regarded as viscoelasticlayer which can be divided into several layers and thedrilling fluids inside and outside drill string have identicalcharacteristics the vibration velocity of drill string is muchlarger than the flow rate of drilling fluid
According to drilling condition the drill string vibrationanalysis model is presented as in Figure 1 wherein ℎ119896 denotesthe location layer of 119896th drill string 119867 is the total lengthof the drill string 119903119875119896 and 119903
1015840
119875119896 respectively denote the outer
and inner radius of 119896th drill string 120591119875119896minus119894 120591119875119896minus119900 respectivelydenote the shear stress of drilling fluid inside and outside119896th drill string 119896119875119896 is the stiffness coefficient and 120575119875119896 meansthe damping coefficient of the 119896th drill string Usually inthe absence of inner and outer mud pressure this vibrationreduces to the classical wave vibration for longitudinal wavesin a uniform rod
21 Model of Drill String without Lateral Inertia Effect Thedeformation produced in a uniform drill string while it
vibrates in a borehole is symmetric about its axis Forestablishing the longitudinal vibration model of drill stringwithout lateral inertia effect regarding the drill string as aone-dimensional sticky elastomer and by using the laws ofmass andmomentum conservation the vibration equation ofthe 119896th drill string segment can be obtained as
119864119875119896119860119896
1205972119908119896
1205971199112minus 2120587120591119875119896minus119900119903119875119896 minus 2120587120591119875119896minus119894119903
1015840
119875119896= 120588119875119896119860119896
1205972119908119896
1205971199052
(V119891 gt VV V119891 gt 0)
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587120591119875119896minus119900119903119875119896 minus 2120587120591119875119896minus119894119903
1015840
119875119896= 120588119875119896119860119896
1205972119908119896
1205971199052
(V119891 lt VV V119891 gt 0)
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587120591119875119896minus119900119903119875119896 + 2120587120591119875119896minus119894119903
1015840
119875119896= 120588119875119896119860119896
1205972119908119896
1205971199052
(V119891 gt VV V119891 lt 0)
119864119875119896119860119896
1205972119908119896
1205971199112minus 2120587120591119875119896minus119900119903119875119896 + 2120587120591119875119896minus119894119903
1015840
119875119896= 120588119875119896119860119896
1205972119908119896
1205971199052
(V119891 lt VV V119891 lt 0)
(1)
wherein the right-hand side of this equation denotes themudpressures which is substituted with axial displacements Theleft-hand side of it denotes inertia force and the shear stressof drilling fluid inside and outside the drill string119908119896 denotesthe longitudinal displacement of the 119896th drill string segmentwhich is function of time 119905 and coordinate 119911 119908119896 = 119908119896(119911 119905)VV V119891 represent the vibration velocity of drill string and theflow rate of drilling fluid respectively Moreover 119864119875119896 denotesthe elastic modulus of the 119896th drill string segment 120588119875119896 is
Shock and Vibration 3
the density119860119896 represents the cross-sectional area of 119896th drillstring segment and the expression is given by
119860119896 = 120587 (1199032
119875119896minus 11990310158402
119875119896) (2)
According to the real working condition of the drillingthe relationship between VV and V119891 is V119891 ≪ VV and themovement of the drill string is always to the right So thecalculation formula should be denoted as
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587120591119875119896minus119900119903119875119896 minus 2120587120591119875119896minus119894119903
1015840
119875119896
= 120588119875119896119860119896
1205972119908119896
1205971199052
(3)
To the shear stress of drilling fluid inside and outside 119896thdrill string 120591119875119896minus119894 120591119875119896minus119900 the expressions can be written as
120591119875119896minus119894 = 119870119896minus119894119908119896 (119911 119905)
120591119875119896minus119900 = 119870119896minus119900119908119896 (119911 119905)
(4)
where 119870119896minus119894 119870119896minus119900 are the longitudinal shear stiffness ofdrilling fluid at the position of inside and outside 119896th drillstring segment With applied Laplace transform of (1) andconsistent with the assumption conditions the followingequation can be obtained
119864119875119896119860119896
1205972119882119896
1205971199112
minus [1205881198751198961198601198961199042minus 2120587119870119896minus119900119903119875119896 + 2120587119870119896minus119894119903
1015840
119875119896]119882119896 = 0
(5)
in which the Laplace transform with respect to time of 119908119896 =119908119896(119911 119905) is represented by119882119896 = 119882119896(119911 119904) and the transformrelation is
119882119896 (119911 119904) = 119871 [119908119896 (119911 119905)] = int
+infin
0
119908119896 (119911 119905) 119890minus119904119905119889119905 (6)
The definition of symbol 120572 is as follows
120572 = radic120588119875119896119860119896119904
2minus 2120587 (119870119896minus119900119903119875119896 minus 119870119896minus119894119903
1015840
119875119896)
119864119875119896119860119896
(7)
Then the general solution of119882119896(119911 119904) can be gained as thefollowing form
119882119896 (119911 119904) = 119862119896119890120572119911+ 119863119896119890
minus120572119911 (8)
where 119862119896 119863119896 denote undetermined coefficients determinedby boundary conditions respectively
During oil and gas field drilling process the vibration ofdrill bit is caused partly because of the uneven bottomholeAssuming that the force of bottomhole to the drill bit is119891119896(119911 119905) according to the transitivity of force the force on thebottom of the drill string can be also denoted as 119891119896(119911 119905) Sothe bottom and top of the drill string can be described as
119864119875119896119860119896
120597119882119896 (119911 119904)
120597119911
10038161003816100381610038161003816100381610038161003816119911=0= 119891119896 (119911 119904) (9)
119864119875119896
120597119882119896 (119911 119904)
120597119911
10038161003816100381610038161003816100381610038161003816119911=119867= minus (119896119875119896 + 120575119875119896119904)119882119896 (119911 119904)
1003816100381610038161003816119911=119867 (10)
where the Laplace transform with respect to time of 119891119896(119911 119905)is represented by 119891119896(119911 119904) Furthermore with (8) and (9)substituted into (5) the following equations can be obtained
119862119896 minus 119863119896 =119891119896 (119911 119904)
120572119864119875119896119860119896
119862119896 =119864119875119896120572 minus (119896119875119896 + 120575119875119896119904)
119864119875119896120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867
119863119896
(11)
Here defining the symbol 120574 its expression is given by
120574 =119864119875119896120572 minus (119896119875119896 + 120575119875119896119904)
119864119875119896120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867
(12)
Then taking the results of 119862119896 119863119896 and definition of 120574into (7) the displacement of the drill string 119882119896(119911 119904) can beobtained as follows
119882119896 (119911 119904) =119891119896 (119911 119904)
(120574 minus 1) 120572119864119875119896119860119896
(120574119890120572119911+ 119890minus120572119911) (13)
Moreover assume that
119904 = 119894120596 (14)
Then the Laplace transform is equivalent to the unilateralFourier transform so the response of displacement frequencycan be expressed as119882119896(119911 119894119908)The complex impedance of thedrill string is derived as follows
119870119875119889 =120572119860119896 (120574 minus 1) 119864119875119896
120574 + 1 (15)
The complex impedance of the drill string bottom is equalto the complex stiffness which can be expressed in the pluralform as follows
119870119875119889 = 119870119903 + 119894119862119894 (16)
in which the plural real component 119870119903 denotes the realdynamic stiffness which reflects the drill string ability oflongitudinal deformation resistance In other words whenthe drill string is pressed by the vibration force the valueof the dynamic stiffness has close association with the defor-mation of the drill string On the other hand the imaginarycomponent 119862119894 denotes the dynamic damping which reflectsthe drill string energy dissipation
22 Model of Considering Lateral Inertia Effect Consideringthe influence of lateral inertia effect on drill string vibrationcharacteristics with actual situation the vibration of the 119896th
4 Shock and Vibration
drill string segment can be described by the theory ofRayleigh-Love rod as follows
119864119875119896119860119896
1205972119908119896
1205971199112minus 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 gt VV V119891 gt 0)
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 lt VV V119891 gt 0)
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 gt VV V119891 lt 0)
119864119875119896119860119896
1205972119908119896
1205971199112minus 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 lt VV V119891 lt 0)
(17)
where ]119875119896 denotes Poissonrsquos ratio of the 119896th drill stringsegment It is also worth noticing that both coupling termsare proportional to Poissonrsquos ratio This is reasonable sincethis ratio describes the radial contraction of a rod subjectedto axial strain According to the drilling field situation theanalysis condition can be obtained as V119891 minus VV ≪ 0 and V119891 gt 0and the above equation with (16) can be simplified to
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(18)
For obtaining the results of the equation similar to themethod which is used for the solution of model without con-sidering the influence of lateral inertia effect on drill stringvibration considering the initial and continuity conditionsthe vibration equation of the drill string is solved by Laplacetransform and the displacement expression of the drill stringwith considering the influence of lateral inertia effect can beobtained
Here defining symbols 120572 120573 and 120574 their expressions aregiven as
120572 = radic120588119875119896119860119896119904
2minus 2120587 (119870119896minus119900119903119875119896 minus 119870119896minus119894119903
1015840
119875119896)
119864119875119896119860119896 + 120588119875119896119860119896]2119875119896 (1199032
119875119896minus 11990310158402119875119896) 1199042
120573 = 120588119875119896]2
119875119896(1199032
119875119896minus 11990310158402
119875119896) 1199042
120574 =(119864119875119896 + 120573) 120572 minus (119896119875119896 + 120575119875119896119904)
(119864119875119896 + 120573) 120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867
(19)
Then the displacement expression of the drill string canbe obtained as
119882119896 (119911 119904) =119891119896 (119911 119904)
(120574 minus 1) 120572 (119864119875119896119860119896 + 119860119896120573)(120574119890120572119911+ 119890minus120572119911) (20)
Moreover the analytical solution of the complex impe-dance at the bottomhole of drill string can be expressed as
119870119875119889 =120572119860119896 (120574 minus 1) (119864119875119896 + 120573)
120574 + 1 (21)
3 Numerical Example and Analysis ofKey Parameters Influence
For contrastively analyzing the influence of design parame-ters on the characteristics of drill string longitudinal vibra-tion the complex impedance results are discussed by usingthe analysis models established above including the modelswith and without lateral inertia effect In the calculationresults figures the horizontal axis is the vibration frequencyand the longitudinal axis is the dynamic stiffness at thebottom of drill string First of all the material parametersof the drill string and drilling fluid including the densityelastic modulus stiffness coefficient and longitudinal shearstiffness are shown in Table 1
31 The Drill String Length The length of the drill string isalso on behalf of the well depth At the same time to theratio of drill string length and diameter the bigger ratio iscorresponding to the poorer vibration stability of drill stringThe influence of drill string length on its dynamic stiffnessat the bottom is analyzed as in Figure 2 including takingaccount of the lateral inertia effect or not The numericalexample parameters are shown in Table 2 including the drillstring Poissonrsquos ratio length radius and damping coefficientand the drill string length value includes 1800m and 2000m
In Figure 2 the symbol119867 denotes the length of drill stringmodel without considering the lateral inertia effect m Thesymbol ℎ denotes the length of drill string with consideringthe lateral inertia effect mThe results of drill string dynamicstiffness are presented corresponding to frequency of 20Hzto 30Hz including different drill string length and modelsof considering the lateral inertia effect or not The followingconclusions can be obtainedwith the increasing of frequencythe amplitude of drill string dynamic stiffness presentsincreasing trend including the results of different length and
Shock and Vibration 5
Table 1 Material parameters of the drilling string
Parameters name ResultsDensity of drill string 120588
119875119896(kgm3) 800000
Elastic modulus of drill string 119864119875119896 (MPa) 206 times 103
Stiffness coefficient 119896119875119896 (Nm) 100 times 105
Longitudinal shear stiffness of drilling fluid inside drill string119870119896minus119894 (Mpam) 03Longitudinal shear stiffness of drilling fluid outside drill string119870119896minus119900 (Mpam) 01
3
2
1
0
minus1
minus2
Kr
(Nm
)
times104
H1 = 1800
h1 = 1800
H2 = 2000
h2 = 2000
22 23 24 25 26 27 28 29 3021f (Hz)
Figure 2 Influence of the drill string length on dynamic stiffness
Table 2 Calculation parameters of the length of drill stringinfluence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string1198671 ℎ1 (m) 1800Length of the drill string1198672 ℎ2 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficient 120575119875119896 (Nsm) 10000
analysis models Meanwhile to the lateral inertia effect theinfluence on drill string dynamic stiffness can be almostnegligible when the frequency is low or zero However withthe increasing of frequency such that the frequency is greaterthan 25Hz the influence of lateral inertia effect becomesmore obvious To the results of models considering lateralinertia effect or not in the increasing period of dynamicstiffness the absolute value of considering lateral inertia effectis greater than that of ignoring lateral inertia effect To theresults of drill string different lengths the influence of lateralinertia effect is bigger when the drill string is shorter
32 The Inside and Outside Radius The inside and outsideradius of drill string are determined by the size specificationThe influence of radius change on the dynamic stiffness isanalyzed as in Figure 3 including the models of consideringthe lateral inertia effect or not As before the numericalexample parameters are shown in Table 3 including the drillstring different inside and outside radius value
543210
minus1
minus3minus2
minus4
Kr
(Nm
)
times104
22 23 24 25 26 27 28 29 302120f (Hz)
RPk1 = 01
rPk1 = 01
RPk2 = 011
rPk2 = 011
Figure 3 Influence of outside radius on dynamic stiffness
Table 3 Calculated parameters of the inside and outside radiusinfluence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string H (m) 2000Outside radii of the drill string 1198771198751198961 1199031198751198961 (m) 010Outside radii of the drill string 119877
1198751198962 1199031198751198962
(m) 011Inside radii of the drill string 1198771015840
1198751198961 1199031198751198961 (m) 008
Inside radii of the drill string 11987710158401198751198962
1199031198751198962 (m) 009Damping coefficient 120575119875119896 (Nsm) 10000
In Figures 3 and 4 the symbols 119877119875119896 and 1198771198751198961 denote theradius of drill string model without considering the lateralinertia effect m The symbols 119903119875119896 and 119903
1015840
119875119896denote the radius
of drill string model considering the lateral inertia effectm The results of drill string dynamic stiffness are presentedcorresponding to frequency of 20Hz to 30Hz as shown inFigures 3 and 4 including different inside and outside radiusandmodels with lateral inertia effect or not As before the fol-lowing conclusions can be got with the frequency increasingthe dynamic stiffness amplitude presents increasing trendMeanwhile when outside radius increases or inside radiusreduces equivalent to the area increasing the increasingtrend of the dynamic stiffness becomes more notable Alsoto the results of models with lateral inertia effect or not inthe increasing period of dynamic stiffness the value of modelwith lateral inertia effect is greater than that without lateralinertia effect Meanwhile with bigger difference between thedrill string outside and inside radius the lateral inertia effecthas a more important influence on the dynamic stiffness ofdrill string
6 Shock and Vibration
3
2
1
0
minus1
minus2
Kr
(Nm
)
times104
22 23 24 25 26 27 28 29 302120f (Hz)
R998400Pk1 = 008
r998400Pk1 = 008
R998400Pk2 = 009
r998400Pk2 = 009
Figure 4 Influence of inside radius on dynamic stiffness
Table 4 Calculation parameters of the damping coefficient influ-ence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficients 12057510158401198751198961
1205751198751198961 (Nsm) 8000Damping coefficients 1205751015840
1198751198962 1205751198751198962 (Nsm) 10000
33 The Damping Coefficient The damping coefficient ofthe bottomhole to drill string is influential to the drillstring longitudinal vibration and the lateral inertia effect isdirectly related to the damping coefficient According to thedrilling field conditions the calculation parameters analyzingdamping coefficient are shown in Table 4 taking the dampingcoefficient as 8000 and 10000Nsm
In Figure 5 the symbol 120575119875119896 denotes the damping coeffi-cient ofmodelwithout lateral inertia effectNsmThe symbol1205751015840
119875119896denotes the damping coefficient of model considering
lateral inertia effect Nsm Figure 5 shows the influence of thedamping coefficient on the drill string dynamic stiffness cor-responding to frequency of 30Hz to 40Hz It can be observedthat the increasing trend of the dynamic stiffness becomesmore notable when the damping coefficient decreased andthe cyclic frequency increased including models with lateralinertia effect or not In a single cycle when it is closer tothe crest or trough the difference of drill string dynamicstiffness is bigger When considering the lateral inertia effectthe absolute value of dynamic stiffness in increasing periodis bigger than that of the model without lateral inertia effectHowever the bigger damping coefficient does notmeanmoreobvious influence in the drill string dynamic stiffness whichindicates that the damping of bottomhole to drill string canweaken the influence of lateral inertia effect on the dynamicstiffness
34 Poissonrsquos Ratio Poissonrsquos ratio is the lateral deformationcoefficient of material which reflects the lateral deformationelastic constants of material In the static force condition
120575Pk1 = 8000
120575998400Pk1 = 8000
120575Pk2 = 10000
120575998400Pk2 = 10000
4
2
0
minus2
minus4
Kr
(Nm
)
32 33 34 35 36 37 38 39 403130f (Hz)
times104
Figure 5 Influence of the damping coefficient on dynamic stiffness
3
4
2
1
0
minus1
minus3
minus2
Kr
(Nm
)
32 33 34 35 36 37 38 39 403130f (Hz)
times104
1 = 01
9984001 = 01
2 = 03
9984002 = 03
Figure 6 Influence of Poissonrsquos ratio on dynamic stiffness
Table 5 Calculated parameters of Poissonrsquos ratio influence
Name of parameters ResultsPoissonrsquos ratios of the drill string ]1015840
1 ]1 010
Poissonrsquos ratios of the drill string ]10158402 ]2 030
Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficient 120575119875119896 (Nsm) 10000
Poissonrsquos ratio change of drill string is very small and it isgenerally taken as ] = 025 However in the dynamic loadcase the dynamic Poissonrsquos ratio is commonly taken as ] =01sim03 According to the analysis above the calculatedparameters analyzing Poissonrsquos ratio are shown in Table 5taking Poissonrsquos ratio as 010 and 030
In Figure 6 the symbol ] denotes Poissonrsquos ratio of modelwithout lateral inertia effect On the other hand the symbol]1015840 denotes Poissonrsquos ratio of model considering lateral inertiaeffect Figure 6 shows the influence of Poissonrsquos ratio on thedrill string dynamic stiffness in the frequency of 30Hz to40Hz It can be seen that the increasing trend of the drillstring dynamic stiffness becomes more obvious when Pois-sonrsquos ratio and cyclic frequency increase both of the modelswith lateral inertia effect or not
Shock and Vibration 7
Table 6 The free vibration natural frequency in different conditions
Boundary conditions Without lateral inertia Lateral inertiaThe first-order natural frequency (rads) 003 004The second-order natural frequency (rads) 005 006The third-order natural frequency (rads) 007 009
However different from other parameters results thechange of the drill string Poissonrsquos ratio has influence on theresults of model considering lateral inertia effect while ithas no influence on the results of model without consideringlateral inertia effect which indicates that the lateral inertiaeffect includes the effect of Poissonrsquos ratio Meanwhile whenconsidering the lateral inertia effect the absolute value ofdynamic stiffness is increasing with the drill string Poissonrsquosratio in the increasing period
35 Natural Frequency Assume that the length of the drillstring is 3500m and the boundary conditions are prescribedby the above as well as the structure parameters of the drillstring The longitudinal vibration model of the drill stringis analyzed with different conditions The first three-ordernatural frequencies of the drill string longitudinal vibrationare shown in Table 6 whether considering the lateral inertiaeffect or not
The calculation results illustrate that the natural fre-quency increases with the increase of frequency order Andwhen there is no inertia it is less than the situation withlateral inertia effect When considering the inertia effect thevibration is coupled with the longitudinal vibration whichwill cause the vibration difference According to adjusting thedrilling parameters the drill string resonance can be avoidedand make drilling optimization possible
4 Conclusion
According to the drilling field conditions the analysis modelof drill string longitudinal vibration is established with thelateral inertia effect and the Laplace transform is used forthe complex impedance solution of drill string bottom Byusing the established equations and analyzing the resultsof numerical examples the influence of design parameterson drill string dynamic stiffness is discussed including theresults of models considering the lateral inertia effect or notWithin the frequency range of drilling field conditions theseconclusions can be drawn
(1) The cyclical transformation amplitude of drill stringdynamic stiffness whether considering the radialinertia effect or not presents increasing with fre-quency and the length of the drill string Thereforefor resisting drill string longitudinal deformation andexcessive vibration the appropriate length of drillstring should be considered in the well structuredesign
(2) For the drill string dynamic stiffness increasing withthe cross-sectional area combining the relationshipof sectional area and inside and outside radius
the influence of sectional area on drill string dynamicstiffness can provide reference for the design of BHA(Bottomhole Assembly)
(3) With the damping coefficient increasing the ampli-tude of drill string dynamic stiffness is reducingwhich would lead to the vibration force increasingThe results are coincidingwith the drilling actual con-dition Moreover the consequences can provide basicdata for the design of downhole friction reductiontools
(4) To the influence of Poissonrsquos ratio on drill stringdynamic stiffness the results are some differencecomparing with other parameters Based on themodel without considering the lateral inertia effectthe change of Poissonrsquos ratio has no influence onthe results of drill string dynamic stiffness Howeverto the model including the lateral inertia effect theinfluence of Poissonrsquos ratio on drill string dynamicstiffness cannot be ignored According to the drillingfield condition the results are important for drillingdynamics analysis especially in deep and ultradeepdrilling or some new wells which is the basis forimproving the rock-breaking efficiency and ROP
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by Open Fund (OGE201403-05) ofKey Laboratory of Oil amp Gas Equipment Ministry of Edu-cation (Southwest Petroleum University) National NaturalScience Foundation of China (no 51074202 and no 11102173)Major Cultivation Foundation of Sichuan Education Depart-ment (12ZZ003 no 667) and Graduate Student InnovationFund of Southwest Petroleum University (CX2015SY06)
References
[1] J J Bailey and I Finnie ldquoAn analytical study of drill-stringvibrationrdquo Journal of Engineering for Industry vol 82 no 2 pp122ndash127 1960
[2] S A Al-Hiddabi B Samanta and A Seibi ldquoNon-linear controlof torsional and bending vibrations of oilwell drillstringsrdquoJournal of Sound andVibration vol 265 no 2 pp 401ndash415 2003
[3] Q Xue R Wang F Sun and Z Huang ldquoChaotic vibrationanalysis of the bottom rotating drill stringrdquo Shock andVibrationvol 2014 Article ID 429164 8 pages 2014
8 Shock and Vibration
[4] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010
[5] W Wang Q Hu H Liu andW Zhang ldquoAnalysis on longitudi-nal vibration of drill string and the corresponding applicationrdquoJournal of Vibration and Shock vol 6 pp 229ndash233 2011
[6] J Li T Yan X Sun and S Peng ldquoFinite element analysis ondrilling string axial vibration in a crooked holerdquo in Proceedingsof the International Conference on Pipelines and TrenchlessTechnology (ICPTT rsquo12) vol 10 pp 1328ndash1334 Wuhan ChinaOctober 2012
[7] T Yan XWang S Li X Bi andCHan ldquoFinite element analysisof longitudinal and lateral coupled vibration of drillstringrdquo OilField Equipment vol 3 p 12 2012
[8] X Yan J Sun M Zhang et al ldquoAnalysis of longitudinalvibration of drill strings using equivalent networkrdquo Oil FieldEquipment vol 4 p 5 2010
[9] S M Sahebkar M R Ghazavi S E Khadem and M HGhayesh ldquoNonlinear vibration analysis of an axially movingdrillstring system with time dependent axial load and axialvelocity in inclined wellrdquoMechanism and Machine Theory vol46 no 5 pp 743ndash760 2011
[10] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011
[11] T Ding andC Li ldquoAnalysis of coupling axial vibrations betweendrilling fluids and drillstringrdquo Chinese Journal of MechanicalEngineering vol 43 no 9 pp 215ndash219 2007
[12] V I Gulyayev and O I Borshch ldquoFree vibrations of drill stringsin hyper deep vertical bore-wellsrdquo Journal of Petroleum Scienceand Engineering vol 78 no 3-4 pp 759ndash764 2011
[13] C Bu X Li L Sun and B Xia ldquoArithmetic solution for the axialvibration of drill string coupling with a down-the-hole hammerin rock drillingrdquo Journal of Vibration and Control 2014
[14] Y A Khulief F A Al-Sulaiman and S Bashmal ldquoVibrationanalysis of drillstrings with self-excited stick-slip oscillationsrdquoJournal of Sound and Vibration vol 299 no 3 pp 540ndash5582007
[15] W-L Wang Q-A Hu H-W Liu and W Zhang ldquoAnalysison longitudinal vibration of drill string and the correspondingapplicationrdquo Journal of Vibration and Shock vol 30 no 6 pp229ndash233 2011
[16] A A Jafari R Kazemi and M F Mahyari ldquoThe effects ofdrilling mud and weight bit on stability and vibration of a drillstringrdquo Journal of Vibration and Acoustics Transactions of theASME vol 134 no 1 Article ID 011014 2012
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Shock and Vibration
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Shock and Vibration 3
the density119860119896 represents the cross-sectional area of 119896th drillstring segment and the expression is given by
119860119896 = 120587 (1199032
119875119896minus 11990310158402
119875119896) (2)
According to the real working condition of the drillingthe relationship between VV and V119891 is V119891 ≪ VV and themovement of the drill string is always to the right So thecalculation formula should be denoted as
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587120591119875119896minus119900119903119875119896 minus 2120587120591119875119896minus119894119903
1015840
119875119896
= 120588119875119896119860119896
1205972119908119896
1205971199052
(3)
To the shear stress of drilling fluid inside and outside 119896thdrill string 120591119875119896minus119894 120591119875119896minus119900 the expressions can be written as
120591119875119896minus119894 = 119870119896minus119894119908119896 (119911 119905)
120591119875119896minus119900 = 119870119896minus119900119908119896 (119911 119905)
(4)
where 119870119896minus119894 119870119896minus119900 are the longitudinal shear stiffness ofdrilling fluid at the position of inside and outside 119896th drillstring segment With applied Laplace transform of (1) andconsistent with the assumption conditions the followingequation can be obtained
119864119875119896119860119896
1205972119882119896
1205971199112
minus [1205881198751198961198601198961199042minus 2120587119870119896minus119900119903119875119896 + 2120587119870119896minus119894119903
1015840
119875119896]119882119896 = 0
(5)
in which the Laplace transform with respect to time of 119908119896 =119908119896(119911 119905) is represented by119882119896 = 119882119896(119911 119904) and the transformrelation is
119882119896 (119911 119904) = 119871 [119908119896 (119911 119905)] = int
+infin
0
119908119896 (119911 119905) 119890minus119904119905119889119905 (6)
The definition of symbol 120572 is as follows
120572 = radic120588119875119896119860119896119904
2minus 2120587 (119870119896minus119900119903119875119896 minus 119870119896minus119894119903
1015840
119875119896)
119864119875119896119860119896
(7)
Then the general solution of119882119896(119911 119904) can be gained as thefollowing form
119882119896 (119911 119904) = 119862119896119890120572119911+ 119863119896119890
minus120572119911 (8)
where 119862119896 119863119896 denote undetermined coefficients determinedby boundary conditions respectively
During oil and gas field drilling process the vibration ofdrill bit is caused partly because of the uneven bottomholeAssuming that the force of bottomhole to the drill bit is119891119896(119911 119905) according to the transitivity of force the force on thebottom of the drill string can be also denoted as 119891119896(119911 119905) Sothe bottom and top of the drill string can be described as
119864119875119896119860119896
120597119882119896 (119911 119904)
120597119911
10038161003816100381610038161003816100381610038161003816119911=0= 119891119896 (119911 119904) (9)
119864119875119896
120597119882119896 (119911 119904)
120597119911
10038161003816100381610038161003816100381610038161003816119911=119867= minus (119896119875119896 + 120575119875119896119904)119882119896 (119911 119904)
1003816100381610038161003816119911=119867 (10)
where the Laplace transform with respect to time of 119891119896(119911 119905)is represented by 119891119896(119911 119904) Furthermore with (8) and (9)substituted into (5) the following equations can be obtained
119862119896 minus 119863119896 =119891119896 (119911 119904)
120572119864119875119896119860119896
119862119896 =119864119875119896120572 minus (119896119875119896 + 120575119875119896119904)
119864119875119896120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867
119863119896
(11)
Here defining the symbol 120574 its expression is given by
120574 =119864119875119896120572 minus (119896119875119896 + 120575119875119896119904)
119864119875119896120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867
(12)
Then taking the results of 119862119896 119863119896 and definition of 120574into (7) the displacement of the drill string 119882119896(119911 119904) can beobtained as follows
119882119896 (119911 119904) =119891119896 (119911 119904)
(120574 minus 1) 120572119864119875119896119860119896
(120574119890120572119911+ 119890minus120572119911) (13)
Moreover assume that
119904 = 119894120596 (14)
Then the Laplace transform is equivalent to the unilateralFourier transform so the response of displacement frequencycan be expressed as119882119896(119911 119894119908)The complex impedance of thedrill string is derived as follows
119870119875119889 =120572119860119896 (120574 minus 1) 119864119875119896
120574 + 1 (15)
The complex impedance of the drill string bottom is equalto the complex stiffness which can be expressed in the pluralform as follows
119870119875119889 = 119870119903 + 119894119862119894 (16)
in which the plural real component 119870119903 denotes the realdynamic stiffness which reflects the drill string ability oflongitudinal deformation resistance In other words whenthe drill string is pressed by the vibration force the valueof the dynamic stiffness has close association with the defor-mation of the drill string On the other hand the imaginarycomponent 119862119894 denotes the dynamic damping which reflectsthe drill string energy dissipation
22 Model of Considering Lateral Inertia Effect Consideringthe influence of lateral inertia effect on drill string vibrationcharacteristics with actual situation the vibration of the 119896th
4 Shock and Vibration
drill string segment can be described by the theory ofRayleigh-Love rod as follows
119864119875119896119860119896
1205972119908119896
1205971199112minus 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 gt VV V119891 gt 0)
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 lt VV V119891 gt 0)
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 gt VV V119891 lt 0)
119864119875119896119860119896
1205972119908119896
1205971199112minus 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 lt VV V119891 lt 0)
(17)
where ]119875119896 denotes Poissonrsquos ratio of the 119896th drill stringsegment It is also worth noticing that both coupling termsare proportional to Poissonrsquos ratio This is reasonable sincethis ratio describes the radial contraction of a rod subjectedto axial strain According to the drilling field situation theanalysis condition can be obtained as V119891 minus VV ≪ 0 and V119891 gt 0and the above equation with (16) can be simplified to
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(18)
For obtaining the results of the equation similar to themethod which is used for the solution of model without con-sidering the influence of lateral inertia effect on drill stringvibration considering the initial and continuity conditionsthe vibration equation of the drill string is solved by Laplacetransform and the displacement expression of the drill stringwith considering the influence of lateral inertia effect can beobtained
Here defining symbols 120572 120573 and 120574 their expressions aregiven as
120572 = radic120588119875119896119860119896119904
2minus 2120587 (119870119896minus119900119903119875119896 minus 119870119896minus119894119903
1015840
119875119896)
119864119875119896119860119896 + 120588119875119896119860119896]2119875119896 (1199032
119875119896minus 11990310158402119875119896) 1199042
120573 = 120588119875119896]2
119875119896(1199032
119875119896minus 11990310158402
119875119896) 1199042
120574 =(119864119875119896 + 120573) 120572 minus (119896119875119896 + 120575119875119896119904)
(119864119875119896 + 120573) 120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867
(19)
Then the displacement expression of the drill string canbe obtained as
119882119896 (119911 119904) =119891119896 (119911 119904)
(120574 minus 1) 120572 (119864119875119896119860119896 + 119860119896120573)(120574119890120572119911+ 119890minus120572119911) (20)
Moreover the analytical solution of the complex impe-dance at the bottomhole of drill string can be expressed as
119870119875119889 =120572119860119896 (120574 minus 1) (119864119875119896 + 120573)
120574 + 1 (21)
3 Numerical Example and Analysis ofKey Parameters Influence
For contrastively analyzing the influence of design parame-ters on the characteristics of drill string longitudinal vibra-tion the complex impedance results are discussed by usingthe analysis models established above including the modelswith and without lateral inertia effect In the calculationresults figures the horizontal axis is the vibration frequencyand the longitudinal axis is the dynamic stiffness at thebottom of drill string First of all the material parametersof the drill string and drilling fluid including the densityelastic modulus stiffness coefficient and longitudinal shearstiffness are shown in Table 1
31 The Drill String Length The length of the drill string isalso on behalf of the well depth At the same time to theratio of drill string length and diameter the bigger ratio iscorresponding to the poorer vibration stability of drill stringThe influence of drill string length on its dynamic stiffnessat the bottom is analyzed as in Figure 2 including takingaccount of the lateral inertia effect or not The numericalexample parameters are shown in Table 2 including the drillstring Poissonrsquos ratio length radius and damping coefficientand the drill string length value includes 1800m and 2000m
In Figure 2 the symbol119867 denotes the length of drill stringmodel without considering the lateral inertia effect m Thesymbol ℎ denotes the length of drill string with consideringthe lateral inertia effect mThe results of drill string dynamicstiffness are presented corresponding to frequency of 20Hzto 30Hz including different drill string length and modelsof considering the lateral inertia effect or not The followingconclusions can be obtainedwith the increasing of frequencythe amplitude of drill string dynamic stiffness presentsincreasing trend including the results of different length and
Shock and Vibration 5
Table 1 Material parameters of the drilling string
Parameters name ResultsDensity of drill string 120588
119875119896(kgm3) 800000
Elastic modulus of drill string 119864119875119896 (MPa) 206 times 103
Stiffness coefficient 119896119875119896 (Nm) 100 times 105
Longitudinal shear stiffness of drilling fluid inside drill string119870119896minus119894 (Mpam) 03Longitudinal shear stiffness of drilling fluid outside drill string119870119896minus119900 (Mpam) 01
3
2
1
0
minus1
minus2
Kr
(Nm
)
times104
H1 = 1800
h1 = 1800
H2 = 2000
h2 = 2000
22 23 24 25 26 27 28 29 3021f (Hz)
Figure 2 Influence of the drill string length on dynamic stiffness
Table 2 Calculation parameters of the length of drill stringinfluence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string1198671 ℎ1 (m) 1800Length of the drill string1198672 ℎ2 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficient 120575119875119896 (Nsm) 10000
analysis models Meanwhile to the lateral inertia effect theinfluence on drill string dynamic stiffness can be almostnegligible when the frequency is low or zero However withthe increasing of frequency such that the frequency is greaterthan 25Hz the influence of lateral inertia effect becomesmore obvious To the results of models considering lateralinertia effect or not in the increasing period of dynamicstiffness the absolute value of considering lateral inertia effectis greater than that of ignoring lateral inertia effect To theresults of drill string different lengths the influence of lateralinertia effect is bigger when the drill string is shorter
32 The Inside and Outside Radius The inside and outsideradius of drill string are determined by the size specificationThe influence of radius change on the dynamic stiffness isanalyzed as in Figure 3 including the models of consideringthe lateral inertia effect or not As before the numericalexample parameters are shown in Table 3 including the drillstring different inside and outside radius value
543210
minus1
minus3minus2
minus4
Kr
(Nm
)
times104
22 23 24 25 26 27 28 29 302120f (Hz)
RPk1 = 01
rPk1 = 01
RPk2 = 011
rPk2 = 011
Figure 3 Influence of outside radius on dynamic stiffness
Table 3 Calculated parameters of the inside and outside radiusinfluence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string H (m) 2000Outside radii of the drill string 1198771198751198961 1199031198751198961 (m) 010Outside radii of the drill string 119877
1198751198962 1199031198751198962
(m) 011Inside radii of the drill string 1198771015840
1198751198961 1199031198751198961 (m) 008
Inside radii of the drill string 11987710158401198751198962
1199031198751198962 (m) 009Damping coefficient 120575119875119896 (Nsm) 10000
In Figures 3 and 4 the symbols 119877119875119896 and 1198771198751198961 denote theradius of drill string model without considering the lateralinertia effect m The symbols 119903119875119896 and 119903
1015840
119875119896denote the radius
of drill string model considering the lateral inertia effectm The results of drill string dynamic stiffness are presentedcorresponding to frequency of 20Hz to 30Hz as shown inFigures 3 and 4 including different inside and outside radiusandmodels with lateral inertia effect or not As before the fol-lowing conclusions can be got with the frequency increasingthe dynamic stiffness amplitude presents increasing trendMeanwhile when outside radius increases or inside radiusreduces equivalent to the area increasing the increasingtrend of the dynamic stiffness becomes more notable Alsoto the results of models with lateral inertia effect or not inthe increasing period of dynamic stiffness the value of modelwith lateral inertia effect is greater than that without lateralinertia effect Meanwhile with bigger difference between thedrill string outside and inside radius the lateral inertia effecthas a more important influence on the dynamic stiffness ofdrill string
6 Shock and Vibration
3
2
1
0
minus1
minus2
Kr
(Nm
)
times104
22 23 24 25 26 27 28 29 302120f (Hz)
R998400Pk1 = 008
r998400Pk1 = 008
R998400Pk2 = 009
r998400Pk2 = 009
Figure 4 Influence of inside radius on dynamic stiffness
Table 4 Calculation parameters of the damping coefficient influ-ence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficients 12057510158401198751198961
1205751198751198961 (Nsm) 8000Damping coefficients 1205751015840
1198751198962 1205751198751198962 (Nsm) 10000
33 The Damping Coefficient The damping coefficient ofthe bottomhole to drill string is influential to the drillstring longitudinal vibration and the lateral inertia effect isdirectly related to the damping coefficient According to thedrilling field conditions the calculation parameters analyzingdamping coefficient are shown in Table 4 taking the dampingcoefficient as 8000 and 10000Nsm
In Figure 5 the symbol 120575119875119896 denotes the damping coeffi-cient ofmodelwithout lateral inertia effectNsmThe symbol1205751015840
119875119896denotes the damping coefficient of model considering
lateral inertia effect Nsm Figure 5 shows the influence of thedamping coefficient on the drill string dynamic stiffness cor-responding to frequency of 30Hz to 40Hz It can be observedthat the increasing trend of the dynamic stiffness becomesmore notable when the damping coefficient decreased andthe cyclic frequency increased including models with lateralinertia effect or not In a single cycle when it is closer tothe crest or trough the difference of drill string dynamicstiffness is bigger When considering the lateral inertia effectthe absolute value of dynamic stiffness in increasing periodis bigger than that of the model without lateral inertia effectHowever the bigger damping coefficient does notmeanmoreobvious influence in the drill string dynamic stiffness whichindicates that the damping of bottomhole to drill string canweaken the influence of lateral inertia effect on the dynamicstiffness
34 Poissonrsquos Ratio Poissonrsquos ratio is the lateral deformationcoefficient of material which reflects the lateral deformationelastic constants of material In the static force condition
120575Pk1 = 8000
120575998400Pk1 = 8000
120575Pk2 = 10000
120575998400Pk2 = 10000
4
2
0
minus2
minus4
Kr
(Nm
)
32 33 34 35 36 37 38 39 403130f (Hz)
times104
Figure 5 Influence of the damping coefficient on dynamic stiffness
3
4
2
1
0
minus1
minus3
minus2
Kr
(Nm
)
32 33 34 35 36 37 38 39 403130f (Hz)
times104
1 = 01
9984001 = 01
2 = 03
9984002 = 03
Figure 6 Influence of Poissonrsquos ratio on dynamic stiffness
Table 5 Calculated parameters of Poissonrsquos ratio influence
Name of parameters ResultsPoissonrsquos ratios of the drill string ]1015840
1 ]1 010
Poissonrsquos ratios of the drill string ]10158402 ]2 030
Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficient 120575119875119896 (Nsm) 10000
Poissonrsquos ratio change of drill string is very small and it isgenerally taken as ] = 025 However in the dynamic loadcase the dynamic Poissonrsquos ratio is commonly taken as ] =01sim03 According to the analysis above the calculatedparameters analyzing Poissonrsquos ratio are shown in Table 5taking Poissonrsquos ratio as 010 and 030
In Figure 6 the symbol ] denotes Poissonrsquos ratio of modelwithout lateral inertia effect On the other hand the symbol]1015840 denotes Poissonrsquos ratio of model considering lateral inertiaeffect Figure 6 shows the influence of Poissonrsquos ratio on thedrill string dynamic stiffness in the frequency of 30Hz to40Hz It can be seen that the increasing trend of the drillstring dynamic stiffness becomes more obvious when Pois-sonrsquos ratio and cyclic frequency increase both of the modelswith lateral inertia effect or not
Shock and Vibration 7
Table 6 The free vibration natural frequency in different conditions
Boundary conditions Without lateral inertia Lateral inertiaThe first-order natural frequency (rads) 003 004The second-order natural frequency (rads) 005 006The third-order natural frequency (rads) 007 009
However different from other parameters results thechange of the drill string Poissonrsquos ratio has influence on theresults of model considering lateral inertia effect while ithas no influence on the results of model without consideringlateral inertia effect which indicates that the lateral inertiaeffect includes the effect of Poissonrsquos ratio Meanwhile whenconsidering the lateral inertia effect the absolute value ofdynamic stiffness is increasing with the drill string Poissonrsquosratio in the increasing period
35 Natural Frequency Assume that the length of the drillstring is 3500m and the boundary conditions are prescribedby the above as well as the structure parameters of the drillstring The longitudinal vibration model of the drill stringis analyzed with different conditions The first three-ordernatural frequencies of the drill string longitudinal vibrationare shown in Table 6 whether considering the lateral inertiaeffect or not
The calculation results illustrate that the natural fre-quency increases with the increase of frequency order Andwhen there is no inertia it is less than the situation withlateral inertia effect When considering the inertia effect thevibration is coupled with the longitudinal vibration whichwill cause the vibration difference According to adjusting thedrilling parameters the drill string resonance can be avoidedand make drilling optimization possible
4 Conclusion
According to the drilling field conditions the analysis modelof drill string longitudinal vibration is established with thelateral inertia effect and the Laplace transform is used forthe complex impedance solution of drill string bottom Byusing the established equations and analyzing the resultsof numerical examples the influence of design parameterson drill string dynamic stiffness is discussed including theresults of models considering the lateral inertia effect or notWithin the frequency range of drilling field conditions theseconclusions can be drawn
(1) The cyclical transformation amplitude of drill stringdynamic stiffness whether considering the radialinertia effect or not presents increasing with fre-quency and the length of the drill string Thereforefor resisting drill string longitudinal deformation andexcessive vibration the appropriate length of drillstring should be considered in the well structuredesign
(2) For the drill string dynamic stiffness increasing withthe cross-sectional area combining the relationshipof sectional area and inside and outside radius
the influence of sectional area on drill string dynamicstiffness can provide reference for the design of BHA(Bottomhole Assembly)
(3) With the damping coefficient increasing the ampli-tude of drill string dynamic stiffness is reducingwhich would lead to the vibration force increasingThe results are coincidingwith the drilling actual con-dition Moreover the consequences can provide basicdata for the design of downhole friction reductiontools
(4) To the influence of Poissonrsquos ratio on drill stringdynamic stiffness the results are some differencecomparing with other parameters Based on themodel without considering the lateral inertia effectthe change of Poissonrsquos ratio has no influence onthe results of drill string dynamic stiffness Howeverto the model including the lateral inertia effect theinfluence of Poissonrsquos ratio on drill string dynamicstiffness cannot be ignored According to the drillingfield condition the results are important for drillingdynamics analysis especially in deep and ultradeepdrilling or some new wells which is the basis forimproving the rock-breaking efficiency and ROP
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by Open Fund (OGE201403-05) ofKey Laboratory of Oil amp Gas Equipment Ministry of Edu-cation (Southwest Petroleum University) National NaturalScience Foundation of China (no 51074202 and no 11102173)Major Cultivation Foundation of Sichuan Education Depart-ment (12ZZ003 no 667) and Graduate Student InnovationFund of Southwest Petroleum University (CX2015SY06)
References
[1] J J Bailey and I Finnie ldquoAn analytical study of drill-stringvibrationrdquo Journal of Engineering for Industry vol 82 no 2 pp122ndash127 1960
[2] S A Al-Hiddabi B Samanta and A Seibi ldquoNon-linear controlof torsional and bending vibrations of oilwell drillstringsrdquoJournal of Sound andVibration vol 265 no 2 pp 401ndash415 2003
[3] Q Xue R Wang F Sun and Z Huang ldquoChaotic vibrationanalysis of the bottom rotating drill stringrdquo Shock andVibrationvol 2014 Article ID 429164 8 pages 2014
8 Shock and Vibration
[4] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010
[5] W Wang Q Hu H Liu andW Zhang ldquoAnalysis on longitudi-nal vibration of drill string and the corresponding applicationrdquoJournal of Vibration and Shock vol 6 pp 229ndash233 2011
[6] J Li T Yan X Sun and S Peng ldquoFinite element analysis ondrilling string axial vibration in a crooked holerdquo in Proceedingsof the International Conference on Pipelines and TrenchlessTechnology (ICPTT rsquo12) vol 10 pp 1328ndash1334 Wuhan ChinaOctober 2012
[7] T Yan XWang S Li X Bi andCHan ldquoFinite element analysisof longitudinal and lateral coupled vibration of drillstringrdquo OilField Equipment vol 3 p 12 2012
[8] X Yan J Sun M Zhang et al ldquoAnalysis of longitudinalvibration of drill strings using equivalent networkrdquo Oil FieldEquipment vol 4 p 5 2010
[9] S M Sahebkar M R Ghazavi S E Khadem and M HGhayesh ldquoNonlinear vibration analysis of an axially movingdrillstring system with time dependent axial load and axialvelocity in inclined wellrdquoMechanism and Machine Theory vol46 no 5 pp 743ndash760 2011
[10] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011
[11] T Ding andC Li ldquoAnalysis of coupling axial vibrations betweendrilling fluids and drillstringrdquo Chinese Journal of MechanicalEngineering vol 43 no 9 pp 215ndash219 2007
[12] V I Gulyayev and O I Borshch ldquoFree vibrations of drill stringsin hyper deep vertical bore-wellsrdquo Journal of Petroleum Scienceand Engineering vol 78 no 3-4 pp 759ndash764 2011
[13] C Bu X Li L Sun and B Xia ldquoArithmetic solution for the axialvibration of drill string coupling with a down-the-hole hammerin rock drillingrdquo Journal of Vibration and Control 2014
[14] Y A Khulief F A Al-Sulaiman and S Bashmal ldquoVibrationanalysis of drillstrings with self-excited stick-slip oscillationsrdquoJournal of Sound and Vibration vol 299 no 3 pp 540ndash5582007
[15] W-L Wang Q-A Hu H-W Liu and W Zhang ldquoAnalysison longitudinal vibration of drill string and the correspondingapplicationrdquo Journal of Vibration and Shock vol 30 no 6 pp229ndash233 2011
[16] A A Jafari R Kazemi and M F Mahyari ldquoThe effects ofdrilling mud and weight bit on stability and vibration of a drillstringrdquo Journal of Vibration and Acoustics Transactions of theASME vol 134 no 1 Article ID 011014 2012
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Shock and Vibration
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4 Shock and Vibration
drill string segment can be described by the theory ofRayleigh-Love rod as follows
119864119875119896119860119896
1205972119908119896
1205971199112minus 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 gt VV V119891 gt 0)
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 lt VV V119891 gt 0)
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 + 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 gt VV V119891 lt 0)
119864119875119896119860119896
1205972119908119896
1205971199112minus 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(V119891 lt VV V119891 lt 0)
(17)
where ]119875119896 denotes Poissonrsquos ratio of the 119896th drill stringsegment It is also worth noticing that both coupling termsare proportional to Poissonrsquos ratio This is reasonable sincethis ratio describes the radial contraction of a rod subjectedto axial strain According to the drilling field situation theanalysis condition can be obtained as V119891 minus VV ≪ 0 and V119891 gt 0and the above equation with (16) can be simplified to
119864119875119896119860119896
1205972119908119896
1205971199112+ 2120587 (120591119875119896minus119894119903119875119896 minus 120591119875119896minus119900119903
1015840
119875119896)
= 120588119875119896119860119896 (1205972119908119896
1205971199052minus ]2119875119896(1199032
119875119896minus 11990310158402
119875119896)1205974119908119896
12059711991121205971199052)
(18)
For obtaining the results of the equation similar to themethod which is used for the solution of model without con-sidering the influence of lateral inertia effect on drill stringvibration considering the initial and continuity conditionsthe vibration equation of the drill string is solved by Laplacetransform and the displacement expression of the drill stringwith considering the influence of lateral inertia effect can beobtained
Here defining symbols 120572 120573 and 120574 their expressions aregiven as
120572 = radic120588119875119896119860119896119904
2minus 2120587 (119870119896minus119900119903119875119896 minus 119870119896minus119894119903
1015840
119875119896)
119864119875119896119860119896 + 120588119875119896119860119896]2119875119896 (1199032
119875119896minus 11990310158402119875119896) 1199042
120573 = 120588119875119896]2
119875119896(1199032
119875119896minus 11990310158402
119875119896) 1199042
120574 =(119864119875119896 + 120573) 120572 minus (119896119875119896 + 120575119875119896119904)
(119864119875119896 + 120573) 120572 + (119896119875119896 + 120575119875119896119904)119890minus2120572119867
(19)
Then the displacement expression of the drill string canbe obtained as
119882119896 (119911 119904) =119891119896 (119911 119904)
(120574 minus 1) 120572 (119864119875119896119860119896 + 119860119896120573)(120574119890120572119911+ 119890minus120572119911) (20)
Moreover the analytical solution of the complex impe-dance at the bottomhole of drill string can be expressed as
119870119875119889 =120572119860119896 (120574 minus 1) (119864119875119896 + 120573)
120574 + 1 (21)
3 Numerical Example and Analysis ofKey Parameters Influence
For contrastively analyzing the influence of design parame-ters on the characteristics of drill string longitudinal vibra-tion the complex impedance results are discussed by usingthe analysis models established above including the modelswith and without lateral inertia effect In the calculationresults figures the horizontal axis is the vibration frequencyand the longitudinal axis is the dynamic stiffness at thebottom of drill string First of all the material parametersof the drill string and drilling fluid including the densityelastic modulus stiffness coefficient and longitudinal shearstiffness are shown in Table 1
31 The Drill String Length The length of the drill string isalso on behalf of the well depth At the same time to theratio of drill string length and diameter the bigger ratio iscorresponding to the poorer vibration stability of drill stringThe influence of drill string length on its dynamic stiffnessat the bottom is analyzed as in Figure 2 including takingaccount of the lateral inertia effect or not The numericalexample parameters are shown in Table 2 including the drillstring Poissonrsquos ratio length radius and damping coefficientand the drill string length value includes 1800m and 2000m
In Figure 2 the symbol119867 denotes the length of drill stringmodel without considering the lateral inertia effect m Thesymbol ℎ denotes the length of drill string with consideringthe lateral inertia effect mThe results of drill string dynamicstiffness are presented corresponding to frequency of 20Hzto 30Hz including different drill string length and modelsof considering the lateral inertia effect or not The followingconclusions can be obtainedwith the increasing of frequencythe amplitude of drill string dynamic stiffness presentsincreasing trend including the results of different length and
Shock and Vibration 5
Table 1 Material parameters of the drilling string
Parameters name ResultsDensity of drill string 120588
119875119896(kgm3) 800000
Elastic modulus of drill string 119864119875119896 (MPa) 206 times 103
Stiffness coefficient 119896119875119896 (Nm) 100 times 105
Longitudinal shear stiffness of drilling fluid inside drill string119870119896minus119894 (Mpam) 03Longitudinal shear stiffness of drilling fluid outside drill string119870119896minus119900 (Mpam) 01
3
2
1
0
minus1
minus2
Kr
(Nm
)
times104
H1 = 1800
h1 = 1800
H2 = 2000
h2 = 2000
22 23 24 25 26 27 28 29 3021f (Hz)
Figure 2 Influence of the drill string length on dynamic stiffness
Table 2 Calculation parameters of the length of drill stringinfluence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string1198671 ℎ1 (m) 1800Length of the drill string1198672 ℎ2 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficient 120575119875119896 (Nsm) 10000
analysis models Meanwhile to the lateral inertia effect theinfluence on drill string dynamic stiffness can be almostnegligible when the frequency is low or zero However withthe increasing of frequency such that the frequency is greaterthan 25Hz the influence of lateral inertia effect becomesmore obvious To the results of models considering lateralinertia effect or not in the increasing period of dynamicstiffness the absolute value of considering lateral inertia effectis greater than that of ignoring lateral inertia effect To theresults of drill string different lengths the influence of lateralinertia effect is bigger when the drill string is shorter
32 The Inside and Outside Radius The inside and outsideradius of drill string are determined by the size specificationThe influence of radius change on the dynamic stiffness isanalyzed as in Figure 3 including the models of consideringthe lateral inertia effect or not As before the numericalexample parameters are shown in Table 3 including the drillstring different inside and outside radius value
543210
minus1
minus3minus2
minus4
Kr
(Nm
)
times104
22 23 24 25 26 27 28 29 302120f (Hz)
RPk1 = 01
rPk1 = 01
RPk2 = 011
rPk2 = 011
Figure 3 Influence of outside radius on dynamic stiffness
Table 3 Calculated parameters of the inside and outside radiusinfluence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string H (m) 2000Outside radii of the drill string 1198771198751198961 1199031198751198961 (m) 010Outside radii of the drill string 119877
1198751198962 1199031198751198962
(m) 011Inside radii of the drill string 1198771015840
1198751198961 1199031198751198961 (m) 008
Inside radii of the drill string 11987710158401198751198962
1199031198751198962 (m) 009Damping coefficient 120575119875119896 (Nsm) 10000
In Figures 3 and 4 the symbols 119877119875119896 and 1198771198751198961 denote theradius of drill string model without considering the lateralinertia effect m The symbols 119903119875119896 and 119903
1015840
119875119896denote the radius
of drill string model considering the lateral inertia effectm The results of drill string dynamic stiffness are presentedcorresponding to frequency of 20Hz to 30Hz as shown inFigures 3 and 4 including different inside and outside radiusandmodels with lateral inertia effect or not As before the fol-lowing conclusions can be got with the frequency increasingthe dynamic stiffness amplitude presents increasing trendMeanwhile when outside radius increases or inside radiusreduces equivalent to the area increasing the increasingtrend of the dynamic stiffness becomes more notable Alsoto the results of models with lateral inertia effect or not inthe increasing period of dynamic stiffness the value of modelwith lateral inertia effect is greater than that without lateralinertia effect Meanwhile with bigger difference between thedrill string outside and inside radius the lateral inertia effecthas a more important influence on the dynamic stiffness ofdrill string
6 Shock and Vibration
3
2
1
0
minus1
minus2
Kr
(Nm
)
times104
22 23 24 25 26 27 28 29 302120f (Hz)
R998400Pk1 = 008
r998400Pk1 = 008
R998400Pk2 = 009
r998400Pk2 = 009
Figure 4 Influence of inside radius on dynamic stiffness
Table 4 Calculation parameters of the damping coefficient influ-ence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficients 12057510158401198751198961
1205751198751198961 (Nsm) 8000Damping coefficients 1205751015840
1198751198962 1205751198751198962 (Nsm) 10000
33 The Damping Coefficient The damping coefficient ofthe bottomhole to drill string is influential to the drillstring longitudinal vibration and the lateral inertia effect isdirectly related to the damping coefficient According to thedrilling field conditions the calculation parameters analyzingdamping coefficient are shown in Table 4 taking the dampingcoefficient as 8000 and 10000Nsm
In Figure 5 the symbol 120575119875119896 denotes the damping coeffi-cient ofmodelwithout lateral inertia effectNsmThe symbol1205751015840
119875119896denotes the damping coefficient of model considering
lateral inertia effect Nsm Figure 5 shows the influence of thedamping coefficient on the drill string dynamic stiffness cor-responding to frequency of 30Hz to 40Hz It can be observedthat the increasing trend of the dynamic stiffness becomesmore notable when the damping coefficient decreased andthe cyclic frequency increased including models with lateralinertia effect or not In a single cycle when it is closer tothe crest or trough the difference of drill string dynamicstiffness is bigger When considering the lateral inertia effectthe absolute value of dynamic stiffness in increasing periodis bigger than that of the model without lateral inertia effectHowever the bigger damping coefficient does notmeanmoreobvious influence in the drill string dynamic stiffness whichindicates that the damping of bottomhole to drill string canweaken the influence of lateral inertia effect on the dynamicstiffness
34 Poissonrsquos Ratio Poissonrsquos ratio is the lateral deformationcoefficient of material which reflects the lateral deformationelastic constants of material In the static force condition
120575Pk1 = 8000
120575998400Pk1 = 8000
120575Pk2 = 10000
120575998400Pk2 = 10000
4
2
0
minus2
minus4
Kr
(Nm
)
32 33 34 35 36 37 38 39 403130f (Hz)
times104
Figure 5 Influence of the damping coefficient on dynamic stiffness
3
4
2
1
0
minus1
minus3
minus2
Kr
(Nm
)
32 33 34 35 36 37 38 39 403130f (Hz)
times104
1 = 01
9984001 = 01
2 = 03
9984002 = 03
Figure 6 Influence of Poissonrsquos ratio on dynamic stiffness
Table 5 Calculated parameters of Poissonrsquos ratio influence
Name of parameters ResultsPoissonrsquos ratios of the drill string ]1015840
1 ]1 010
Poissonrsquos ratios of the drill string ]10158402 ]2 030
Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficient 120575119875119896 (Nsm) 10000
Poissonrsquos ratio change of drill string is very small and it isgenerally taken as ] = 025 However in the dynamic loadcase the dynamic Poissonrsquos ratio is commonly taken as ] =01sim03 According to the analysis above the calculatedparameters analyzing Poissonrsquos ratio are shown in Table 5taking Poissonrsquos ratio as 010 and 030
In Figure 6 the symbol ] denotes Poissonrsquos ratio of modelwithout lateral inertia effect On the other hand the symbol]1015840 denotes Poissonrsquos ratio of model considering lateral inertiaeffect Figure 6 shows the influence of Poissonrsquos ratio on thedrill string dynamic stiffness in the frequency of 30Hz to40Hz It can be seen that the increasing trend of the drillstring dynamic stiffness becomes more obvious when Pois-sonrsquos ratio and cyclic frequency increase both of the modelswith lateral inertia effect or not
Shock and Vibration 7
Table 6 The free vibration natural frequency in different conditions
Boundary conditions Without lateral inertia Lateral inertiaThe first-order natural frequency (rads) 003 004The second-order natural frequency (rads) 005 006The third-order natural frequency (rads) 007 009
However different from other parameters results thechange of the drill string Poissonrsquos ratio has influence on theresults of model considering lateral inertia effect while ithas no influence on the results of model without consideringlateral inertia effect which indicates that the lateral inertiaeffect includes the effect of Poissonrsquos ratio Meanwhile whenconsidering the lateral inertia effect the absolute value ofdynamic stiffness is increasing with the drill string Poissonrsquosratio in the increasing period
35 Natural Frequency Assume that the length of the drillstring is 3500m and the boundary conditions are prescribedby the above as well as the structure parameters of the drillstring The longitudinal vibration model of the drill stringis analyzed with different conditions The first three-ordernatural frequencies of the drill string longitudinal vibrationare shown in Table 6 whether considering the lateral inertiaeffect or not
The calculation results illustrate that the natural fre-quency increases with the increase of frequency order Andwhen there is no inertia it is less than the situation withlateral inertia effect When considering the inertia effect thevibration is coupled with the longitudinal vibration whichwill cause the vibration difference According to adjusting thedrilling parameters the drill string resonance can be avoidedand make drilling optimization possible
4 Conclusion
According to the drilling field conditions the analysis modelof drill string longitudinal vibration is established with thelateral inertia effect and the Laplace transform is used forthe complex impedance solution of drill string bottom Byusing the established equations and analyzing the resultsof numerical examples the influence of design parameterson drill string dynamic stiffness is discussed including theresults of models considering the lateral inertia effect or notWithin the frequency range of drilling field conditions theseconclusions can be drawn
(1) The cyclical transformation amplitude of drill stringdynamic stiffness whether considering the radialinertia effect or not presents increasing with fre-quency and the length of the drill string Thereforefor resisting drill string longitudinal deformation andexcessive vibration the appropriate length of drillstring should be considered in the well structuredesign
(2) For the drill string dynamic stiffness increasing withthe cross-sectional area combining the relationshipof sectional area and inside and outside radius
the influence of sectional area on drill string dynamicstiffness can provide reference for the design of BHA(Bottomhole Assembly)
(3) With the damping coefficient increasing the ampli-tude of drill string dynamic stiffness is reducingwhich would lead to the vibration force increasingThe results are coincidingwith the drilling actual con-dition Moreover the consequences can provide basicdata for the design of downhole friction reductiontools
(4) To the influence of Poissonrsquos ratio on drill stringdynamic stiffness the results are some differencecomparing with other parameters Based on themodel without considering the lateral inertia effectthe change of Poissonrsquos ratio has no influence onthe results of drill string dynamic stiffness Howeverto the model including the lateral inertia effect theinfluence of Poissonrsquos ratio on drill string dynamicstiffness cannot be ignored According to the drillingfield condition the results are important for drillingdynamics analysis especially in deep and ultradeepdrilling or some new wells which is the basis forimproving the rock-breaking efficiency and ROP
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by Open Fund (OGE201403-05) ofKey Laboratory of Oil amp Gas Equipment Ministry of Edu-cation (Southwest Petroleum University) National NaturalScience Foundation of China (no 51074202 and no 11102173)Major Cultivation Foundation of Sichuan Education Depart-ment (12ZZ003 no 667) and Graduate Student InnovationFund of Southwest Petroleum University (CX2015SY06)
References
[1] J J Bailey and I Finnie ldquoAn analytical study of drill-stringvibrationrdquo Journal of Engineering for Industry vol 82 no 2 pp122ndash127 1960
[2] S A Al-Hiddabi B Samanta and A Seibi ldquoNon-linear controlof torsional and bending vibrations of oilwell drillstringsrdquoJournal of Sound andVibration vol 265 no 2 pp 401ndash415 2003
[3] Q Xue R Wang F Sun and Z Huang ldquoChaotic vibrationanalysis of the bottom rotating drill stringrdquo Shock andVibrationvol 2014 Article ID 429164 8 pages 2014
8 Shock and Vibration
[4] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010
[5] W Wang Q Hu H Liu andW Zhang ldquoAnalysis on longitudi-nal vibration of drill string and the corresponding applicationrdquoJournal of Vibration and Shock vol 6 pp 229ndash233 2011
[6] J Li T Yan X Sun and S Peng ldquoFinite element analysis ondrilling string axial vibration in a crooked holerdquo in Proceedingsof the International Conference on Pipelines and TrenchlessTechnology (ICPTT rsquo12) vol 10 pp 1328ndash1334 Wuhan ChinaOctober 2012
[7] T Yan XWang S Li X Bi andCHan ldquoFinite element analysisof longitudinal and lateral coupled vibration of drillstringrdquo OilField Equipment vol 3 p 12 2012
[8] X Yan J Sun M Zhang et al ldquoAnalysis of longitudinalvibration of drill strings using equivalent networkrdquo Oil FieldEquipment vol 4 p 5 2010
[9] S M Sahebkar M R Ghazavi S E Khadem and M HGhayesh ldquoNonlinear vibration analysis of an axially movingdrillstring system with time dependent axial load and axialvelocity in inclined wellrdquoMechanism and Machine Theory vol46 no 5 pp 743ndash760 2011
[10] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011
[11] T Ding andC Li ldquoAnalysis of coupling axial vibrations betweendrilling fluids and drillstringrdquo Chinese Journal of MechanicalEngineering vol 43 no 9 pp 215ndash219 2007
[12] V I Gulyayev and O I Borshch ldquoFree vibrations of drill stringsin hyper deep vertical bore-wellsrdquo Journal of Petroleum Scienceand Engineering vol 78 no 3-4 pp 759ndash764 2011
[13] C Bu X Li L Sun and B Xia ldquoArithmetic solution for the axialvibration of drill string coupling with a down-the-hole hammerin rock drillingrdquo Journal of Vibration and Control 2014
[14] Y A Khulief F A Al-Sulaiman and S Bashmal ldquoVibrationanalysis of drillstrings with self-excited stick-slip oscillationsrdquoJournal of Sound and Vibration vol 299 no 3 pp 540ndash5582007
[15] W-L Wang Q-A Hu H-W Liu and W Zhang ldquoAnalysison longitudinal vibration of drill string and the correspondingapplicationrdquo Journal of Vibration and Shock vol 30 no 6 pp229ndash233 2011
[16] A A Jafari R Kazemi and M F Mahyari ldquoThe effects ofdrilling mud and weight bit on stability and vibration of a drillstringrdquo Journal of Vibration and Acoustics Transactions of theASME vol 134 no 1 Article ID 011014 2012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 5
Table 1 Material parameters of the drilling string
Parameters name ResultsDensity of drill string 120588
119875119896(kgm3) 800000
Elastic modulus of drill string 119864119875119896 (MPa) 206 times 103
Stiffness coefficient 119896119875119896 (Nm) 100 times 105
Longitudinal shear stiffness of drilling fluid inside drill string119870119896minus119894 (Mpam) 03Longitudinal shear stiffness of drilling fluid outside drill string119870119896minus119900 (Mpam) 01
3
2
1
0
minus1
minus2
Kr
(Nm
)
times104
H1 = 1800
h1 = 1800
H2 = 2000
h2 = 2000
22 23 24 25 26 27 28 29 3021f (Hz)
Figure 2 Influence of the drill string length on dynamic stiffness
Table 2 Calculation parameters of the length of drill stringinfluence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string1198671 ℎ1 (m) 1800Length of the drill string1198672 ℎ2 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficient 120575119875119896 (Nsm) 10000
analysis models Meanwhile to the lateral inertia effect theinfluence on drill string dynamic stiffness can be almostnegligible when the frequency is low or zero However withthe increasing of frequency such that the frequency is greaterthan 25Hz the influence of lateral inertia effect becomesmore obvious To the results of models considering lateralinertia effect or not in the increasing period of dynamicstiffness the absolute value of considering lateral inertia effectis greater than that of ignoring lateral inertia effect To theresults of drill string different lengths the influence of lateralinertia effect is bigger when the drill string is shorter
32 The Inside and Outside Radius The inside and outsideradius of drill string are determined by the size specificationThe influence of radius change on the dynamic stiffness isanalyzed as in Figure 3 including the models of consideringthe lateral inertia effect or not As before the numericalexample parameters are shown in Table 3 including the drillstring different inside and outside radius value
543210
minus1
minus3minus2
minus4
Kr
(Nm
)
times104
22 23 24 25 26 27 28 29 302120f (Hz)
RPk1 = 01
rPk1 = 01
RPk2 = 011
rPk2 = 011
Figure 3 Influence of outside radius on dynamic stiffness
Table 3 Calculated parameters of the inside and outside radiusinfluence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string H (m) 2000Outside radii of the drill string 1198771198751198961 1199031198751198961 (m) 010Outside radii of the drill string 119877
1198751198962 1199031198751198962
(m) 011Inside radii of the drill string 1198771015840
1198751198961 1199031198751198961 (m) 008
Inside radii of the drill string 11987710158401198751198962
1199031198751198962 (m) 009Damping coefficient 120575119875119896 (Nsm) 10000
In Figures 3 and 4 the symbols 119877119875119896 and 1198771198751198961 denote theradius of drill string model without considering the lateralinertia effect m The symbols 119903119875119896 and 119903
1015840
119875119896denote the radius
of drill string model considering the lateral inertia effectm The results of drill string dynamic stiffness are presentedcorresponding to frequency of 20Hz to 30Hz as shown inFigures 3 and 4 including different inside and outside radiusandmodels with lateral inertia effect or not As before the fol-lowing conclusions can be got with the frequency increasingthe dynamic stiffness amplitude presents increasing trendMeanwhile when outside radius increases or inside radiusreduces equivalent to the area increasing the increasingtrend of the dynamic stiffness becomes more notable Alsoto the results of models with lateral inertia effect or not inthe increasing period of dynamic stiffness the value of modelwith lateral inertia effect is greater than that without lateralinertia effect Meanwhile with bigger difference between thedrill string outside and inside radius the lateral inertia effecthas a more important influence on the dynamic stiffness ofdrill string
6 Shock and Vibration
3
2
1
0
minus1
minus2
Kr
(Nm
)
times104
22 23 24 25 26 27 28 29 302120f (Hz)
R998400Pk1 = 008
r998400Pk1 = 008
R998400Pk2 = 009
r998400Pk2 = 009
Figure 4 Influence of inside radius on dynamic stiffness
Table 4 Calculation parameters of the damping coefficient influ-ence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficients 12057510158401198751198961
1205751198751198961 (Nsm) 8000Damping coefficients 1205751015840
1198751198962 1205751198751198962 (Nsm) 10000
33 The Damping Coefficient The damping coefficient ofthe bottomhole to drill string is influential to the drillstring longitudinal vibration and the lateral inertia effect isdirectly related to the damping coefficient According to thedrilling field conditions the calculation parameters analyzingdamping coefficient are shown in Table 4 taking the dampingcoefficient as 8000 and 10000Nsm
In Figure 5 the symbol 120575119875119896 denotes the damping coeffi-cient ofmodelwithout lateral inertia effectNsmThe symbol1205751015840
119875119896denotes the damping coefficient of model considering
lateral inertia effect Nsm Figure 5 shows the influence of thedamping coefficient on the drill string dynamic stiffness cor-responding to frequency of 30Hz to 40Hz It can be observedthat the increasing trend of the dynamic stiffness becomesmore notable when the damping coefficient decreased andthe cyclic frequency increased including models with lateralinertia effect or not In a single cycle when it is closer tothe crest or trough the difference of drill string dynamicstiffness is bigger When considering the lateral inertia effectthe absolute value of dynamic stiffness in increasing periodis bigger than that of the model without lateral inertia effectHowever the bigger damping coefficient does notmeanmoreobvious influence in the drill string dynamic stiffness whichindicates that the damping of bottomhole to drill string canweaken the influence of lateral inertia effect on the dynamicstiffness
34 Poissonrsquos Ratio Poissonrsquos ratio is the lateral deformationcoefficient of material which reflects the lateral deformationelastic constants of material In the static force condition
120575Pk1 = 8000
120575998400Pk1 = 8000
120575Pk2 = 10000
120575998400Pk2 = 10000
4
2
0
minus2
minus4
Kr
(Nm
)
32 33 34 35 36 37 38 39 403130f (Hz)
times104
Figure 5 Influence of the damping coefficient on dynamic stiffness
3
4
2
1
0
minus1
minus3
minus2
Kr
(Nm
)
32 33 34 35 36 37 38 39 403130f (Hz)
times104
1 = 01
9984001 = 01
2 = 03
9984002 = 03
Figure 6 Influence of Poissonrsquos ratio on dynamic stiffness
Table 5 Calculated parameters of Poissonrsquos ratio influence
Name of parameters ResultsPoissonrsquos ratios of the drill string ]1015840
1 ]1 010
Poissonrsquos ratios of the drill string ]10158402 ]2 030
Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficient 120575119875119896 (Nsm) 10000
Poissonrsquos ratio change of drill string is very small and it isgenerally taken as ] = 025 However in the dynamic loadcase the dynamic Poissonrsquos ratio is commonly taken as ] =01sim03 According to the analysis above the calculatedparameters analyzing Poissonrsquos ratio are shown in Table 5taking Poissonrsquos ratio as 010 and 030
In Figure 6 the symbol ] denotes Poissonrsquos ratio of modelwithout lateral inertia effect On the other hand the symbol]1015840 denotes Poissonrsquos ratio of model considering lateral inertiaeffect Figure 6 shows the influence of Poissonrsquos ratio on thedrill string dynamic stiffness in the frequency of 30Hz to40Hz It can be seen that the increasing trend of the drillstring dynamic stiffness becomes more obvious when Pois-sonrsquos ratio and cyclic frequency increase both of the modelswith lateral inertia effect or not
Shock and Vibration 7
Table 6 The free vibration natural frequency in different conditions
Boundary conditions Without lateral inertia Lateral inertiaThe first-order natural frequency (rads) 003 004The second-order natural frequency (rads) 005 006The third-order natural frequency (rads) 007 009
However different from other parameters results thechange of the drill string Poissonrsquos ratio has influence on theresults of model considering lateral inertia effect while ithas no influence on the results of model without consideringlateral inertia effect which indicates that the lateral inertiaeffect includes the effect of Poissonrsquos ratio Meanwhile whenconsidering the lateral inertia effect the absolute value ofdynamic stiffness is increasing with the drill string Poissonrsquosratio in the increasing period
35 Natural Frequency Assume that the length of the drillstring is 3500m and the boundary conditions are prescribedby the above as well as the structure parameters of the drillstring The longitudinal vibration model of the drill stringis analyzed with different conditions The first three-ordernatural frequencies of the drill string longitudinal vibrationare shown in Table 6 whether considering the lateral inertiaeffect or not
The calculation results illustrate that the natural fre-quency increases with the increase of frequency order Andwhen there is no inertia it is less than the situation withlateral inertia effect When considering the inertia effect thevibration is coupled with the longitudinal vibration whichwill cause the vibration difference According to adjusting thedrilling parameters the drill string resonance can be avoidedand make drilling optimization possible
4 Conclusion
According to the drilling field conditions the analysis modelof drill string longitudinal vibration is established with thelateral inertia effect and the Laplace transform is used forthe complex impedance solution of drill string bottom Byusing the established equations and analyzing the resultsof numerical examples the influence of design parameterson drill string dynamic stiffness is discussed including theresults of models considering the lateral inertia effect or notWithin the frequency range of drilling field conditions theseconclusions can be drawn
(1) The cyclical transformation amplitude of drill stringdynamic stiffness whether considering the radialinertia effect or not presents increasing with fre-quency and the length of the drill string Thereforefor resisting drill string longitudinal deformation andexcessive vibration the appropriate length of drillstring should be considered in the well structuredesign
(2) For the drill string dynamic stiffness increasing withthe cross-sectional area combining the relationshipof sectional area and inside and outside radius
the influence of sectional area on drill string dynamicstiffness can provide reference for the design of BHA(Bottomhole Assembly)
(3) With the damping coefficient increasing the ampli-tude of drill string dynamic stiffness is reducingwhich would lead to the vibration force increasingThe results are coincidingwith the drilling actual con-dition Moreover the consequences can provide basicdata for the design of downhole friction reductiontools
(4) To the influence of Poissonrsquos ratio on drill stringdynamic stiffness the results are some differencecomparing with other parameters Based on themodel without considering the lateral inertia effectthe change of Poissonrsquos ratio has no influence onthe results of drill string dynamic stiffness Howeverto the model including the lateral inertia effect theinfluence of Poissonrsquos ratio on drill string dynamicstiffness cannot be ignored According to the drillingfield condition the results are important for drillingdynamics analysis especially in deep and ultradeepdrilling or some new wells which is the basis forimproving the rock-breaking efficiency and ROP
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by Open Fund (OGE201403-05) ofKey Laboratory of Oil amp Gas Equipment Ministry of Edu-cation (Southwest Petroleum University) National NaturalScience Foundation of China (no 51074202 and no 11102173)Major Cultivation Foundation of Sichuan Education Depart-ment (12ZZ003 no 667) and Graduate Student InnovationFund of Southwest Petroleum University (CX2015SY06)
References
[1] J J Bailey and I Finnie ldquoAn analytical study of drill-stringvibrationrdquo Journal of Engineering for Industry vol 82 no 2 pp122ndash127 1960
[2] S A Al-Hiddabi B Samanta and A Seibi ldquoNon-linear controlof torsional and bending vibrations of oilwell drillstringsrdquoJournal of Sound andVibration vol 265 no 2 pp 401ndash415 2003
[3] Q Xue R Wang F Sun and Z Huang ldquoChaotic vibrationanalysis of the bottom rotating drill stringrdquo Shock andVibrationvol 2014 Article ID 429164 8 pages 2014
8 Shock and Vibration
[4] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010
[5] W Wang Q Hu H Liu andW Zhang ldquoAnalysis on longitudi-nal vibration of drill string and the corresponding applicationrdquoJournal of Vibration and Shock vol 6 pp 229ndash233 2011
[6] J Li T Yan X Sun and S Peng ldquoFinite element analysis ondrilling string axial vibration in a crooked holerdquo in Proceedingsof the International Conference on Pipelines and TrenchlessTechnology (ICPTT rsquo12) vol 10 pp 1328ndash1334 Wuhan ChinaOctober 2012
[7] T Yan XWang S Li X Bi andCHan ldquoFinite element analysisof longitudinal and lateral coupled vibration of drillstringrdquo OilField Equipment vol 3 p 12 2012
[8] X Yan J Sun M Zhang et al ldquoAnalysis of longitudinalvibration of drill strings using equivalent networkrdquo Oil FieldEquipment vol 4 p 5 2010
[9] S M Sahebkar M R Ghazavi S E Khadem and M HGhayesh ldquoNonlinear vibration analysis of an axially movingdrillstring system with time dependent axial load and axialvelocity in inclined wellrdquoMechanism and Machine Theory vol46 no 5 pp 743ndash760 2011
[10] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011
[11] T Ding andC Li ldquoAnalysis of coupling axial vibrations betweendrilling fluids and drillstringrdquo Chinese Journal of MechanicalEngineering vol 43 no 9 pp 215ndash219 2007
[12] V I Gulyayev and O I Borshch ldquoFree vibrations of drill stringsin hyper deep vertical bore-wellsrdquo Journal of Petroleum Scienceand Engineering vol 78 no 3-4 pp 759ndash764 2011
[13] C Bu X Li L Sun and B Xia ldquoArithmetic solution for the axialvibration of drill string coupling with a down-the-hole hammerin rock drillingrdquo Journal of Vibration and Control 2014
[14] Y A Khulief F A Al-Sulaiman and S Bashmal ldquoVibrationanalysis of drillstrings with self-excited stick-slip oscillationsrdquoJournal of Sound and Vibration vol 299 no 3 pp 540ndash5582007
[15] W-L Wang Q-A Hu H-W Liu and W Zhang ldquoAnalysison longitudinal vibration of drill string and the correspondingapplicationrdquo Journal of Vibration and Shock vol 30 no 6 pp229ndash233 2011
[16] A A Jafari R Kazemi and M F Mahyari ldquoThe effects ofdrilling mud and weight bit on stability and vibration of a drillstringrdquo Journal of Vibration and Acoustics Transactions of theASME vol 134 no 1 Article ID 011014 2012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Shock and Vibration
3
2
1
0
minus1
minus2
Kr
(Nm
)
times104
22 23 24 25 26 27 28 29 302120f (Hz)
R998400Pk1 = 008
r998400Pk1 = 008
R998400Pk2 = 009
r998400Pk2 = 009
Figure 4 Influence of inside radius on dynamic stiffness
Table 4 Calculation parameters of the damping coefficient influ-ence
Name of parameters ResultsPoissonrsquos ratio of the drill string ] 030Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficients 12057510158401198751198961
1205751198751198961 (Nsm) 8000Damping coefficients 1205751015840
1198751198962 1205751198751198962 (Nsm) 10000
33 The Damping Coefficient The damping coefficient ofthe bottomhole to drill string is influential to the drillstring longitudinal vibration and the lateral inertia effect isdirectly related to the damping coefficient According to thedrilling field conditions the calculation parameters analyzingdamping coefficient are shown in Table 4 taking the dampingcoefficient as 8000 and 10000Nsm
In Figure 5 the symbol 120575119875119896 denotes the damping coeffi-cient ofmodelwithout lateral inertia effectNsmThe symbol1205751015840
119875119896denotes the damping coefficient of model considering
lateral inertia effect Nsm Figure 5 shows the influence of thedamping coefficient on the drill string dynamic stiffness cor-responding to frequency of 30Hz to 40Hz It can be observedthat the increasing trend of the dynamic stiffness becomesmore notable when the damping coefficient decreased andthe cyclic frequency increased including models with lateralinertia effect or not In a single cycle when it is closer tothe crest or trough the difference of drill string dynamicstiffness is bigger When considering the lateral inertia effectthe absolute value of dynamic stiffness in increasing periodis bigger than that of the model without lateral inertia effectHowever the bigger damping coefficient does notmeanmoreobvious influence in the drill string dynamic stiffness whichindicates that the damping of bottomhole to drill string canweaken the influence of lateral inertia effect on the dynamicstiffness
34 Poissonrsquos Ratio Poissonrsquos ratio is the lateral deformationcoefficient of material which reflects the lateral deformationelastic constants of material In the static force condition
120575Pk1 = 8000
120575998400Pk1 = 8000
120575Pk2 = 10000
120575998400Pk2 = 10000
4
2
0
minus2
minus4
Kr
(Nm
)
32 33 34 35 36 37 38 39 403130f (Hz)
times104
Figure 5 Influence of the damping coefficient on dynamic stiffness
3
4
2
1
0
minus1
minus3
minus2
Kr
(Nm
)
32 33 34 35 36 37 38 39 403130f (Hz)
times104
1 = 01
9984001 = 01
2 = 03
9984002 = 03
Figure 6 Influence of Poissonrsquos ratio on dynamic stiffness
Table 5 Calculated parameters of Poissonrsquos ratio influence
Name of parameters ResultsPoissonrsquos ratios of the drill string ]1015840
1 ]1 010
Poissonrsquos ratios of the drill string ]10158402 ]2 030
Length of the drill string119867 (m) 2000Outside radius of the drill string 119903119875119896 (m) 010Inside radius of the drill string 1199031015840
119875119896(m) 008
Damping coefficient 120575119875119896 (Nsm) 10000
Poissonrsquos ratio change of drill string is very small and it isgenerally taken as ] = 025 However in the dynamic loadcase the dynamic Poissonrsquos ratio is commonly taken as ] =01sim03 According to the analysis above the calculatedparameters analyzing Poissonrsquos ratio are shown in Table 5taking Poissonrsquos ratio as 010 and 030
In Figure 6 the symbol ] denotes Poissonrsquos ratio of modelwithout lateral inertia effect On the other hand the symbol]1015840 denotes Poissonrsquos ratio of model considering lateral inertiaeffect Figure 6 shows the influence of Poissonrsquos ratio on thedrill string dynamic stiffness in the frequency of 30Hz to40Hz It can be seen that the increasing trend of the drillstring dynamic stiffness becomes more obvious when Pois-sonrsquos ratio and cyclic frequency increase both of the modelswith lateral inertia effect or not
Shock and Vibration 7
Table 6 The free vibration natural frequency in different conditions
Boundary conditions Without lateral inertia Lateral inertiaThe first-order natural frequency (rads) 003 004The second-order natural frequency (rads) 005 006The third-order natural frequency (rads) 007 009
However different from other parameters results thechange of the drill string Poissonrsquos ratio has influence on theresults of model considering lateral inertia effect while ithas no influence on the results of model without consideringlateral inertia effect which indicates that the lateral inertiaeffect includes the effect of Poissonrsquos ratio Meanwhile whenconsidering the lateral inertia effect the absolute value ofdynamic stiffness is increasing with the drill string Poissonrsquosratio in the increasing period
35 Natural Frequency Assume that the length of the drillstring is 3500m and the boundary conditions are prescribedby the above as well as the structure parameters of the drillstring The longitudinal vibration model of the drill stringis analyzed with different conditions The first three-ordernatural frequencies of the drill string longitudinal vibrationare shown in Table 6 whether considering the lateral inertiaeffect or not
The calculation results illustrate that the natural fre-quency increases with the increase of frequency order Andwhen there is no inertia it is less than the situation withlateral inertia effect When considering the inertia effect thevibration is coupled with the longitudinal vibration whichwill cause the vibration difference According to adjusting thedrilling parameters the drill string resonance can be avoidedand make drilling optimization possible
4 Conclusion
According to the drilling field conditions the analysis modelof drill string longitudinal vibration is established with thelateral inertia effect and the Laplace transform is used forthe complex impedance solution of drill string bottom Byusing the established equations and analyzing the resultsof numerical examples the influence of design parameterson drill string dynamic stiffness is discussed including theresults of models considering the lateral inertia effect or notWithin the frequency range of drilling field conditions theseconclusions can be drawn
(1) The cyclical transformation amplitude of drill stringdynamic stiffness whether considering the radialinertia effect or not presents increasing with fre-quency and the length of the drill string Thereforefor resisting drill string longitudinal deformation andexcessive vibration the appropriate length of drillstring should be considered in the well structuredesign
(2) For the drill string dynamic stiffness increasing withthe cross-sectional area combining the relationshipof sectional area and inside and outside radius
the influence of sectional area on drill string dynamicstiffness can provide reference for the design of BHA(Bottomhole Assembly)
(3) With the damping coefficient increasing the ampli-tude of drill string dynamic stiffness is reducingwhich would lead to the vibration force increasingThe results are coincidingwith the drilling actual con-dition Moreover the consequences can provide basicdata for the design of downhole friction reductiontools
(4) To the influence of Poissonrsquos ratio on drill stringdynamic stiffness the results are some differencecomparing with other parameters Based on themodel without considering the lateral inertia effectthe change of Poissonrsquos ratio has no influence onthe results of drill string dynamic stiffness Howeverto the model including the lateral inertia effect theinfluence of Poissonrsquos ratio on drill string dynamicstiffness cannot be ignored According to the drillingfield condition the results are important for drillingdynamics analysis especially in deep and ultradeepdrilling or some new wells which is the basis forimproving the rock-breaking efficiency and ROP
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by Open Fund (OGE201403-05) ofKey Laboratory of Oil amp Gas Equipment Ministry of Edu-cation (Southwest Petroleum University) National NaturalScience Foundation of China (no 51074202 and no 11102173)Major Cultivation Foundation of Sichuan Education Depart-ment (12ZZ003 no 667) and Graduate Student InnovationFund of Southwest Petroleum University (CX2015SY06)
References
[1] J J Bailey and I Finnie ldquoAn analytical study of drill-stringvibrationrdquo Journal of Engineering for Industry vol 82 no 2 pp122ndash127 1960
[2] S A Al-Hiddabi B Samanta and A Seibi ldquoNon-linear controlof torsional and bending vibrations of oilwell drillstringsrdquoJournal of Sound andVibration vol 265 no 2 pp 401ndash415 2003
[3] Q Xue R Wang F Sun and Z Huang ldquoChaotic vibrationanalysis of the bottom rotating drill stringrdquo Shock andVibrationvol 2014 Article ID 429164 8 pages 2014
8 Shock and Vibration
[4] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010
[5] W Wang Q Hu H Liu andW Zhang ldquoAnalysis on longitudi-nal vibration of drill string and the corresponding applicationrdquoJournal of Vibration and Shock vol 6 pp 229ndash233 2011
[6] J Li T Yan X Sun and S Peng ldquoFinite element analysis ondrilling string axial vibration in a crooked holerdquo in Proceedingsof the International Conference on Pipelines and TrenchlessTechnology (ICPTT rsquo12) vol 10 pp 1328ndash1334 Wuhan ChinaOctober 2012
[7] T Yan XWang S Li X Bi andCHan ldquoFinite element analysisof longitudinal and lateral coupled vibration of drillstringrdquo OilField Equipment vol 3 p 12 2012
[8] X Yan J Sun M Zhang et al ldquoAnalysis of longitudinalvibration of drill strings using equivalent networkrdquo Oil FieldEquipment vol 4 p 5 2010
[9] S M Sahebkar M R Ghazavi S E Khadem and M HGhayesh ldquoNonlinear vibration analysis of an axially movingdrillstring system with time dependent axial load and axialvelocity in inclined wellrdquoMechanism and Machine Theory vol46 no 5 pp 743ndash760 2011
[10] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011
[11] T Ding andC Li ldquoAnalysis of coupling axial vibrations betweendrilling fluids and drillstringrdquo Chinese Journal of MechanicalEngineering vol 43 no 9 pp 215ndash219 2007
[12] V I Gulyayev and O I Borshch ldquoFree vibrations of drill stringsin hyper deep vertical bore-wellsrdquo Journal of Petroleum Scienceand Engineering vol 78 no 3-4 pp 759ndash764 2011
[13] C Bu X Li L Sun and B Xia ldquoArithmetic solution for the axialvibration of drill string coupling with a down-the-hole hammerin rock drillingrdquo Journal of Vibration and Control 2014
[14] Y A Khulief F A Al-Sulaiman and S Bashmal ldquoVibrationanalysis of drillstrings with self-excited stick-slip oscillationsrdquoJournal of Sound and Vibration vol 299 no 3 pp 540ndash5582007
[15] W-L Wang Q-A Hu H-W Liu and W Zhang ldquoAnalysison longitudinal vibration of drill string and the correspondingapplicationrdquo Journal of Vibration and Shock vol 30 no 6 pp229ndash233 2011
[16] A A Jafari R Kazemi and M F Mahyari ldquoThe effects ofdrilling mud and weight bit on stability and vibration of a drillstringrdquo Journal of Vibration and Acoustics Transactions of theASME vol 134 no 1 Article ID 011014 2012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 7
Table 6 The free vibration natural frequency in different conditions
Boundary conditions Without lateral inertia Lateral inertiaThe first-order natural frequency (rads) 003 004The second-order natural frequency (rads) 005 006The third-order natural frequency (rads) 007 009
However different from other parameters results thechange of the drill string Poissonrsquos ratio has influence on theresults of model considering lateral inertia effect while ithas no influence on the results of model without consideringlateral inertia effect which indicates that the lateral inertiaeffect includes the effect of Poissonrsquos ratio Meanwhile whenconsidering the lateral inertia effect the absolute value ofdynamic stiffness is increasing with the drill string Poissonrsquosratio in the increasing period
35 Natural Frequency Assume that the length of the drillstring is 3500m and the boundary conditions are prescribedby the above as well as the structure parameters of the drillstring The longitudinal vibration model of the drill stringis analyzed with different conditions The first three-ordernatural frequencies of the drill string longitudinal vibrationare shown in Table 6 whether considering the lateral inertiaeffect or not
The calculation results illustrate that the natural fre-quency increases with the increase of frequency order Andwhen there is no inertia it is less than the situation withlateral inertia effect When considering the inertia effect thevibration is coupled with the longitudinal vibration whichwill cause the vibration difference According to adjusting thedrilling parameters the drill string resonance can be avoidedand make drilling optimization possible
4 Conclusion
According to the drilling field conditions the analysis modelof drill string longitudinal vibration is established with thelateral inertia effect and the Laplace transform is used forthe complex impedance solution of drill string bottom Byusing the established equations and analyzing the resultsof numerical examples the influence of design parameterson drill string dynamic stiffness is discussed including theresults of models considering the lateral inertia effect or notWithin the frequency range of drilling field conditions theseconclusions can be drawn
(1) The cyclical transformation amplitude of drill stringdynamic stiffness whether considering the radialinertia effect or not presents increasing with fre-quency and the length of the drill string Thereforefor resisting drill string longitudinal deformation andexcessive vibration the appropriate length of drillstring should be considered in the well structuredesign
(2) For the drill string dynamic stiffness increasing withthe cross-sectional area combining the relationshipof sectional area and inside and outside radius
the influence of sectional area on drill string dynamicstiffness can provide reference for the design of BHA(Bottomhole Assembly)
(3) With the damping coefficient increasing the ampli-tude of drill string dynamic stiffness is reducingwhich would lead to the vibration force increasingThe results are coincidingwith the drilling actual con-dition Moreover the consequences can provide basicdata for the design of downhole friction reductiontools
(4) To the influence of Poissonrsquos ratio on drill stringdynamic stiffness the results are some differencecomparing with other parameters Based on themodel without considering the lateral inertia effectthe change of Poissonrsquos ratio has no influence onthe results of drill string dynamic stiffness Howeverto the model including the lateral inertia effect theinfluence of Poissonrsquos ratio on drill string dynamicstiffness cannot be ignored According to the drillingfield condition the results are important for drillingdynamics analysis especially in deep and ultradeepdrilling or some new wells which is the basis forimproving the rock-breaking efficiency and ROP
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by Open Fund (OGE201403-05) ofKey Laboratory of Oil amp Gas Equipment Ministry of Edu-cation (Southwest Petroleum University) National NaturalScience Foundation of China (no 51074202 and no 11102173)Major Cultivation Foundation of Sichuan Education Depart-ment (12ZZ003 no 667) and Graduate Student InnovationFund of Southwest Petroleum University (CX2015SY06)
References
[1] J J Bailey and I Finnie ldquoAn analytical study of drill-stringvibrationrdquo Journal of Engineering for Industry vol 82 no 2 pp122ndash127 1960
[2] S A Al-Hiddabi B Samanta and A Seibi ldquoNon-linear controlof torsional and bending vibrations of oilwell drillstringsrdquoJournal of Sound andVibration vol 265 no 2 pp 401ndash415 2003
[3] Q Xue R Wang F Sun and Z Huang ldquoChaotic vibrationanalysis of the bottom rotating drill stringrdquo Shock andVibrationvol 2014 Article ID 429164 8 pages 2014
8 Shock and Vibration
[4] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010
[5] W Wang Q Hu H Liu andW Zhang ldquoAnalysis on longitudi-nal vibration of drill string and the corresponding applicationrdquoJournal of Vibration and Shock vol 6 pp 229ndash233 2011
[6] J Li T Yan X Sun and S Peng ldquoFinite element analysis ondrilling string axial vibration in a crooked holerdquo in Proceedingsof the International Conference on Pipelines and TrenchlessTechnology (ICPTT rsquo12) vol 10 pp 1328ndash1334 Wuhan ChinaOctober 2012
[7] T Yan XWang S Li X Bi andCHan ldquoFinite element analysisof longitudinal and lateral coupled vibration of drillstringrdquo OilField Equipment vol 3 p 12 2012
[8] X Yan J Sun M Zhang et al ldquoAnalysis of longitudinalvibration of drill strings using equivalent networkrdquo Oil FieldEquipment vol 4 p 5 2010
[9] S M Sahebkar M R Ghazavi S E Khadem and M HGhayesh ldquoNonlinear vibration analysis of an axially movingdrillstring system with time dependent axial load and axialvelocity in inclined wellrdquoMechanism and Machine Theory vol46 no 5 pp 743ndash760 2011
[10] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011
[11] T Ding andC Li ldquoAnalysis of coupling axial vibrations betweendrilling fluids and drillstringrdquo Chinese Journal of MechanicalEngineering vol 43 no 9 pp 215ndash219 2007
[12] V I Gulyayev and O I Borshch ldquoFree vibrations of drill stringsin hyper deep vertical bore-wellsrdquo Journal of Petroleum Scienceand Engineering vol 78 no 3-4 pp 759ndash764 2011
[13] C Bu X Li L Sun and B Xia ldquoArithmetic solution for the axialvibration of drill string coupling with a down-the-hole hammerin rock drillingrdquo Journal of Vibration and Control 2014
[14] Y A Khulief F A Al-Sulaiman and S Bashmal ldquoVibrationanalysis of drillstrings with self-excited stick-slip oscillationsrdquoJournal of Sound and Vibration vol 299 no 3 pp 540ndash5582007
[15] W-L Wang Q-A Hu H-W Liu and W Zhang ldquoAnalysison longitudinal vibration of drill string and the correspondingapplicationrdquo Journal of Vibration and Shock vol 30 no 6 pp229ndash233 2011
[16] A A Jafari R Kazemi and M F Mahyari ldquoThe effects ofdrilling mud and weight bit on stability and vibration of a drillstringrdquo Journal of Vibration and Acoustics Transactions of theASME vol 134 no 1 Article ID 011014 2012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 Shock and Vibration
[4] H Hakimi and S Moradi ldquoDrillstring vibration analysis usingdifferential quadrature methodrdquo Journal of Petroleum Scienceand Engineering vol 70 no 3-4 pp 235ndash242 2010
[5] W Wang Q Hu H Liu andW Zhang ldquoAnalysis on longitudi-nal vibration of drill string and the corresponding applicationrdquoJournal of Vibration and Shock vol 6 pp 229ndash233 2011
[6] J Li T Yan X Sun and S Peng ldquoFinite element analysis ondrilling string axial vibration in a crooked holerdquo in Proceedingsof the International Conference on Pipelines and TrenchlessTechnology (ICPTT rsquo12) vol 10 pp 1328ndash1334 Wuhan ChinaOctober 2012
[7] T Yan XWang S Li X Bi andCHan ldquoFinite element analysisof longitudinal and lateral coupled vibration of drillstringrdquo OilField Equipment vol 3 p 12 2012
[8] X Yan J Sun M Zhang et al ldquoAnalysis of longitudinalvibration of drill strings using equivalent networkrdquo Oil FieldEquipment vol 4 p 5 2010
[9] S M Sahebkar M R Ghazavi S E Khadem and M HGhayesh ldquoNonlinear vibration analysis of an axially movingdrillstring system with time dependent axial load and axialvelocity in inclined wellrdquoMechanism and Machine Theory vol46 no 5 pp 743ndash760 2011
[10] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011
[11] T Ding andC Li ldquoAnalysis of coupling axial vibrations betweendrilling fluids and drillstringrdquo Chinese Journal of MechanicalEngineering vol 43 no 9 pp 215ndash219 2007
[12] V I Gulyayev and O I Borshch ldquoFree vibrations of drill stringsin hyper deep vertical bore-wellsrdquo Journal of Petroleum Scienceand Engineering vol 78 no 3-4 pp 759ndash764 2011
[13] C Bu X Li L Sun and B Xia ldquoArithmetic solution for the axialvibration of drill string coupling with a down-the-hole hammerin rock drillingrdquo Journal of Vibration and Control 2014
[14] Y A Khulief F A Al-Sulaiman and S Bashmal ldquoVibrationanalysis of drillstrings with self-excited stick-slip oscillationsrdquoJournal of Sound and Vibration vol 299 no 3 pp 540ndash5582007
[15] W-L Wang Q-A Hu H-W Liu and W Zhang ldquoAnalysison longitudinal vibration of drill string and the correspondingapplicationrdquo Journal of Vibration and Shock vol 30 no 6 pp229ndash233 2011
[16] A A Jafari R Kazemi and M F Mahyari ldquoThe effects ofdrilling mud and weight bit on stability and vibration of a drillstringrdquo Journal of Vibration and Acoustics Transactions of theASME vol 134 no 1 Article ID 011014 2012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of