Analysis of PDC Wobbling & DS Buckling

8/2/2019 Analysis of PDC Wobbling & DS Buckling

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ANALYSIS OF PDC BIT WOBBLING AND DRILLINGSTRING BUCKLING

byRAMKAMAL BHAGAVATULA, B.Tech., M.S.M.E.

A THESISIN

PETROLEUM ENGINEERINGSubmitted to the Graduate Faculty

of Texas Tech University inPartial Fulfillment ofthe Requirements forthe Degree ofMASTER OF SCIENCE

INPETROLEUM ENGINEERING

Approved

Chairperbn of the Co Hte

Accepted

Dean of the Graduate SchoolMay, 2004


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Copyright 200 4, Ramkamal Bhagavatula


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ACKNOWLEDGEMENTSI would hke to express my sincere thanks and appreciation to my academic

advisor Dr. Lloyd Heinze and graduate advisor Dr. Akanni Lawal for their excellentguidance and constant support during my graduate studies, without which my effortswould not have been complete and fruitful. I would also hke to express my thanks to Dr.Paulus A disoem arta and D r. Akanni Lawal for their time and effort in reviewing thismanuscript and making valuable suggestions as my thesis comm ittee m embers.

I am ex tremely grateflil to m y parents and in-laws for their support and co nstantencouragement throughout my academic career. I thank my wife, Saroja for being withme against all odds and providing the moral support to pursue higher studies. I wouldalso like to thank D r. J. F. Lea, all the facuhy me mb ers, and my friends w ho have helpedme directly or indirectly during my graduate studies.

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TABLE OF CONTENTSACKNOWLEDGEMENTS i iABSTRACT vLIST OF TABLES viLIST OF FIGUR ES viiCHAPTERI. INTRODUC TION 1

1.1 PDC Bit WobbUng 11.2 DrilUng String BuckUng 41.3 Research Objective and Scope of Study 6

II. CUTTING FORCES ON A PDC BIT 82.1 Orthogonal Cutting Principle 82.2 Emst-Merchant Minimum Energy Criterion 102.3 PDC Bit Force Evaluation 13

III. BIT WOB BLING MODEL 153.1 Strain Energies in a Stressed Bod y 153.2 CastigUano 's theorem and M axweU 's reciprocal theorem 213.3 Forces acting upon Bit and Bottom Hole Assem bly (BHA ) 253.4 Elastic Constants of Bottom Hole Assem bly 31

3.4.1 Static Conditions 313.4.2 Dynam ic Cond itions 37

3.5 Form ulation of Bit Vibrations 413.5.1 Lateral Vibrations 413.5.2 Ang ular Vibrations 47IV. BIT WO BBLING-CALCU LATIONS, RESULTS AND DISCUSSION 50

4.1 Exam ple Calculations 504.2 Results and Discussion 55

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V. DRILL-STRING BUCK LING: GENERALIZED ANALY TICAL 66SOLUTIONVI. DRILL-STRING BUCK LING-ANALYSIS AND DISCUSSION 78

6.1 CriticalBuck Un gCo nditionoftheF irst-Order 786. L1 Point of Tangen cy for First-Order Critical Con dition 856.1.2 Equ ation Coefficients for First-Order Critical Condition 85

6.2 Critical Buckling Cond itions above First-Order 876.2.1 Equ ation Coefficients for Critical Co nditions above First- 91Order

6.3 Shap eoftheB uckled Dril l ingStr ing 926.4 Bend ing M om ent Diagram s for First and Higher Buc kling 96Orders6.5 Force Applied by Buckled Drill-String on bore-hole wall 99

VII. CONCLU SIONS AND RECOM MEND ATIONS 102REFERENCES 104APPENDDCA BIT WO BBLING DATA 106B MA TLAB SOURCE CODE 116

IV


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ABSTRACTEarlier study of failure of PolycrystaUine-diamond-com pact (PDC) bits was

attributed to "bit-whirUng" theory which caused cutter chipping due to down-hole bitvibrations. Based on the bit-whiri theory, the PDC bit design was modified by changingthe cutters orientation, introduction of low-friction pads around the bit so that the netimbalance forces from the cutters are minimized. The "bit-whirl" theory by itself was notsufficient to address the failure m echanism as it considered only the kinem atics of the bitand the geometric aspect of the bit dynamics was neglected. The study in the paperfocuses on another theory known as PDC "bit-wobbUng" which takes into account the bitdown-hole dynamics. Based on this theory, a kinetic model of the bit and the bottom-holeassemb ly (BH A) is developed. The various forces acting on the model are presented andanalyzed. Sensitivity analysis is carried out on the model to study the effects of stabiUzerposition, phase angle, bit velocity, bit weight, driU-coUar stiffiiess etc. on the backwardcutter velocity. This study identifies p ossible solutions for reducing the bit-wobbUng.

The theory of buckUng is appUed to derive the analytical solutions to driU-stringbuckUng. Based o n the analytical solutions, the different buckling orders are modeled andanalyzed. The bu ckled driU-string and bending m om ent profiles of first and high er-orderbuckUng are generated through compu ter programs given in the Appen dix. The effect ofvarious param eters on driU-string buckUng is studied and p resented.


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LIST OF TABLES4.1 Basic Input Param eters for bit wobbUng analysis 506.1 Va lues of x^ and xs for Critical Cond ition above First-Order 90A .l Variation of AmpUtude with Frequency ratio for different viscous 107dam ping factor (AmpUtude in inches)A.2 Variation of AmpUtude Ratio with Frequen cy ratio for different 108viscous damping factor (AmpUtude in inches)A.3 Variation of Phase Angle with Frequency Ratio (Phase Ang le in 109degrees)A.4 Cutter Velocity at Bit Center (Er = 5x\0^psi) 110A.5 Cutter Velocity at Bit Center (, = IxlO^ psi) 111A.6 Cutter Velocity variation with Lateral Contact Area 112A.7 Effect of Bit W eight on Cutter Backw ard Velocity 113A.8 Effect of StabiUzer Position on Cutter Backward Ve locity 114A.9 Effect of DriU Collar Stiffhess on Cutter Back ward Velo city for 115given d istance of StabiUzer from BitA.IO Effect of Dam ping Factor on Cutter Backw ard Velocity 115

VI


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LIST OF FIGURES1.1 Test-weU RO P com parison in Oswego Umestone 11.2 General Bit W hirl condition 31.3 Typical cutter paths on a whirUng bit 42.1 Typical configuration showing shear plane 82.2 Forces acting on a Chip 92.3 Relation between forces in metal cutting 102.4 Com pon ents of forces in cutting face of tool 102.5 Relation between forces and angles in orthogonal metal cutting 113.1 No rmal and shear stresses at a point in a body 153.2 No rmal forces and strain energy stored in a bod y 173.3 Shear forces and shear strain energy stored in a bod y 183.4 Energy in beam s subjected to a uniform bending mo men t 203.5 Influence coefficients for beam deflections 223.6 W ork done on a beam in sequence (a) and (b) 233.7 GeneraUzed forces and corresponding displacements in an elastic bod y 243.8 Free-Bo dy Diagram of a BHA in vertical hole 273.9 Free-Body diagram of a bit in a vertical hole 273.10 Dyn amic equiUbrium of a BH A in a vertical hole 373.11 Cantilever beam with an end load 384.1 Path of bit center at resonance condition 564.2 Path of bit center at non-resonan ce condition (Frequency ratio=0.88) 564.3 Path of bit center at non-resonance condition (Frequency ratio=1.30) 574.4 Variation of AmpU tude with Frequency Ratio 574.5 Variation of AmpU tude Ratio with Frequency Ratio 584.6 Variation of Phase Ang le with Frequency Ratio 584.7 Cutter Velocifies on a PDC bit face 614.8 Veloc ity of cutter at bit center {Er = 5x10^psi) 62

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4.9 Veloc ity of cutter at bit center {Er = IxlO^psi) 624.10 Effect of Lateral Contact Area on Cutter Backw ard Velocity 634.11 Effect of W eight of Bit on Cutter Backw ard Velocity 644.12 Effect of StabiUzer position on Cutter Backw ard Veloc ity 644.13 Effect of StabiUzer position on Cutter Backw ard Velocity 654.14 Effect of Viscou s Dam ping Factor on Cutter Backw ard Velocity 655.1 Extem al forces acting upon a DrilUng String 665.2 Ex tem al Forces acting on a driUing string section 685.3 Vectorial representation of forces on a drilling string section 695.4 Fun ctions of F(x), P(x), and S{x) 765.5 Functions ofG{x), Q{x), and T(x) 765.6 Functions of/ /(x ), R(x), and U(x) 776.1 Critical Cond ition of the First-Order 796.2 Length of a dimen sionless unit with change in drilling mud density 806.3 Critical W eights on bit for First-Ord er Critical BuckUng 816.4 Buc kling Cond itions for Com bination DriUing Strings 836.5 Influence of Drill-CoUar size on BuckUng (I.D=1.875-m ) 836.6 Influence of Drill-CoUar size on BuckUng (I.D=2.5-/>z) 846.7 Influence of DriU-CoUar size on Bu ckl ing (LD=3.0-z>2) 846.8 Shape of Buck led Curves for Different BuckUng Orders 936.9 Tangen cy and Neu tral Points Variation with different BuckUng Orders 946.10 Critical W eights on Bit for Second-Order Buckling 966.11 Ben ding M om ent Coefficient Profile for First-Order BuckUng 976.12 Ben ding M om ent Coefficient Profile for Second-Order BuckUng 986.13 Bend ing M om ent Coefficient Profile: Second buck le contacts bore-hole 98waU6.14 Co effic ient /for calculating force on bore-hole waU due to driU string 100buckUng6.15 Force of Buckled DriUing String (Second Buckle Contacts Bore Ho le 100WaU)

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C H A P T E R 1I N T R O D U C T I O N

l . l P D C B i t W o b b U n gPDC bits were introduced in the early 1970s and have then almost replaced three-

cone bits for use in relatively soft , non-abrasive formations. The main Umitation of PDCbits is when driUing in harder formations or even in softer formations with infrequenthard streaks. The usuaUy high polycrystaUine-diamond compact (PDC) bit wear Umits i tsUfe for use in hard formations even though higher penetration rates can be achieved withthem. Figure 1.1 shows the performance analysis^ of a three-cone and a PDC bit whiledril l ing through a section of the Oswego limestone at the Catoosa test faciUty near Tulsa.Th e li thology at the test is described by W inters and On yia . Fro m the figure, i t isobserv ed th at the PD C b it driUed the Umestone at a rate three to four t imes that of thethree-cone baseline. However, after driUing 20-ft the PDC bit rate of penetration (ROP)fell to nearly the three-cone baseUne. Glowka and Stone"^ and Zijsling^ also confirm thatPD C bit wea r and not the init ial RO P limit i ts appUcabiUty in harder, more abrasiverocks .

3 0 0

cr:Xoir

2 0 0 -

1 0 0 -

12 0 HPU6-10 KIPS WO TKRte-CONCIASUNE

: *J\

260 270 280 290 300 310 320 330 340DEPTH (FT)

Figure 1.1: Test-well ROP comparison in Oswego Umestone^


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From analysis of the drag-bit wear model by Warren,^ it is shown that theperformance of a sUghtly duU PDC bit can actuaUy be m uch worse than that of a three-cone bit. This is because the bearing area increases as the wear-flats grows and thisreduces the stresses induced in the rocks. It has been observed by Zijsling^ that anapparently duU PDC bit, i.e., one with significant wear flats can stiU driU almost as fast asa new bit because of diamond table lips provided above the tungsten carbide. Thisdiamon d Ups acts as the contact area and not the com plete tungsten-carbide w ear-flat du eto which the high contact rock stresses are stiU maintained. T he presence of diam ond Upsis important for efficient driUing in hard formations. However, they are not critical whiledriUing softer formations. A sharp bit is one which has diamon d Ups provided on eno ughcutters to cut the bottom of the hole comp letely. A duU bit is one that does not haveenough Ups on the cutters to cover the entire hole. PDC b its with diam ond Ups are able tooutdriU P DC bits by two m echanism s: (1) they create higher contact stresses becauseonly the diam ond contacts the formation on a sharp PD C bit, and (2) they are able toclean the bottom of the hole mechanically and are therefore not as greatly affected bymud, hydrauUcs and rotary speed.

PDC bits fail mainly due to cutter chipping. Warren and Sinor^''^ have shown thatsteady-state loads applied to PDC cutters under m ost normal driUing situations are toolow to cause cu tters to chip. Even fatigue from cycUc loading of the bit is not attributed tothe cutter chipping. It was observed from laboratory driUing^ that failure occurs mainlydue to impact loading caused b y bit vibrations which w ere so severe that it caused PDCstuds to break and numerous cutters chipped. Even though force balancing on the PDCbits was carried out, the primary cutter chipping mechanism due to bit vibrations couldnot be overcome initiaUy. W arren et al^ has shown that cutters that chip develop wearflats very qu ickly. On ce a diamond table is lost because of chipp ing, the tun gsten-carbidewear process proceeds very quickly. They also observed that a "low-friction" bit designcan substantiaUy eliminate bit wh iri. The lowfirictiondesign is based on placing thecutters so that the net imbalance force from the cutters is directed towards a smooth pa dthat slides along the well bore waU. The detrimental effects caused by im pacts loads are


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attributed to a phenom enon caUed "bit w hiri" or bit "backward wh iri." Bad bearingdesign w as know n to produc e the bit wh iri. W ith bit whiri as shown in Figure 1.2, the bitmo ves prim arily laterally around the ho le. The bit acts as a pinion in a hole that acts as agear. The drilUng imb alance push es one side of the bit against the borehole w all creatinga new fiictional force. A clockw ise torque on the bit comb ined with this frictional forcemoves the instantaneous center of rotation away from the geometric center and towardsthe wellbore waU.

Hole

Hole Center

Bit Centr

Figure 1.2: General Bit Whirl cond ition'Figure 1.3 shows the typical cutter paths on a whirUng bit. It is observed that the

cutters can move backward, sideways, and farther per revolution than those on a tmerotating bit. As a result, they are subjected to high impact load s. On a whirUng bit, acutter impacts the side of the well bore ma ny times per revolution. The detrimen tal effectof bit whirl is that it cannot be stopped once started. Hen ce, bit whirl occurs when thedynam ic forces on a bit cause the center of rotation to move as the bit rotates. A w hirlingbit cuts an overgauge hole, and the cutters move faster, backwards, and sideways andconsequ ently are subjected to high impact loads. Once a bit begins to wh irl, forces aregenerated that reinforce this tendency. Bit whirl is worse at higher rotary speeds becaus ethe centrifiigal forces are squared as rotary speeds d ouble. Bit whirUng becom es severe at


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low weight on bit (WOB). Extemal stabilization by providing stabiUzer bars and shocksubs can limit the extent of bit whirl but not eUm inate it. Greater driUing force imb alance,longer driU-bit tapers, m ore agg ressive cutting structures and gauge c utters increase abi t's ten denc y to whirl. Higher back rake ang les, flatter profiles and smo oth gauges tendto reduce bit w hirl. Bit whirUng is not severe while driUing soft form ations. Ro cks thatare driUed slow ly prom ote bit whirl because the bit has more time to create a regenerativestmctu re on the bo ttom of the hole. Formations that cause bit baUing also reduce b itwhirling tendencies by creating a less aggressive gauge.

4

2

;-direction and the following relation is obtained:

Fjy -Fcy -Fhy -(Ff,)y = m dt^ . . . ( 3 . 5 7 )From equation 3.19, ignoring the small value oFfya, Ffy Ffy^ sm((Oyt) andsubstituting equaton 3.56, 3.23 and 3.25 into equation 3.57, we get

FfyoSm{co,t)-kdyi- yi-cf dyi d yidt m dt (3.58)

d yi I ^ / dyi ,dt^ w dtE^A' '- + k.

V L > , F.m yi = ^sm{crt)m

w h e re Fo = F,yo -

andZ) = , the above equat ion becomes

) ^ + i ) + (co^f yi=-^sin{(s>y.t).m m . . ( 3 . 5 9 )

As the right hand side of equation 3.59 is not equal to zero, its com plete solution equalsthe sum of the complementary solution and the particular integral.

Equ ating the left han d side of equation 3,59 to zero, the equation is quadratic andits solution is given b y:

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1 + r / ^E.A.^k^D,D = m m Dj.m

CfIf ^ = r wh ich is defined as the viscous dam ping factor, then eq uation 3.59 can be2mo)y^written in terms of , and .

Z)^+2 ^y) + y = 0 . . (3.60)and its solution is given by:

Z) = - ^ y V ^ COn - n= -4(0^+1^0)^ -^ ( (3.61)

From the roots given by equation 3.61, the comp lementary fimction of the differentialequation 3.60 is given by

C.F = e ^< t C\ co s ^ll -^^c^ +C2SnU/ y - ^ y . . . (3.62)

The Particular Integral is given by:F

P.I = ^ 1m r\2 sm{co,t)D +2i^co D + a)1

f" 2^co'D- )^ + col sm {co,t)

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_Fo \o)y, -co'i]-24)^Dco ^ -CD^j -4^ô)^ D^ sm {cD^t)

FQ K -C0r]-2^C0nDm ( '^ 2V . ^2 " 2)^ -)f. j -\-4g )^ j.\n{o)^t)sm

m 0) n -^rf 2 '2 2+ 4 ^ 0)j^ 0)y. o)jj - o)j. Isin(2>^r)- 2 ^ )yj o)y. cos{p)y.t)

I f Rcos = o)yj -j. ' / ' 2 1 2 '^ 2y. dinRsin = 2ô)yô)j.,ih.cn R = ^\^^ -^rj + 4 ^ 0)^^ ^. .Sub stituting the v alue o f 7? in the above e quation, w e get:

FP.I = ^ [sin(y^r)cos 6 - cos(y^/)sin 6\mR

i2 2 r 2 ' 2m^j\o)f^ -o)j.] + 4 ^ o)y^ cOyFQ s\n{p)y.t-6)

s\n{o)yt-0)

s\n{cOy.t-6)cOy^m. 1 _ J ^

V ^n j2ô)y

\ ^n J

fyo EyAy + A:.sm^ 2 ^

^ i22 / {co^t-)

+2^co,

V ^n y= 7/, sin(y^ - d) (3.63)

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^ fyowhere Y^ = D. and tan 6 =A2l_fC +n J

f \22^C0.\ ^n j

2^0)y,CO,CfJ COy.

OT 6 = tan 2^0) yjO)^^n 0) r J

The term YL is the constant am plitude term and is the steady state am plitude. Thecom plete solution of the differential equation 3.59 is the sum of equations 3.62 and 3.63The constan ts defmed in equation 3.62 need to be determined based on the boundaryconditions given by:At/ = 0;j^/,=jîoandAt dyLdt t= o = ô where Vo is the initial linear veloc ity of the bit.Applying the first boundary condition to equation 3.62, we get

yLo =c\= VLO 'Differentiating equation 3.62 with respect to t, we get^^-a^ê-i".-dt : )j cosly^ -(^ o)j^ r + c^sinly^ -^ ) i +

ô)'t V \2 2 '^ / ~^ 2 ^y - ^ y sin -^y - ^ co ^ t + C203n - ^ 0)n COS ^^(Oj, -, Applying the second boundary condition to the above equation, we get

dyidt =^=^0= - # 0)n C \ + C2 ^C n - ^ ^ ^ |2 2 '2

C2

C2

ô+^(^n yuCO, ^ 0)n

ô^ + ^) ô-l^ ^cco Vw^ yVw C ,

AA


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with the assumption that ^(l + ^)/(l - ^) 1 as (f is very small.Lci o)^=o)j^ - ^ ^ ^ =>o)^ =o)yj^j\-


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yi = Y L sm{(o^t - e) . . . (3.66)As 0)j. approaches ty^, resonance occurs. Near resonance the magnitude Yi of the

steady-state so lution is a strong function of the viscous dam ping factor ^ and the non-dime nsional frequency ratio y / y . Und er the resonance condition, the bit wob blecauses the early failure of the PDC b it. The critical 0)r that causes resonance can bedetermined using the following equation:

k^ + '^ ^D,COj. = (3.67)mwhere kd is defmed by equation 3.56

Since the term kd is very sm all, it can be neglected in the above equation. Thecondition at resonance, then becomes

0)y = Ef.AyDj.m . . (3.68)The amplitude at resonance can be determined by substituting co ^ = in the term YL ofequation 3.63.

'fyoYT =

^E A ^n . . (3.69)1 _ _ ^

V y+ 2 > ^

V ^ yThe denominator of above equation equals 2,^ and the amplitude Yi at resonance equalsFfyog,^ f- 22go)j-m where, gc is the gravitational constant for the following units;

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Ffyo =M Ay. =in^kd = Ibflft D, = inEj. = psiThe time period at resonance equals ^njcj.3.5.2 Angular Vibrations

For a drill bit under p enetrating cond ition, the un-balanced lateral forces actingupon the bit will cause a lateral vibration of the bit in a maximum deflection plane (x-_yplane ). The u nbalanced torsional loads result in the angular vibration of the bit. The bit istreated as a particle in the lateral motion and equation of motion is applied in the j ^ -direction which yields the following differential equation:

Tc-Tjb-Tf-Fhz =I m . ^ - ( S .VO)where:Fc =Ts -Tfc=TbTf=Tf^+TfoSn{o)Lt)Fhz = M FhySubstitutng the above equations in equation 3.70, we get

2Fh Du d 6Ts -Tfc -Tfb -Tfa-Tf,n{coLt)-^^:^ = Imxf - ^^-^^^

But we have from the relation: T^ = Tf^ + Tf^ + TfbSubstituting this relation into equation 3.71 , we get

2r , s i K , ) - ^ ^ n = / . ^ -("2)^E), dt^

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Considering o nly the steady state component ofyi, we getyi ^YiSn{o)j.t).

Substituting above equation into equation 3.72, we get2-Tf,sm{o),t)- ^Y,sm{a),t)=I ^^^ . ... (3.73)^^r dt

T rr. jUDuEj.Aj.L e t i ^ = Yi, then equation 3.73 becom es:rd^6,TfoSn{o)i^t)-T^sin{o)j.t) = Ijj,^ -^ . ... (3.74)dt^

Equ ation 3.74 is the govem ing differential equation of the angular motion of the bit in thex-y plane.Integration equation 3.74 with respect to time t, we get

^ = ^^cos{o)j.t)-\- ^^cos{o)f.t)-^C2 ... (3.75)at COj. 1 yj^j^ 0)j. 1 jyij^

where cs is the constant of integration.dAtt = 0 ^ = 0)^, where coo is the initial angular velocity of the bit.dt

Substituting this bou ndary condition into equation 3.75, we get:Tfo + T^ Tfy + T^

0^ 0 = ~ ^ + 3 ^ ^ 3 = 0 j

d6j Tfy + r ^ \ Lfo+ T^ ^ = 0) = 0)^ -\- cos(y^/j .at o)j. 1 jj^-^ o)j. 1 ^-jj-Considering the absolute angular velocity, we get

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Tr -[-To) = o)r.^ [1 - cos(y^ t)\0)j,I mx (3.76)

Integrating equation 3.76 with respect to time t, we get

OL =>ot + ^ ^^r^mxsm {o),t)

co . + C 4 (3.77)

At /=0 C4 = 0Hence equation 3.77 becomes

QL =o)ot + Tfo + T^^r^mx\n{o)f.t)sm

0), . . . (3.78)

The ang ular velocity and angular displacement of the bit motion due to the un-balancedtorsional loads are given by eq uations 3.76 and 3.78, respectively.

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CHAPTER 4BIT WOBBLING-CALCULATIONS, RESULTS AND DISCUSSION

4.1 Examnle CalculationsThe following basic input parameters sho wn in Table 4.1 are considered for

analyzing the bit wo bbling m odel. Based on these param eters, sensitivity analysis is alsocarried out to determine the various factors that influence bit wobbling.

Table 4.1: Basic Input Parameters for bit wobbling analysisNo.

1234567891011121314

DescriptioiiLocation of the bottom stabihzer from bit, Lfriner diameter of the bottom drill collar, DOuter diam eter of the bottom drill collar, dYoung's Modulus of the drill coUar, EcBit diameter, D/,Weightofbi t , WbBit lateral contact area to formation rock, ArYo ung 's Modu lus of the formation rock, ErEffective depth of rock deformation, DrFriction coeffcient between bit and rock, ^Rotary Speed, A'Viscous Damping Factor, ^Variation in cutting force, FfyoVariation in cutting torque, Tfa

Magnitude60 ft2.5-in6.25-in3x10 ' ps i7.875-in96 Ibl in^5x10' 'psi4 ft0.36 0 R P M0.5600 Ib25 Ib-ft

Density of steel considered, p = 490.0 Ibm/ftCross-sectional areaof drill coIlar,/ = - ( D ^ -d^]= - ( 6 . 2 5 ^ - 2 .5 ^ )= 25.7708/^

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T V 1 1 1 . 2nN 2 X ; T X 6 0Initial angular velocity = o)o= = = 6.2832 rad/sec.60 60

AreaMomentoflnertia, /^ = ( D ^ - ^ " ^ 1 = ( 6 . 2 5 ^ - 2 . 5 " ^ ) / ? ' ^ = 0.0035197//"^.64 ^ ^ 64 ^ ^12^

From equation 3.53; C = T";; ~ ft6E,I,Writing the above equation in terms of individual units, we get

^n pAcol i ^ . - V . . 2 V , A6 c ^z 6Ibm

yfi jfi144,V J vsec j

1 1^ Ibf 144^fi2 1

/^ 'V . / " J

1 1 Ibm \6 144^ /6/ / ^ sec^ ^c 6 x 144^ x 32.2^ /r"^ =2.496134 X10"^/r"^

Hence,C = 2.496134x10"^ x ^ft'^' ^ 6 c / z ...(4.1)

Substituting the known values into equation 4.1, we get

3x10^x0.0035197

From equation 3.56, k^ = EJz

3 35 " 34650 "

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If the units ofEc, L, Cn, and L are psi,ft'',ft'\ andft, respectively, and th e un it of ^rf isIbf/ft, then

k^=^3 11+ 35

\44EJ,

" 34650 C^^n ' Ibflft (4.2)

Substituting the given values in kd, we get

kd 144x3x10 ' xO.003519760 +3 35 1.178x10-^ x 6 0 ^ + ^ ^ ( l . 1 7 8 x 1 0 - ^ ^ x 6 0 ^ ^34650

= 3.0721 Ibflft

Fromy = checking for the un its, we get

Ibf Ibm ftIbm ft ibf sec^ sec

deglsec. (4.3)

Substituting the given values in above equation, we get^ 5 x 1 0 ^ x 1

(y ,+ 3.072196

x32.2= 647.5 deglsec^ 11.301 radlsec.

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The visco us dam ping factor is dimensionless and is given by


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The m ass mom ent of inertia, Imx assuming the bit as a sphere is given by:lyy,^ =~Ma^ =-mD^ = - x 9 6 x - ^ = 4.\34 Ibmft^5 5 ^ 5 1 4 4 x 4 ^_ _ juDi.Ej.Aj. ^^tvom 1^= Yi, writmg m terms of individual units, we get

T =r Ibf , 2ftx^^xin'^^ ^ft=lbfft

Rcncc. T.,=^^^^^Y,ft' -"w 2D , (4.6)

Substituting the individual values in T^, we getx 4.8x10 = 59.0625 Ibfft0 . 3 x ^ ^ x 5 x 1 0 ^ x 1T = 122 x 4

From equation 3.76, we have

6i =o)^t + Tfo +^w^r^mx t-m{o)y.t)sm

0) , (4.7)

Each term in the right hand side of equation 4.7 should be d imension less. Considering thedimensions of the second term in the above equation we get:Ibf-ft 1 Ibf-scc^^ X X s e c = f 1^secy Ibm- ft Ibm - ft

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Introducing the gravitational constant, gc in the above equation we get:Ibf - sec Ibf - sec Ibm - ft~Ti T~^g c = ~ X T = dimensionlessIbm-ft Ibm-ft ibf-sQC^

Hence equation 4.7 re-arranged with the proper units becomes:

6i =o)ot-\- ^Tfo -\-T^^\ ^r^mx jg< sm{o),t)

0), . . . ( 4 . 8 )

Substituting the given values in equation 4.8, we get:(25 + 5 9.0625 )x32.26J =2nt-^ '-647.5x4.134

t sin(647.5/)647.5 ^

= 6.2832-17.7622 - 1 . 5 4 4 4 x l 0 " - ' s mn(647.5r)J. (4.9)

4.2 Results and DiscussonFrom equation 3.63, it is observed that resonance occurs wh en thefirequencyratio\^y. IcOyj I is equal to unit in the Yi term. Using equation 3.66 and substituting the values,the lateral displacement of the bit is plotted with time as the parameter under resonancecondition. The correspond ing data for all the figures shown in this section is enclosed inApp endix A. The path of the bit center is shown in Figure 4.1. The plot indicates that thebit undergo es a wob bling m otion. As a result of this motion, a lobed pattem of the bottomhole is observed when resonance occurs. The maximum amplitude of vibration atresonance is calculated to be 0.00576-zn for the given co nditions. For n on-resonancecond itions, Figures 4.2 and 4.3 indicate the path of bit center for frequency ratios of lessthan one and greater than one, respectively.

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0.008

O .OOB o.ooa

0.008

Figure 4.1: Path of bit center at resonance condition

Figure 4.2: Path of bit center at non-resonance condition (Frequency ratio=0.88)

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. oa

-0.008-^Figure 4.3: Path of bit center at non-resonance condition (Frequency ratio=1.30)The effect of lateral vibration amplitude w ith fi'equency ratio for a given viscous

dam ping factor is shown in Figure 4.4. Figure 4.5 shows the variation of am plitude ratio(ratio of the observed amp litude to the amplitude at resonance condition) with frequencyratio. The variation of phase angle with frequency ratio and damp ing factor is shown inFigure 4.6.

0.4 0.6 0.8 1 1.2FrequAncy Ratio (wr/Wn*)

1.4

Figure 4.4: Variation of Am pUtude with Frequen cy R atio

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2.8 -2 . 8 -2 . 4 -2 52 . 0 -oe l ^ rotE: 1.6 -u 1.4-tL1 1^^.0 -0.8 -0.6 -OA '0,2 -0.0 -

t=2J

, , j

M . 6-1.0

1

/ " 0 . 2

- - - ^ . ^ . ^ . l E ^ O . O ^ ^ ' ' * " * ' ' ' ' * ^ , , , , . . ^

"-"fc..^.^^^^^^ * * ,

s.

\ -O.D

' ^ ' " - - ^ ' ^ ^ ^ ^ N ^

^ ^ ^0.2 0.4 0.6 0.8 1 15

Fre qu enc y Ratio (wrAn/n*)1.4 1. 6 1.8

Figure 4.5: Variation of Amplitude Ratio with Frequency Ratio180160

0.0 0.5 1.0 1 2.0Fre


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The amplitude ratio is given by:Amplitude Ratio =

1 _ ^ 2 ^2 / AV ^n j +

2^co,K ^n j

(l-r^f+(2^ry . (4.10)

The foUowing observations^^ can be made from Figures 4.5 and 4.6:1. For an un-damped system, the viscous damp ing factor is equal to zero and the

amplitude ratio becomes - . -.2. The d amp ing redu ces the am plitude ratio for all the values of the frequency ratio.3. The reduction of the amplitude ratio in the presence of dam ping is very significant

at or near resonance.4. The m axim um amplitude ratio is obtained by differentiating equation 4.10 w ith

respect to r and equating the resulting equation to zero.-3/d(A.R) - 1 ^ = Xdr 2 [\-r^J+{2^rf 2\\-r ) ( - 2 r ) + 8 # ^ r = 0

2 ^ ^ = 1 - / - ^ ^r = Vl-2^'0) . = co'J\-2^^ . . . (4.11)

The damped frequency Or is less than the un-damp ed n atural frequency o)y^ fromequation 4.11 and also less than the damped n atural frequency defined byequation 3.64, i.e. o)^ =o)yj^j\-


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The above equation can be used to determine the amount of damping present in asystem if the maximum amplitude of the response is measured. Conversely, if themagnitude of damping in a system is known, an estimate of the maximumamplitude of vibration can be made.

6. Below reso nance , the phase angle increases with increase in dam ping. Aboveresonance, the phase angle decreases with increase in damping.Brett et al. concluded that the failure of PDC cu tters in hard rock drilling is due

to cutter chipping (impacts from vibrations). Cutter chipping occurs when the cuttermoves backward due to bit wobbling. Figure 4.7 shows the schematic of the bit face. Thewo bbling mo tion of the bit in the bit-face plane can be resolved into two m otions, i.e.,lateral and rotational m otion. hi F igure 4.7, thick arrows represent cutter velocities in thelateral m otion of the bit, while thin arrows represent cutter velocities in the rotationalmo tion of the bit. The vector sum of velocities due to these two m otions is the totalvelocity of the cu tter. At points B, C, and D, the magnitudes of the total velocities aregreater than the magnitudes of their component velocities. At po in t^ , the magnitude ofthe total velocity is less than that of its compo nent velocities. The cutter velocity at pointA can either be a forward velocity or be a backward velocity depen ding upo n themag nitude of the velocities due to the lateral and rotational m otion. If the magnitude ofthe velocity d ue to lateral mo tion is greater than that of the velocity due to rotation, thecutter at point A will have a backward velocity. This backward velocity may cause cutterchipping if the formation rock is hard.

Taking derivative of the steady state part of equation 3.65, the lateral velocity ofthe bit center is obtained and given by:

Vc = h ^r cos{o)j .t-(^). ... (4.12)The rotational velocity of the cutter at a distance rl from the bit center is given by:Vryc=co^ . . . ( 4 . 13 )

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Figure 4.7: Cutter Velocities on a PDC bit faceSource: "Bit Wobble: A Kinematic Interpretation of PDC Bit Failure"SP EP aper 28313^^

wh ich is the v elocity relative to the bit center. The total forward velocity of a cutterlocated between point A and the bit center can be found out from the following relation:

K=Vrl/c-~V,. . . . (4.14)

If Vr is negative, the cutter has a backward velocity. Equations 4.13 and 4.14indicate that the rotary speed has a positive effect on reducing the backward cuttervelocity. Since bit rotation h as no effect on the velocity of a cutter located at the bitcenter, the backward velocity of a cutter at bit center is higher than that of other cutters. Itis usually found from field observations that the cutters near the bit center are moreseverely chipped than other cutters.

Using the given d ata, the total forward v elocity Vr is plotted versus time andshown in Figure 4 .8. The plot indicates that the cutter at the bit center will move forwardand backward for the same amount of time. Figure 4.9 is a similar plot but for a harderformation. Althoug h the mag nitude of backward velocity is lower compared to that of a

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softer formation, the frequency o f the vibration is increased in this case. Hence in hardformation drilling, more cutter chipping is expected.

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10Time (sec)

Figure 4 .8: Velocity of cutter at bit center (Er = 5x\0^psi)0.15

0.00 0.01 0.02 0.03 0.04 0.05 0.06Tme (sec)

0.10

Figure 4 .9: Velocity of cutter at bit center (* = IxlO^ psi)To m inimize cutter failure of PDC bits, it is necessary to analyze the factors

affecting the backward velocities of the cutters. The pseudo-natural circular frequency isgiven by:

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0) , (4.15)

If the stabiUzer position is high, then the value okd will be low and the pseudo-natural frequency is dependent mainly on the formation rock property Er- It is alsoobserved that the drill-string vibration is sensitive to the lithology of the formation b eingdrilled. The effect of lateral contact area on backward cutter velocity at bit center isshown in Figure 4.10. This plot indicates that the backw ard cutter velocity can beeffectively reduced by increasing the lateral contact area of the bit. This is in fact put intopractice in field by utiUzing a wider low friction pad.

0.80OJO0.60

J" 0.50o 0.40>L.^ 0.30

0.20

(Frequency Ratio = 0.5, Damping Factor = 0.5)

0.100.00 3 4 5 6 7

Lateral C onact Area, in**28 9

Figure 4.10: Effect o f Lateral Contact Area on Cu tter Backw ard V elocityThe effect of weight of bit on backward cutter velocity is shown in Figure 4.1 1.

From the plot it is observed that a heavy b it is better than a lighter bit for controllingbackw ard cutter velocity. The effect of stabilizer position on backw ard cutter velocity is

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show n in Fig ure 4.12 . It is observed that the nearer the stabilizer is to the bit, the lowerthe backward cutter velocity can be achieved.

0.50

50 100 150 200 250 300 360 400 450 600W e igh to fB i t ( l b f )

Figure 4.11: Effect of Weight of Bit on Cutter Backw ard V elocity

0.18 -0.160.14 -u01tfl 0.12

o 0.10 -4>>S 0.08 -

0.06 -0.04 -0.02 -0.00 -

//

//

/

1 1 1 1 110 20 30 40

Stabllzer dstance from Bit, ft60 60

Figure 4.12: Effect of Stabilizer position on C utter Backw ard Velocity

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Figure 4.13 shows the effect of drill-collar stiffness on backward cutter velocityindicating that increasing drilI-coUar stiffness reduces backw ard cutter velocity wh en thestabilizer is placed close to the bit. The effect of viscous damp ing factor on backw ardcutter velocity is shown in Figure 4.14. It indicates that viscous drilling fluids are helpfulin reducing the cutter backward velocity.

0,16

0.15 ^0.14

0.13

Dhrahc-e of Stsbiliaer from Sit = 5 ft

Distance of Stab l ise r f r om B i t 3 ft

Dri l l -Col iar Stl ffnss {x +7 ibf/in^)

Figure 4.13: Effect of DriII CoIIar Stifftiess on Cutter Backward Velocity

0.6 0.8Damping Factor

1.6

Figure 4.14: Effect of Viscous Dam ping Factor on C utter Backward Velocity

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CHAPTER 5DRILL-STRING BUCKLING: GENERALIZED ANALYTICAL SOLUTION

For analyzing the drill-string buck ling, both the ends of the drill-string areconsidered h inged. As a result, there are reactions at both the ends which in tum haveboth v ertical and horizontal com ponen ts. There are also reactions of the wall at the pointswh ere the buckled p ipe con tacts the wall. The drilling string is assumed as a continuouspipe with no tool joints.

Figure 5.1: Extem al forces acting upon a Drilling StringSource: Arthur Lubinski^"^ Study ofBuckling ofRotary Drilling Strings, 1950.The ex temal forces acting on the d rilling string are shown in Figure 5.1 in which:1. The upward force Wi is the reaction at the top hinged end.2. The upward force W2 is the vertical com ponent of the reaction of the bottom

of the hole on the drilling string and represents the "w eight on b it."3. The force F2 is the horizontal com ponent of the reaction of the bottom of the

hole on the drilling string.

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4. The horizon tal force F} is the reaction of the wall o f the hole on the drillingstring if the pipe is buc kled.

Two additional forces not shown in the above figure are the weight of the pipewh ich is a vertical dow nward force, and the buoyancy wh ich is a vertical upward force,both appUed at the center of gravity of the drilling string. The influence of viscous forceson the drill-string and the jet forces on the bit are neglected as they are sm all whencompared to the weight on the bit. When W^^Q, there is no w eight on the bit and thedrilling string is straight resulting in an elastic equilibrium. If a lateral force is appliedand a small deflection is produced, this deflection disappears when the lateral force isremoved and the pipe becomes straight again. If W2 is increased, but maintained below acertain critical value, the straight form remains stable. If the critical value of W2 isreached, th e straight form of the pipe becom es unstable; i.e., if a lateral force, howeversmall is appUed and a small deflection is produced, this deflection does not disappearwhe n the lateral force is remo ved. O n the contrary, the deflection increases until a bentform of stable equilibrium know n as buckling is reached. The X ax is is defined as the axisof the hole and taken to be positive dow nwards. The point of origin is defined as theneutral point. The 7a xi s is taken as positive to the right. The plane of th eX an d Fax es isthe plane of sm allest flexural rigidity in which the bu ckling occu rs. The units of X and Yare in feet. The bending moment Mof the buckled string expressed in units oflbfft canbe given by:

d^YM = E I ^ . . . ( 5 . 1 )dX ^where E is the You ng's Mod ulus of elasticity of steel an d / is the moment of inertia of thecross section. The un its oE and /a re Ibf/ft^ and//'^, respectively. The rate of change ofthe bending m om ent is defined as the shear force and is obtained by differentiatingequation 5.1 w ith respect to X .

dhF,=EI^ . . . ( 5 . 2 )dX ^

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The shear forces along any cross section of the drilling string, such as MNinFigure 5.1 can be determined and substituted in equation 5.2 from which the bendingmoment M can be obtained by integration. The forces acting on the portion of the stringMN are represented vectorially in Figure 5 .2. The weight of the drilling string below thesection MN is represented by W and the buoyant force due to the drilling m ud isrepresented by force B}. The hydrostatic pressure B2 does not act upon the section MNand hen ce this must b e vectorially subtracted from the buoyant force B} to get the tmebuo yanc y. Since the considered section of the drilling string is in equilibrium, the sum ofall forces equals to zero . As shown in Figure 5 .3, AB is the weight on the bit; BC is thehorizon tal com ponent of the reaction of the bottom of the hole, or Ff, CD is the weight Wof the part of the string located under MN\ DE is the buoyancy force Bj\ and EF is thebuoyan cy force B2.

^ N

Figure 5.2: Extem al Forces acting on a drilling string sectionSource: Arthur Lubinski^"^ - Study ofBuckling ofRotary Drilling Strings, 1950.

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M. " JB C

^ N

Figure 5.3: Vectorial representation of forces on a drilling string sectionSource: Arthur Lubinski^"* - Study ofBuckling ofRotary Drilling Strings, 1950.In the initial an alysis, the effect of force F is not considered and set equal to zero

to simplify the analysis. From constm ction of Figure 5 .3, the force FA which representsthe reaction of that part of the d rilling string above MN on the portion below. This forcehas two components: FG is the shearing force and GA is the compressive or tensile forceaccording to its direction. The vector CD represents the actual weight of the drill-stringsection and DE represents the buoy ant force due to the drilling mu d. For determining theshear forces, the different forces shown in Figure 5.3 are resolved along MN. Thevectorial equation is represented by:

AB-\-BC-\-CE-\-EF-\-FG + GA = 0 . (5.3)and the projection of all the vectors along the axis MN is given by:

ABsina- BCcosa-CEsina -FG = 0The shearing force FG is given by:

(5.4)

FG = {AB - CE)sma-BCcosa . (5.5)

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Under actual conditions in the hole, the angle a is very small. Therefore cos a=\ and sina = tan a. Then, equation 5.5 reduces to

FG = {AB-CE)\'xia-BC (5.6)The buoyancy factor is given by:

B.F = \ r \Pmud\Psteel j

. . . (5 .7)

Letjo represent the weight of the drilling string in mud (units olbflft) which is equal tothe produ ct of the actual weight of the drilling string in air and equation 5.7. LetXy andX 2 designate, respectively, the values of X for the two ends of the drilling string. Th en:

X,=- W,Xo =

PW2 (5.8)

Equation 5.5 can be re-written as follows:F, = [W2-p{X 2 -X)tana]-F2 (5.9)

Substituting in equation 5.9 the value 0X2 from equation 5.8 and replacing tan a with-dYldX, we get:

F, =-pX -F2' dX ^ (5.10)

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Substituting equation 5.10 into equation 5.2, we get

d^Y dYEI- + pX + F2=0 . . . (5 .11)dX^ dXEqu ation 5.11 represents the differential equation of the buckled drilling string. Let

X = mx . . . (5.12)an d Y = my . . . (5.13)wh ere m is a constant wh ich will be defined later.W e have from above two equations

dY _dydX dx

d^Y _ 1 d^ydX^ m dx^

d^Y ^ 1 d^ydX^ m^ dx^

Substituting equations 5.12, 5.14, and 5.16 into equation 5 .11, we get

(5.14)

. . . (5.15)

(5.16)

d y p 3 dy F2 2 r\ c< \n \ ^ + m ^ x + m ^ O . .. (5 .1 7)j;^3 EJ dx EI

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The value o f m is chosen such that

L e t cbede f inedas :3 EIm^ = . . . (5 .18)

Foc = -^ . . . (5.19)pmSubstituting equations 5.18 and 5.19 into equation 5.17, we get

^ + x ^ + c = 0 . .. ( 5 .20)dx^ dx

Substituting equations 5.15 and 5.18 into equation 5.1, we get,2M = pm ^f . . . (5.21)dx^

Equation 5.18 shows that m is expressed inft and equations 5.12, 5.13, 5.14, 5.19, and5.21 show that;:,,)^, dyldx^ (^yldx ^ and c are dimen sionless. Consequen tly, the analysisma de with these factors w ill be altogether general and independen t of the drilling stringand drilling fluid.

The solution to the differential equation defmed by equation 5.21 and shownbelow is required.

d y dy . f + JC + c = 0dx^ dx

Let z = ^ . . . (5.22)dx

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Sub stituting equation 5.22 into the above differential equation, we get

dh_dx + xz + c = 0 . . . (5.23)

The solution to equation 5.23 can be expressed in terms of a pow er series defined b y:00z= ^ a x " .=0 . . . (5.24)

Substituting equation 5.24 into equation 5.23, we get

7Z-2 00X ( - l ) o x " - ^ + ^f l ;c " -^ ^ + c = 0 .=0 =0 (5.25)

Equ ation 5.25 is a polyn om ial fimction in pow ers of x and satisfies for any value ox. Ifthis is tme then the coefficients of x' , x\ x ^ x^, etc. mu st all be equal to zero.Coefficient of xCoefficient of .xCoefficient of xCoefficient of xCoefficient of x'Coefficien t of x"Coefficien t of x'

2^ 2 + c = 0 => 32 - ^ 2

aQ + 2.3^3 = 0 => 3 = -ai +3 .4^4 = 0 => 4 = -2 + 4.5(35 = 0 =^ 5 = -

ao_2.3fL3.4^ 2 ^4.5 2.4.5^3 ^O^3 + 5.6^5 = 0 => a^ = =

4 + 6.7^7 = 0 => ay =5 +7.8^8 =0=> ag =

5.6 2.3.5.6^ 4 i6.1 3.4.6.7a^ c7.8 2.4.5.7.8

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Substituting the above coefficients into equation 5.25, we get

C 2 ^O 3 ^l 4 C 5 0 f ^] 7 C 8z = ao +a tX x^ ^x^ ^-x^ + x^ + ^ x ' ' + x ' x +2 2.3 3.4 2.4.5 2.3.5.6 3.4.6.7 2.4.5.7.8

0 1 - ^ +2.3 2.3.5.6 2.3.5.6.8.9 + ... + aix 1 + +3.4 3.4.6.7 3.4.6.7.9.10^x '2

, x' x'1 +4.5 4.5.7.8 4.5.7.8.10.11 + . . (5.26)

z = ^ = aF(x)-\-bG(x) + cH(x)dx . . . (5.27)

where F(x), G(x), and H(x) are the terms within brackets as defined in equation 5.26Integrating equation 5.27, gives

y = aS(x) + bT(x)-\-cU(x)-\-g . . (5.28)

Differentiating equation 5.27 gives

d^dx^ = aP(x) + bQ(x) + cR(x) (5.29)

The fiinctions in equafions 5.27, 5.28, and 5.29 are defined as foUows:F{x) = \- + "" +2.3 2.3.5.6 2.3.5.6.8.9 (5.30)

G(x) = X , x' x'1 +3.4 3.4.6.7 3.4.6.7.9.10 + ... (5.31)

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H (x) = - ~2, x' x'1 +4.5 4.5.7.8 4.5.7.8.10.11 + ., (5.32)

S(x) = X

T(x) = X

, x' x'1 +2.3.4 2.3.5.6.7 2.3.5.6.8.9.101 x' x'+

+ ..,

2 3.4.5 3.4.6.7.8 3.4.6.7.9.10.11 +

. . (5.33)

(5.34)

U(x) = - X 1 J\ X X+ X3 4.5.6 4.5.7.8.9 4.5.7.8.10.11.12 + . (5.35)

P(x) = - , x' x'I +3.5 3.5.6.8 3.5.6.8,9.11 . . . (5.36)

3 6Q(x) = \-^+ ^3 3.4.6 3.4.6.7.9 + ... (5.37)

R(x) = -X 3 6X X1 + X2.4 2.4.5.7 2.4.5.7.8.10 + (5.38)

The fiinction values for equations from 5.31 through 5.38 have been calculated bywriting computer program using MA TLA B. The com puter source code is enclosed in B-1of Appen dix B. The curves generated for above fianctions are plotted in d imensionlessunits and are show n in Figures 5.4, 5.5, and 5.6, respectively. T hese curves can be usedas reference for calculating the deflections once the equation coefficients are determined.

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-M

Figure 5.4: Functions of F(x), P(x), and S(x)

Figure 5.5: Functions of G(x), Q(x), and T(x)

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Figure 5.5: Functions ofH(x), R(x), and U(x)

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CHAPTER 6DRILL-STRING BUCKLING - ANALYSIS AND DISCUSSION

6.1 Critical Buck ling C ondition of the First OrderLet X} and x^ designate the values of x for the upp er and low er ends of the drilling

string respectively. The distance X} represents the distance between the neutral point andthe top of the hole. The distance x^ represents the distance between the bit and neutralpoint. Let P}, Q}, R}, Sj etc. deno te the values of P(jc), Q(x), R(x), S(x), etc , at x=X} andPi Qi Ri S2, etc, denote the values of P(x), Q(x), R(x), S(x), etc, at x=X2. As both theends of the d rilling string are hinged, the bend ing m om ents at these two ends are equal tozero. From equations 5.21 and 5.29, we get

aPi+bQi-\-cRi =0 . . . (6.1)

aP2-\-bQ2-\-cR2 =0. (6.2)At both the ends, y= 0 and equ ation 5.28 gives for both the ends:

aSi-\-bTi-\-cUi-\-g = 0 . (6.3)

aS2-^bT2-\-cU2-\-g = 0 . . (6.4)Eliminating g from equations 5.41 and 5.42, we get

a{Si-S2)^b{T^-T2) + c{U^-U2)=0 . . . (6.5)

Equations 6.1 , 6.2, and 6.5 together yield the relationship b etween X} and x^ for bucklingto occur. This is obtained b y eliminating the coefficients and equating the determinant tozero.

^i i ^iPl Ql ^2{8,-82) {TI-T2) {UI-U2)

(6.6)

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Equation 6.6 is solved by iteratively using computer program and the values of x/ and X2determined. These values are plotted and shown in Figure 6.1. The computer source codefor determining the solution is enclosed in B.2 of Appendix B.

X2, Dlmensionless Units-S.O-7.5-7.0-G.5e.o-6.5-6.0-4.5-4.0-3.5-3.0-2.5-2.0-1.6-1.0-0.50.0

\ \ \ \ , sN.\ ,^

^

>\ sl\ N- ^

~. ~

1.80 1,85 1.90 1S5 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70Figure 6.1: Critical Co ndition of the First-Order

From Figure 6.1 , it is observed that w hen the absolute value of x; is sm all, i.e.,wh en the ho le is very sh allow, the pipe requires a larger weight on the bit in order tobuck le. W hen the hole is deeper, the critical value of the weight on the bit decreases andapproaches asym ptotically to a certain value. Under actual drilling con ditions, xy is verylarge and the value of x is equal to its asymptotic limit. As seen from Figure 6.1 , theasym ptotic limit is reached when the value of x; is equal to -6.0 and the correspondingvalue of X2 is equal to 1.94. Hen ce, it may be co nsidered with negligible error thatX2=1.94 is the critical con dition of the first-order.

The critical length of stands of pipe stacked vertically in the derrick can beobtained by referring to Figure 6.1. The critical length corresponds to the condition whenx/=0 and the corresponding value of X2=2.65.

The critical buckling of the first-order was investigated in Chapter 5 and it wasfound that at this condition, the distance x between the bit and the neutral point is equal

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to 1.94 dimensionless units. From equation 5.18, the length in feet of one dimensionalunit equals:

m = 3,EI . , . (6 .7)

The weight, in Ib s of the length of the drilling string equal to one dim ensionless unitequals:

mp = ^^EIp (6.8)

The critical length of the driUing string for critical buckling of the first order is obtainedby mu ltiplying equation 6.8 with 1.94. Figure 6.2 shows the plot of length in feet of onedimensionless unit for different drill-pipes and drill-collars with change in drilling muddensity.

8 10 12Flujd Dens ty (Ibm/gai)

20

Figure 6.2: Length of a dimension less unit with change in driUing mud densityFrom the above figure, it is observed that as the drilling fluid density increases,

the length of on e dimen sionless unit also increases. Howev er, the increase in length is notgreat enough when compared with the length of the drilling string, AIso, for a given

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drilling fluid den sity, the range of length of on e-dimensionless unit for drill-pipe or drill-collars is very sm all. Hen ce, for an alysis purpo se, it can be assumed that the length ofone-dimensionless unit for drill-pipe is approximately the same for both drill-pipe anddrill-coUar. The critical weight on bit for critical buckling of the first-order is obtained byusing equation 6.2. The critical weights on bit for first-order critical buckling are plottedand sho wn in Figure 6.3. From Figu re 6.3, it is observed that heavier the mu d, the sm allerare the critical weights on the bit. How ever, the influence of the drill fluid density onbuck ling is not very significant.

14000

12000

10000

o 6000

4000

2000

1 DC;7.0"O4),3.0"l.D |

1 DC: 6.25** OS), 2 ^ 5 " W

|DP:4^"b j ) .3 .826" I .D1 1 DP; 2.875" OJ), 2.151" 1.0.

==*- . - ^ . = ^

1 DC 4.75" OJ), 1^75" LO | ^1 DP: 5.5" 0,D, 4,67" U) |

.-.-

' ' " " '

8 10 12riuld Densi ty {ibm/^3\)

14 16 18 20

Figure 6 ,3: Critical W eights on bit for First-Order Critical Bu cklingDrill-Strings used for d rilling are usually a comb ination of d rill-pipes and driU-

coUars, and henc e their comb ined effect m ust be taken into account wh en analyzingbuck ling. Two different scenarios are possible with this com bination. One scenario iswhe re the neutral point is assumed to be located in the drill-pipe and the other scenarioconsiders the neu tral point to be located in the d rill-collars. Considering the first scenario,the critical w eight on the bit in this case w ill be equal to the w eight of the drill collars andthe portion of the drill pipe below the neutral point. If the length of the driU coUars isrepresented by Zc, then the length of the collars in dimensionless u nits is given by LJm.

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For critical cond itions of the first-order, th e length in dimension less units of the portionof the driU-pipe below the neutral point is given by:

\.94-Ljm. . . . (6 .9)

The w eights of these two portions in the presence of drilling fluid are LcPc and(l .94 - L ^ /m)m Pp , respectively. The sum of the weights equal to the weight W} andequals:

Wi=L,(p,-pp)+\.94mpp. . . . (6.10)

From equation 6.10, it is observed that the critical weights are linearfiinctionsofthe total length of the drill-collars. The slope of the straight line and the>'-intercept beingequal to Pc-Pp and \.94mpp, respectively. This is represented by the inclined line inFigure 6.4. For the second scenario, the who le weight on the bit is given by the drillcollars. The critical values of the weight on bit cannot be increased by adding mo re drillcoUars. The second scenario corresponds to the horizontal Une of Figure 6.4. The do ttedlines in Figure 6.4 represent similar conditions in the presence of higher drilling fluiddensity. From Figure 6.4 to prevent first-order buckling in the presence of 10 Ibm/galdrilling fluid, the w eight on the bit should not exceed 8600 Ibs. In the presence of 16Ibm/gal drilling fluid, the w eight on bit should not exceed 7800 Ibs. If the weight on bitexceeds these va lues, then the drill string will buck le once until the weight on bit exceeds16600 and 15400 Ibs, respectively, when the d rill string will buckle a second tim e. Thecritical weight on the bit can be increased b y using d rill-collars w ith bigger size, highstifftiess and elasticity values.

Figure 6.5 shows the effect of driU coUar size on first order buckling with a radialclearance of 2.75-in. betwe en the h ole and the d rill collar and in the presence of 12.0Ibm/gal drilling m ud. T he drill collar inside d iameter is taken as 1.875-in. It is observedthat the critical weight on the bit increases as the drill collar size increases. This is due tothe increase in stifftiess of the drill collars with increa sing size. Similar plots are show n inFigures 6.6 and 6.7 for drill collar inside diameters of 2.5-in and 3.0-in, respectively. As

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the driU collar inside diameter increases, the critical weight on bit also increases andsimilar profiles as in Figu re 6.5 are o bserved.

Drll-Ppe: 4,5-in, 16.6 Ib/ft. Drll-Collar: 6.25-ln O.D,2.25-in LD.S.S Ib/ft180001600014000

12000

co 8000

6000

40002000

lOlbm/galMud// '/ // /

/ /

16lbm/galMud

Dr j l l lng S t r ng Buck led Once

// lOlbm/galMud/'// ' /

/ / '//'

1 1 1 1 ' \

16lbm/galMud

Straight Dr l l ing String

6 8 10 12Number of Drll Coflars

14 16 18 20

Figure 6.4: Buckling Conditions for Combination Drilling Strings160001400012000

kix:o

10000

80006000

4000 2000

! 1m \ Denslty: 12.0 Ibm/galOearance: 2.7S-in.Dr^Cotar 1.0:1.875-111.

y* y ^

12010590 Xto

- 7 515 60 w g

30 'Co15

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0Or i l lCoI larO .D, in .Figure 6.5: Influence of DrilI-CoIIar size on Buckling (I.D=1.875-z^)

83


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20000180


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6.1.1 Point of Tangencv for First-Order Critical ConditionLet the point of contact w here the drill-string contacts the bore-hole wall be

represented in feet by JSG and in dimensionless units by x . Atx=X3, the slope dy/dx=0 andfrom equation 5.27, we get

aF^-\-bG2-\-cH^ =0. . . . (6 .11)

As the bending moments at the ends are equal to zero, equations 5.39 and 5.40 can beused along with equation 6.5.

aPi-\-bQi-\-cRi =0 ... (6.12)

aP2+bQ2+cR2 =0. . . . ( 6 . 13)The abo ve three equations ha ve possible solutions if the following determinant equalszero.

0 . . . (6.14)The above equation gives the relation between x^ at the point of tangency and X} and x^ atthe end of the strings for critical condition. But at critical condition of first order, X}=-6.0and X2=\ .94. Hen ce the corresponding value of xj for which the determinant given byequation 6.14 is equal to zero gives the value ofxs. The computer source code fordetermining the value oxs is enclosed in B .3 of Appendix B. The value ofxs thusobtained is equal to 0.145.6.1.2 Equation Coefficients for First-Order Critical Condition

In order to determine the shape of the buckled string axis, the distribution ofbending mo men ts, etc , the values of a, b, and c from equations 6.1, 6.2, and 6.5 needs tobe d etermined. Bu t it is observed that indeterm inate values of these factors are obtainedfrom these equations. This is because the bending moment defined by equation 5.21 is

85

F3^ lPl

G3QiQi

H3R xR i


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vaUd only for smaU deflections. AIso, when the d rill-string contacts the bore-hole w all,further b ucklin g is stopped until the weight on the bit is increased to a new critical value,He nce, in order to remove the indeterminacy, at the point of tangency the deflection istaken equal to the app arent radius of the hole. The apparent radius of the hole is definedby the following relation.

r= {D-D,) . . . (6.15)

and r= {D-D,) . . . (6.16)

where D is the diameter of the hole, D^ is the outside diameter of the driU collars, and Ais the outside d iameter of the tool join ts.

Hence, aXx=X3, Y=r and according to equation 5.13, j^=r/m. Equation 5.28 gives 5 3 + 6 r 3 + c t / 3 + g = . - . . . (6,17)m

At the lower end of the string, the deflection is zero, i.e., forx=X2=1.94, y=0. Hence,equation 5.28 gives

aS2-\-bT2-\-cU2-^g = 0. . . . (6.18)Eliminating g between equations 6.17 and 6.18 and rewriting equations 6.1 and 6.2, weget

^(^3 -Si)^b{T^ -Ti)+c{U^ -Ui)= - ... (6.19)

aPi-\-bQi-\-cRi =0 . . . (6.1)

aP2-\-bQ2-\-cR2 = 0 . . . . (6.2)

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Substitutmgxy=-6.0, X2=\.94, andx i=0.14 5 in the above equations, the values oa(m/r),b(m/r), and c(m/r) can be calculated. From equations 6.1 and 6.2, we get

a ^ _ ^QxRi -QiRi ~ R\P2 -RiP\ ~ PiQi -PiQi

a = 9l lzQAc . . . (6.20)P\Q2-P2Q\^^R^PIZMLC. . . . (6.21)^ 1 0 2 - ^ 2 0 1

Substituting equations 6.14 and 6.15 into equation 6.13 gives:c = X . . . . (6.22)m (5 _s^)Q^^2-Q2R\ ^ ( y _T^f\P2 -R2P\ ^ ( ^ _ ^ )^ P l 2 - ^ 2 l ^ P l 2 - ^ 2 a

6.2 Critical Buckling Conditions above First-OrderAfter the drill string undergoes first-order buckling and if the weight on the bitgoes on increasing, there is a tendency for the drill string to buckle asecond time. In such

a situation, the differential equation defined by equation 5.11 does not hold true for thewh ole length of the drilling string. For the upper portion of the drilling string (locatedabove the point of tangency), the force F which is the reaction of the wall of the holeagainst the pipe needs to be considered. Henc e, equation 5.11 is modified as:

d^Y dYE I ^ + pX + F2-F = 0. . . . (6.23)dX ^ dX ^

The equ ations 5.19 and 5.23 are also modified as follows:c = ^^/^ ...(6.24)pm

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d'^zand ^ j c z + ci =0. . . . (6 .25)Jx^The integration of equations 5.23 and 6.25 gives the same kind of general

solution, i.e., equations 5.27, 5.28, and 5.29. However, not o nly c but also the integrationconstants a, b, and g becom e different for the lower and u pper portions of the drillingstring. The constants y, 67, c/ and a^, b^, c^ refer to the upper and lower portions of thedrilling strings above the tangen cy point. The equations for the upper portion can bewritten as:

y = aiS{x)-\-biT{x)-\-ciU{x)-\-gl ... (6.26)

^ = aiF (x) + 6iG (x)+ ci/ / (x ) . . . (6.27)dx

^ = aiP{x) + biQ{x) + ciR{x). ... (6.28)dx^The value of cj = F^ /pm .The correspo nding equations for the lower portion can be written as:

y = a28{x)+b2T{x) + C2U{x)+g2 . . . ( 6 . 2 9 )

^ = a2F{x)+b2G{x)+C2H{x) ... (6.30)dx2

2 ^ = a2P{x)-^b2Q{x) + C2R{x). . . . ( 6 . 3 1 )dxThe value of c^ = F^ /pm

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Let X} correspond to the upper end of the dril l ing string as previously stated andvalue of xy=6.0. Let x^ correspond to the lower end of the dril l ing string, and X3 to thepoint at which the pipe is tangent to the wall of the hole. The three boundary conditionsfor the upper portion of the dril l ing string are as follows:

1. Th e ben din g mom ent is equal to zero at the upp er end of the string wh ich leads toequat ion 6.32.

2. A t x=X3, the s lope is equal to zero which leads to equation 6.33.3 . Atx=xy , the de f lec t ion j ;=0 .4. A t x=X3, the deflection j ; is equal to the apparent radius of the hole, i.e., y=r/m.

The third and fourth boundary conditions when substituted into equation 6.26,give two expressions from which upon elimination of g, equation 6.34 is obtained. Thethree boundary conditions for the lower portion of the string are same as for the upperportion of the string which leads to equations 6.35, 6.36, and 6.37, respectively. Oneadditional boundary condition is that at x=jC5, the bending mo men ts are equal. Hence,equa tions 6.28 and 6.31 , w e get equation 6.38. The equations are as follows:

aiPi + biQx + cii?i = 0 . . . (6.32)^1^3 + 61G3 + C1//3 = 0 . . . (6.33)

l f e - ^ l ) + l ( ^ 3 - ^ l ) + q ( ^ 3 - t ^ l ) = - . . . ( 6 . 3 4 )

2 ^ 2 + ^ 2 0 2 + ^ 2 ^ 2 = 0 . . . ( 6 . 3 5 )

2 ^ 3 -^b^Gi^ + C 2 / / 3 = 0 . . . (6,36)a2{S -S2) b2{T -T2) +C2{U^-U2)-- . . . ( 6 . 3 7 )

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1^3 +^103 + q ^ 3 - 2 ^3 ~hQ2> - ^ 2 ^ 3 = 0 . . . (6 .38)

The above seven equations have possible solutions only if their determinantequals ze ro. This cond ition show n in equation 6.39 in which xy=-6.0 gives therelationship between x^ and X 3. The v alues o f x and X3 calculated using this equation isshow n in Tab le 6 .1. The com puter source code for calculating the value of Xi when thesecond buckle contacts the bore-hole wall is enclosed in B.4 of Appendix B. Thecomputer source code for calculating the different values of x^ and X3 using equation 6.39is enclosed in B.5 of Appendix B.

^i Q\ RiF^ C^ H^{8^-8,){T,-T,) ( f / 3 - / i )000

^ 3

000^3

0003

000

P2P3

0002

G3

000

R2H3

00rm00{82-82) {T3-T2) {U3-U 2) L

-P3 -Q3 -R3 'H

0 . . . (6.39)

Table 6.1: Values of x^ an dx j for Critical Cond ition above First Orderx2

1.940002.000002.300002.600003.200003.500003.753004.000004.21800

x30.144500.217100.580200.942801.666502.030802.344702.671003.08550

x2-x31.795501.782901.719801.657201.533501.469201.408301.329001.13250

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6.2.1 Equation Coefficients for Crtical Conditions above Frst-OrderThe values of x^ andxj obtained from Table 6.1 are substituted into equations

6.32 throug h 6.37 and the value of the equations coefficients determined are by so lvingthe simu ltaneous equation s. For determining the equations coefficients for the upperportion of the string, we have

1 bi ciQlH^ - G3R1 i?iF3 - P1//3 P1G3 - F^Qi

For the lowe r portion of the string, from equations 6.35 and 6.36, we get:^2 = ^2 = ^^Q2H2 - G2R2 R2P3 - P2H3 P2G3 - F3Q2

Substituting equations 6.45 and 6.46 into equation 6.37 gives:

91

. (6.40)

Q,H,-G,R,P1G2-F2Q1

RlFj-P^H^ ,^ .r..b\ = - zrTr^i (6-42)P1G3 - F^QxSubstituting equations 6.41 and 6.42 into equation 6.34 gives:c i= X . . .. ( 6 .43)m ^ _^^QiH3z^^(T _T)MlzMl + {u^_U,)" P.G^-F^Q, 'P\G3-F3Qx

(6.44)

Q2H3-G2R2 .....flT = Cj . . . ( 6 . 4 5 )^ 2 ^ 3 - ^ 3 0 2^ MIZM C . . . . (6.46)^2^3-^302


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6.3 Shape of the Buckled D rlUng StringLet the deflection coefficient h be defined b y the following equation:h = y . . . . (6 .48)r

From equations 6.48 and 5.13, we getY = hr . . . . (6 .49)

From above equations, h is equal to unity for the deflection Y equal to the apparent radiusof the hole r. At the upp er end of the string above the tangency po int the deflection iszero and equation 6.26 gives

0 = aiSi + bxTi + cit/i + g i . . . . (6.50)EUminatinggy between equations 6.26 and 6.50, we get:

y = a^{S-8^)+b^{T-T^)+c^{U-U^). . . . (6.51)At the low er end o f the strng, the deflection is equal to zero and equ ation 6.29 gives:

0 = fl2'^2+^2^2+C2t^2+^2^ ...(6 .52 )Eliminating g2 between equations 6.23 and 6.52 gives:

y = a2{8-82)+b2{T-T2)+C2{U-U2). . . . ( 6 . 53 )

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The v alues of the equation coefficients for the upper and lower portions of thedrilling string calculated eariier can be substituted into equations 6.51 and 6.52 to givethe d eflection j/ at each value of x. The shape of the buckled drilling string has beenobtained forxy=-6.0 and forx2=1.94, 3.753, and 4.218, respectively. The distancex2-xrepresents the distance above the bit and the shape of the buckled d rilling string is shownon a plot of y v ersus x -x in Figure 6.8. It has been determined by iterative process that atX2=4.219; the second buck le contacts the wall of the ho le. As the length ofone-dim ensionless unit does not vary much from one type of driUing string to another,the shap e of any buck led d rilling string is almost the same for any order of buckling.How ever, in case of a drill-pipe, it corresponds to a m uch lesser weight on the bit. As theweigh t on the bit increases between the first-order and second-order, the shape of thebuck led string changes and at a particular weight on bit above the critical weight of thesecond order, the drill string buckles again and contacts the wall of the hole at two points.Com parison of these buckled curves indicate that the portion of the strng located close tothe bit is deflected mo re, while the portion located above the tangency po int isprogressively straight.

+1-

* **> c

X

-3.0 1.0 -03 0.0 0.5 1.0D^flectjon. y (Dimensionldss unjt$)Figure 6.8: Shape of Buckled C urves for Different Buck ling O rders

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A plo t of X2-Xi ve rsus x is obtained and show n in Figure 6.9. The d ata for thisplot is obtained from Table 6.1. From this figure, the change of shape of the buckledcurves and varation in position of the neutral and tangency points can be easilyvisualized

c41

4. 0

3.0

f I 2.0* .2

1.0

0.0y

j i*.-'

CritlcalCondition -First Ord*r

^''

4

rt'//

^*f

CrticalCondtion -Second Order

,-''y'

>'

i>

> ' " 1SecondBuckleContcts Wall

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0x , Distane betwftn B i t and Neutra l Po in t . Dmension iess un i ts

4. 5 5,0

Figure 6 .9: Tangency and Neutral Points Variation with different Buck ling O rdersIn Figure 6.9, the abscissa represents the distance between the bit and the neutralpoint in dimen sionless u nits. The curve represents the varation of the distance between

the bit and the tange ncy po int which is taken as the ordinate. In order to determine therelative position of the point of tangency w ith respect to the neutral point, a dashedstraight line is drawn at an angle of 45 degrees with respect to the coordinate axis. Theordinate of any point on this dashed line indicates the distance between the bit and theneutral point. Th e vertical distance between the curve and the dashed line indicates thedistance betw een the tang ency and the neutral points. It is observed from the figure that,as the weight on the bit goes on increasing or the distance x^ goes on increasing, thedistance between the tangency and the neutral points goes on increasing and the distancebetween the bit and tangency point goes on decreasing. In other words, as the weight onthe bit increases, the neutral point moves upwards and the tangency point moves

94


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dow nwards. Th e downw ard displacement of the tangency point becomes m uch fasterwh en the weigh t on the bit is increased betwee n the critical value of the second -order andthe weight for which the second buckle con tacts the wall of the hole.

It is also observed that at higher buckling orders the highest buck le contacts thewall of the hole at the neutral point; and that, as the weight on the bit is increased, thepoint of tangency is slightly displaced downward while the neutral point moves rapidlyupward. Knowledge of the location of the tangency point may be usefiil when drillinghard form ation be low a soft and caving shale. If less weight than the critical weight of thefirst-order is used while drilling in the caving shale in order to avoid caving and once thehard formation is encountered, then the weight on the bit should not be increasedimm ediately. If the weight on the bit is increased imm ediately, the drilling string willcontact the caving shale wall above and caving could occur. One-and-half-dimensionalunits should b e drilled before weight on bit is increased and at that time, the weight o n bitshould be increased immediately rather than progressively in order to obtain a tangencypoint in the hard form ation. It is observed that, drilling shou ld be carried w ith a weight onthe bit corresponding to approximately 3.75 dimensionless units between the bit and theneutral point b ut should not exceed 4.22 dim ensionless units in order to keep the secondbuck le from contacting the caving shale above. Higher values of weight on bit should notbe used before 4.2 dimen sionless units of the hard formation are drilled.

The crtical weight for buckling of the second-order to occur is obtained b ym ultiplying eq uation 6.8 by 3.753 units. Figure 6.10 shows the crtical weight on bit forsecond order buck ling for different drll pipes and drill collars. The ma gnitude of thecritical weigh ts is higher than the co rresponding w eights of the first order buckling. Asthe drlling fluid density increases , the crtical weight on bits decreases. How ever, thedecrease is very sm all when com pared with the change in the drlling fluid density. Thecom puter source cod e for calculating the deflections of the buckled drlling strng at eachposition is enclosed in B.6 of Appendix B.

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14000

2000

Figure 6.10: Crtical Weights on Bit for Second-Order Buckling6.4 Bending Moment Diagrams for First and Higher-Buckling Orders

W hen the d rilling strng buckles, each cross section becom es subjected to abending moment generating a tensile stress on one side and a compressive stress on theother side. A s the d rilling strng rotates, these stresses are reversed in each rotation cycleand can lead to fatigue loading of the drlling strng materal. Let the bending mo mentcoefficient be defined by:

i =2d y m

dx^ r . . . (6.54)

By eliminating Sy/do^ from equation 6.54 and 5.21 , we getM =i pmr. . . . (6.55)

For any given size of drll-pipe or drU collars, equation 6.55 indicates that thebend ing m om ent increases w ith increase in the apparent radius of the hole. If the size ofthe hole is constant, then the bending moment is directiy proportional to the bendingcoefficient /. Sub stituting the equation coefficients v alues into equation s 6.28 and 6.31

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and using equation 6 .54, we get a relation between the bending mo ment coefficient anddistance x.

The d ata thus obtained is plotted and show n in Figures 6.11 , 6.12, and 6.13 forfirst-order buckling, second-order buckling, and when the second buckle contacts thebore hole wall, respectively. From Figure 6.11, it is observed that the bending momentcoefficient has two points at which the bending m om ent is a ma ximu m. The larger ofthese tw o is found to be closer to the bit. A similar pattem is observed in Figures 6.12 and6.13, and it is observed that the maximum bending moment occurs near the bit and isleast at points farthest from the bit. In Figures 6.12 and 6.13 , a discontinuity is o bservedin the profile at a point w here the strng contacts the wall of the hole w hich appUes alateral load on the strng. There is no discontinuity of the bending m om ent coefficient atwh ich the second bu ckle co ntacts the wall of the hole, because there is no lateral loadapplied to the string at this point. The points where the bending m om ents are zeroindicate points of high shear stress forces acting on the drilling strings. The com putersource cod e for calculating the deflections of the buckled drilling string at each positionis enclosed in B.6 of Appendix B.

.t:X *-'

l ls lc ^.? I S

1 "

& "

T

6_5 -4 -^

4 -

/

"''-2.0 -1.5 1.0 -0.5 0.0 0.5 1.0

B end ng Mome n t C oe t f i c l en t i iD imens ion less un l ts f1.5 2.0

Figure 6.11: Bending Moment Coefficient Profile for First-Order Buckling

97


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C* i .t:T c? =o **s iC tf. 1.

"

f 4 - ]y _

' 8-^f-^

Jr

I j ^

4 -

...-f ** '

' 1 .5 1.0 -0.5 0.0B e n d i n g M o m e n t C o e f f c i e n t . I ( D m e n $ o n l e s s u n i t s )

0. 5 1.0

Figure 6.12: Bend ing Mo men t Coefficient Profile for Second-Order Buck ling

m _! => ll ls.iC iAX X

-2.0 1.5 -1.0 -0 .5 0 .0B e n d i n g i v io m e n t C o e f f i c e n t I ( D i m e n s o n l e s s u n i t s f

Figure 6 .13: Bending M omen t Coefficient Profile: Second buck le contacts bore-hole waU

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6.5 Force AppUed bv Buckled Drill-String on bore-hole wallThe reaction F of the wall of the hole on the b uckled drilling string is equal to:

F = F2-Fi. . . . (6 .56)Substituting the values ofF} and F2 from the value of cy and C2 given in equations 6.28and 6.3 1, we get

F = pr ^ m m^V r r j (6.57)

If the term w ithin brackets in equation 6.57 is defined as the coefficient/, we getF = fpr. . . . (6.58)

The force coe ffic ien t/is a fimction of cy and c and hence depends on the distancebetwe en the bit an d the neutral point. This distance is also proportional to the weight onthe bit. The variation o f/ w it h the distance in dimensionless units between the bit andneutral point is show n in Figure 6.14. The comp uter source code for calculating the forcecoe fficien t/is enclosed in B.5 of Appendix B. When the weight on the bit progressivelyincreases and the weight on bit reaches the critical condition of the first order, thecoefficient and, consequently the force F is equal to zero. This is because, wh en b ucklingis on the verge of occurring, the sUghtest increase in the rigidity of the pipe w ouldprevent it. Similarly, if it occurs, the sUghtest force can stop it and the reaction of the wallof the hole on the buck led string is zero.

As the weight on the bit becom es greater than the critical value, the co effic ien t/and consequently the force Fin crea ses . The greatest value of /a t which the secondbuck le contacts the w all of the hole is found to be 2.7 from the figure. The g reatestmag nitude of the force is found by sub stituting the value of /i n to equation 6.58, i.e.,

F = 2.7 pr . . . (6.59)

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3.0

2. 5

cS 1.61,0

0.5

0.0

ChticalConditlon 'Frst Order

I I

P Secondauc ieContactsWall

CriticalCondition -Second Order

/ 0.0 0.5 1.0 1.5 2,0 23 3.0 3.5X2, D imens ion less U n t t s D i s tance be tw een B i t and N eu t r a i P o in t

4.0 4.5

Figure 6.14: C oef ficien t/for calculating force on bore-hole wall due to drill stringbuckling50 0

10 20 30 40 50 60 iOActuo l Hole D imeter . in .$0 100

Figure 6.15: Force of Buckled Drilling String (Second Buckle Contacts Bore Hole Wall)

100


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The values of the force F given by equation 6.59 is plotted in Figure 6.15 forvariou s sizes of drill pipe and drill coUars in a drilling m ud of 12 Ibm/gal density. Fromthe figure it is observed that the mag nitude of this force is not very high when comp aredto the weigh t on the bit. It is also found out that in most formations the hole remainsstraight in spite of the buckled strings. How ever, if the formation has a tendency to cave,the rate at wh ich the cave grows accelerates because the force involved increases with thediameter of the cave.

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CHAPTER 7CONCLUSIONS AND RECOMMENDATIONS

A PDC "bit-wobbling" model has been developed and analyzed for both lateraland angular vibrations. Resonance is responsible for severe bit wobbling which maydamage PDC bits rapidly in hard formation drilling. Resonant frequency is very sensitiveto rock properties. The amplitude of bit vibration at both resonance and non-resonanceconditions has been determined. The factors that influence the backward cutter velocityhave be en d etermined. T he rotational motion of the bit is inversely proportional to theback ward cutter velocity and it is observed that the cutters located at the bit center tend tochip out and wear more. As the rock becomes harder, the vibration amplitude reduces andthe frequency of vibration increases. The lateral bit contact area, damping factor, drill-collar stiffiiess and bit weight are inversely proportional to the backward cutter velocity.The b ackw ard cutter velocity is highly sensitive to the stabiUzer length within 10 feet,and beyon d a length of 10 feet the backw ard velocity stabilizes to a constant value. It isrecom me nded that the stabilizer be placed as close to the bit as possible to prevent bu ild-up of large vibration amplitudes.

The present m odel can befiartherrefined b y cond ucting field tests and correlatingthe expe rimental results w ith the theoretical results obtained from this study. The effectof drilling param eters such as weight on bit and rotary speed, on the backward cuttervelocity wh ich cannot be analyzed using the present mo del can be incorporated as fixturestudy in this area. Further w ork can also include determination of the critical rotaryspeeds and fluctuation am plitudes of the cutting forces and cu tting torque . The effect ofshear forces o nly at the bottom of the BHA has been co nsidered for determining the staticand dy nam ic elastic constants. Th e effect of other forces such as the axial forces, bend ingm om ents and torques acting on the top and bottom of the BHA and their effect on theBH A elastic constants can be taken as a further study.

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The differential equation for drilling-string buckling has been developed and itsanalytical so lution d etermined. Th e critical values of weight on bit at wh ich the first andhigher order buckling occur have been determined. It is observed that the length ofone-d imen sionless unit for first-order buck ling is almost the same for a drill-pipe anddrilI-coUar. The critical weight on bit reduces as the drilling fluid density increases for agiven size of drill-pipe and drilI-coUar. The regions w ithin which the m agnitude of theweigh t on bit can be set to prevent buckling hav e been determined. It is also observedthat the magn itude of the force acting on the bore-hole w all when the second bucklecontacts the b ore-hole w all is very small. This force is directly proportional to theapparent rad ius of the hole and tends to increase while drilling in a caving formation. T heshapes of the buckled drilling string for different buckling orders have been determined.The point of tangen cy for different b uckling orders also has been determined w hich givesinsight into the depth at which the weight on the bit needs to be applied wh en a hardformation is encountered.

Further study in this area includes determining the buckling conditions andbuckled shapes for higher buc kling orders above tw o. As drill-strings are subjected tocom bined reversed b ending stresses due to its rotation that causes fatigue failure, theeffect of fatigue on the strength of a drill-string and its buckling can be studied. Theeffect of a drilling string having v arying sections of material properties and its combinedeffect on buckling can be studied. The effect of drill-pipe vibration on buckling can alsobe taken up as further study.

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REFERENCES1. Brett, J.F., Warren, T.M., and Behr, S.M.: "Bit Whirl: A New Theory of PDC BitFailure," SPE DriUing Eng., December 1990.2. Winters, W.J.: "Roller-Bit Model with Rock Ductihty and Cone Offset," paper SPE16696 presented at the SPE Annual Technical Con ference and Exhibition, Dallas,September 27-30, 1987.3. Oniya, E.C.: "Relationships between Formation Strength, Drilling Strength, andElectric Log Properties," Paper SPE 18166 presented at the SPE Annual TechnicalConference and exhibition, Houston, October 2-5, 1988.4. Glow ka, D.A, and Stone, C.M .: "Effects of Therm al and Mechanical Loading onPDC Bit Life," SPED E; Trans., AIM E, 281, June 19865. Zijsling, D.H.: "Analysis of Temperature Distribution and Performance of PDC Bitsunder Field Drilling Conditions," Paper SPE 13260 presented at the SPE AnnualTechnical Conference and Exhibition, Houston, September 16-19, 1984.6. W arren, T.M. and Sinor, L.A.: "Drag -Bit Wear M od el," SPED E; Trans., AIM E,287, June 19897. W arren, T.M. and Sinor, L.A.: "Drag-Bit Performance M odeling ," SPEDE 119-27;Trans. AIM E, 287, June 19898. Brett, J.F., W arren, T.M., and Behr, S.M.: "Bit W hiri: A New Theo ry of PDC BitFailure," Paper SPE 19571 presented at the SPE Annual Technical Conference andExhibition, San Antonio, October 8-11, 1989.9. Brett, J.F., W arren, T.M., and Sinor, L.A.: "Developm ent of a Whiri-Resistant Bit,"SPE Drilling Eng , December 1990.10. Boyun Guo and Geir Hareland: "Bit Wobble: A Kinetic hiterpretation of PDC BitFailure," Paper SPE 28313 presented at the SPE Annual Technical Conference and

Exhibition, New Orleans, September 25-28, 1994.11. Huang, Tseng and Dareing, D.W.: "Buckling and Lateral Vibration of Drill Pipe,"ASME Paper No. 68-Pet. 31 presented at the Petroleum Mechanical Engineering andFirst Pressure Vessels and Piping Conference, Dallas, Texas, September 22-25,1968.

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12. Huan g, Tseng and D areing, D. W.: "Buckling and Frequencies of Long VerticalPipe s," proceedings of the Am erican Society of Civil Engineers, Vol. 95, No. E M l,pp. 167-181, February 1969.13. Bary shniko v, A ., Calderoni, A., Ligrone, A., Ferrara, P., and Agip , S. P. A., " A New

Approach to the Analysis of Drilling String Fatigue Behavior," SPE 30524 presentedat the SPE An nual Techn ical Conference and Exhibition, Dallas, Oct. 22-2 5, 199814. Arthur Lubinski,: "A Study of the Buckling of Rotary Drilling Strings," presented atthe SPE Spring Meeting, Mid-Continent District, Division of Production, OklahomaCity , M archl9 50.15. Black, H. Paul.: Theory ofMetal Cutting, McGraw-Hill Book Company, Inc, NewYork, 1961.16. Burr, H.. Arthur, Cheatham, B. John.: Mechanical Analysis andDesign, SecondEdition, Easter Economy Edition, Prentice-Hall India, New Delhi, 1999.17. Khu rmi, R. S.: 8trength ofMaterials, Twenty-First Edition, S. Chand and CompanyLtd., 1990.18. Rao, S. Singiresu.: Mechanical Vibrations, Addison-Wesley Pubhshing Company,Reading,M A, 1985.

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APPENDIX ABIT WOBBLING DATA

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oc

tot i-H>

H

Si

00II

Si

Si

Si

Si

Si

8

Si

Si

Si

Si

Freq.

Ratio

05

05

05

05

05

05

05

05

05

05

05

0

05

0

05

05

055

053

0

05

05

0

53

0

58

0.1

0

0

0

0

58

057

05

2699000

05

0

58

05

0

0

035

0

0

3

0

5

0

058

05

05

0

0

7

0

3

0

0

38

035

03

0

0

55

0

9

055

05

0

0

7

0

8

0

0

9

0

9

03

037

0

7

0

8

055

05

0

7

07

078

0

0

39

0

5

0

8

03

035

0

3

0

05

07

0

09

0

0

0

0

57

0

038

03

0

05

075

0

0

9

Documents

Analysis of PDC Wobbling & DS Buckling