Drillpipe Buckling in Inclined Holes Rapier Dawson, SPE, Exxon Production Research CO.

P.R. Paslay, consultant

Summary

In high-angle wells the force of gravity pulls the drill­string against the low side of the hole. This stabilizes the string and allows drillpipe to carry axial compressive loads without buckling. For this reason it is practical to run drillpipe in compression in high-angle drilling where the drillcollar weights, which are needed to avoid com­pression, would cause excessive torque and drag.

Introduction

In drilling vertical wells it is common industry practice to avoid loading drill pipe so that it would be unstable from a simple column-buckling point of view. Drillpipe is kept stable by making sure that the buoyed weight of the drill collars and heavyweight pipe exceeds the weight on bit (WOB). This practice was recommended by Lubinski in 1950. 1

In drilling directional wells it is common practice to use about the same bottomhole assembly (BRA) weight that would be used for a vertical well. Most operators do not add collars as hole angle increases even though they may have several thousand feet of drillpipe in compres­sion in high-angle holes. This practice seems to work fairly well; the high incidence of drillstring failure that would be expected if compression really were harmful does not occur. This paper is intended to justify and sup­port this practice by showing that drillpipe can tolerate significant levels of compression in small-diameter high­angle holes because of the support provided by the low side of the hole.

The benefit of using drillpipe in compression is that the BRA weight can be kept low in high-angle drilling. This, in tum, helps reduce torque and drag, which are often operational constraints in very deep directional drilling.

Earlier Work

The fact that a high hole angle can stabilize drillpipe against buckling seems to have been recognized first by Lubinski and Woods in 1953. 2 They stated that for given values of drill collar weight and compressive load, "there is a value of ¢Ir below which helical buckling may occur and above which it cannot occur." Lubinski and Woods mentioned that they had done some ex­perimental studies to quantify this phenomenon and that the results of these tests had been used in preparing some of the figures in their paper. Unfortunately these brief comments were not noticed widely by the drilling industry.

After Lubinski and Woods' work, the contribution of hole angle to drillpipe stability was ignored in the drill-0149·2136/84/0101·1167 $00.25 Copyright 1984 Society of Petroleum Engineers of AIME

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ing literature for 30 years. In 1983, Dellinger published a crossplot and extrapolation of Lubinski and Woods' results. 3 Dellinger's presentation is helpful but not real­ly clear enough for routine use by drilling engineers. In addition, neither Dellinger nor Lubinski and Woods presented any theoretical analysis of the stability of drillpipe in inclined wellbores.

Stability Analysis

The purpose of stability analysis is to find the boundary between regions of stable and unstable behaviors. For drill pipe in compression, stability analysis can give the maximum compressive load that can be carried without buckling the pipe.

Paslay and Bogy have analyzed the stability of a cir­cular rod lying on the low side of an inclined circular hole. 4 As is shown in the Appendix, their result may be simplified to yield the following expression for the critical compressive load:

( ElpAg sin ()) liz

Fcrit =2 r

................. (1)

This equation should be used to predict the onset of buckling in an inclined hole. It replaces the simple Euler column-buckling equation:

FCrit=EI( ; ) 2 ......................... (2)

The Paslay-Bogy analysis predicts a much higher buckling load than would be calculated from the Euler analysis. Consider, for example, 1,000 ft [305 m] of 5-in. [13-cm], 19.5-lbm/ft [29-kg/m] drillpipe in an 8.5-in. [22-cm] hole at a 45° angle. Eq. 1 gives a critical load of 35,000 Ibm [15 876 kg] in this case while Eq. 2 gives a critical load of only 29 Ibm [13 kg]. This il­lustrates the profound difference between the two types of buckling analysis.

The reason that pipe in an inclined hole is so resistant to buckling is that the hole is supporting and constraining the pipe throughout its length. The low side of the hole forms a trough that resists even a slight displacement of the pipe from its initial straight configuration. The effect of gravity and the curving sides of the hole form an elastic restraint similar to the elastic foundation studied by Timoshenko. 5

Note that Eq. 1 does not always predict high buckling loads. If either the hole angle or the pipe weight is small, the critical load also will be small. Likewise, very large values of radial clearance will give low critical buckling loads. Also note that Eq. 1 is written for long pipe

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TABLE 1-DRILLPIPE DATA"

Nominal Body Body Moment E/t Size Weight 00 ID of Inertia"" (psi-in. 4 pAgt (in.) (lbm/tI) Material ~ ~ (in.4) x 106 ) (Ibmlfl)

5 25.6 steel 5.00 4.00 18.11 543 21.4 5 19.5 steel 5.00 4.28 14.27 429 16.4

4112 16.6 steel 4.50 3.83 9.61 288 13.9 4 14.0 steel 4.00 3.34 6.46 194 12.2

3112 13.3 steel 3.50 2.76 4.50 135 10.7 5 10.0 aluminum 5.15 4.10 20.66 219 6.2

4112 8.4 aluminum 4.60 3.60 13.73 146 4.9

Notes: ' Based on new-pipe dimensions. " 1=0.04909 (0 4 _d 4 ).

tE(steel)=30x10 6 psi; E(aluminum)=10.6x10 6 psi [73x10 B kPal. :t: These are the buoyed weight-per-foot values used to prepare Figs. 1 through 8.

lengths and is therefore independent of pipe length as shown in the Appendix. For short lengths of pipe (e.g., 100 ft [30 mD the actual buckling load would be even higher than given by Eq. 1.

The development of Eq. 1 by Paslay and Bogy does not mention the effects of buoyancy. Nonetheless, Eq. 1 applies rigorously to buoyed pipe provided the following steps are taken.

1. The term pAg must be the buoyed weight per length of the pipe.

2. The compressive load, F, or Fcrit must be the load existing above and beyond any isotropic compression caused by hydrostatic pressure.

The second condition corresponds to using the distributed buoyancy (or buoyancy factor) method to calculate the loads in the pipe. This method is totally ac­curate for pipe stability analysis and is significantly simpler than the pressure-area method.

The critical buckling load from Eq. 1 may be used directly for BRA design. The design requirement for drillpipe stability is

Fcrit > Wb - WBHA cos () ..................... (3)

This equation indicates that either the WOB, W b, must be low enough, or the buoyed BRA weight, WBHA , must be high enough, so that their difference is less than F erit. Eq. 3 is the equation that usually is used for BRA design except that the traditional value for F crit is zero.

Comparison With Earlier Work Eq. 1 can be shown to be similar to the equation for critical-buckling load recently presented by Dellinger et al. 3 If Dellinger's equation is rewritten with SPE sym­bols, the result is

(sin ()) 0.436

Fcrit=2.93 (El) 0.479 (pAg)0.522 -r-

.................................... (4)

In the range of fairly high stability (i.e., high hole angle and/or low radial clearance), this equation gives results close to Eq. 1. In the lower range of stabilities, Eq. 4 gives critical load values up to 50 % higher than Eq. 1. This discrepancy may represent a weakness in the underlying data that were observed originally by Woods and Lubinski and later used by Dellinger to derive Eq. 4. In their experimental tests, Woods and Lubinski ob-

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18

16 BUCKLING OCCURS!N THIS REGION

10 NO BUCKLING OCCURS IN THIS REGION

10 20 30 40 50 60 70 80

HOLE ANGLE !DEGREES I

Fig. 1-Stability analysis results for 5-in. [13-cm], 19.5-lbm/tl [29-kg/m] drill pipe carrying a 20,000 Ibf [89 764 N] axial compressive load.

90

served the minimum loads that were obviously unstable. Since slightly unstable loads produce only small deflec­tions in this type of buckling problem, Woods and Lubinski probably were led to overestimate the value of the critical buckling load. *

Woods and Lubinski themselves fit an equation to their experimental data collected in 1953. This equation was never published, but in SPE nomenclature it is*

(sin ()) 0.511

Fcrit =2.85 (EI )0.504 (pAg)0.496 _r-

.................................... (5)

Presumably the differences between Eq. 4 and 5 reflect the fact that Woods and Lubinski used the data directly, whereas Dellinger et al. attempted to reconstruct the data from Fig. 2 of Ref. 2.

Graphical Presentation Eq. 1 lends itself to simple graphical presentation pro­vided a separate graph is made for each size of drillpipe. Several of these graphs are included in this paper; their preparation and use is discussed briefly here.

'Lubinski, A.: personal communication, Smith Inti. Inc., Tulsa.

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HOLE ANGLE (DEGREES)

Fig.2-Critical buckling loads for 5-in. [13-cm], 19.5-lbm/ft [29-kg/mj drillpipe.

60 70 80

HOLE ANGLE IDEGREES~

Fig. 3-Critical buckling loads for 5-in. [13-cmj, 25.6-lbmlft [38-kg/mj drillpipe.

30,000 Ibf

60 70 80

HOLE ANGLE (DEGREES)

Fig. 4-Critical buckling loads for 4.5-in. [11-cmj, 16.6-lbm/ft [25-kg/mj drillpipe.

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90

Consider, for the moment, that we wish to analyze the stability of 5-in. [13-cm), I9.5-lbm/ft [29-kg/m) steel drillpipe in I5.5-lbm/gal [I546-g/m 3 ) mud. Table 14 shows that this pipe has an EI value of 4.29 x 108 psi-in. [0.75 x 108 kPa·m). In I5.5-lbm/gal [I546-g/m 3 ) mud this pipe has pAg value of about 16.32 lbm/ft [24.28 kg/m]. If these two values are substituted into Eq. 1 the result is

. = (23.4XI08

SinX) '/2 F ent

r .............. (6)

If some particular value of critical load is chosen then this equation can be solved for radial clearance as a func­tion of hole angle. Fig. 1 was prepared in this fashion for a critical load of 20,000 Ibm [9077 kg]. In Fig. 1 the graph of Eq. 6 divides the hole-angle/hole-diameter plane into two regions; the lower region is stable and the upper region is unstable. Fig. 1 makes it easy to evaluate the stability for 5-in. [13-cm], I9.5-lbm/ft [29-kg/m) drillpipe that has a 20,000-lbm [9077-kg) compressive load; if the hole size and inclination specify a point below the curve then the pipe is stable.

The chief limitation to Fig. 1 is that it applies only to one particular load value. This defect is removed by plot­ting curves for several loads on the same graph; Fig. 2 is an example of this. By interpolating between the curves in Fig. 2 it is possible to estimate the critical buckling load for 5-in. [13-cm) pipe at any combination of hole size and inclination. For example, a I2IA-in. [31-cm) hole at 60° angle yields critical buckling load of roughly 25,000 Ibm [11 340 kg]. If the actual compressive load is lower than that value the pipe will be stable.

Fig. 2 contains several minor numerical approxima­tions that were necessary to prepare a single figure for all 5-in. [13-cm], I9.5-1bm/ft [29-kg/m) drillpipe. These approximations primarily involve selection of a buoyed weight-per-foot value for the pipe in question. In prac­tice, three factors can influence the buoyed weight of 5-in. [13-cm), I9.5-1bm/ft [29-kg/m) drillpipe: (1) mud density, (2) tool joint dimensions, and (3) wear of the pipe and tool joints.

If all these variables were included it would be necessary to start over with Eq. 1 every time a new situa­tion was encountered. Since such precision is usually un­necessary, we have selected a single value of buoyed weight per foot to use for each size of drillpipe. For 5-in. [13-cm], I9.5-1bm/ft [29-kg/m] pipe, this value is 16.3 Ibm/ft [24.25 kg/m], which corresponds to Grade X pipe in I5.5-1bm/gal [1546-g/m 3) mud. In a similar fashion, typical buoyed weight-per-foot values were chosen for other sizes of drillpipe. These arbitrarily chosen values were used to prepare our figures.

Figs. 3 through 8 show critical buckling loads for other common sizes of drillpipe besides 5 in. [13 cm], I9.5lbmlft [29 kg/m]. The values of buoyed weight per foot that were used to prepare these figures are shown in Table 1.

90 Discussion of Approximations

A number of idealizations and approximations were made in the analysis scheme presented in the preceding section.

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20

18

16

14

12

~ 0°

° ,0'

80

HOLE ANGLE IDEGREESI

Fig. 5-Critical buckling loads for 4-in. [10-cm), 14.0-lbm/ft [21-kg/m) drillpipe.

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Favorable Approximations. The following approxima­tions tend to make the analysis err on the side of safety and reduce the chances of unexpected pipe buckling.

1. The radial clearance used is based on the pipe-body diameter. Presence of tool joints will reduce this clearance and increase the critical buckling loads.

2. The mud weight used in the analysis is 15.5 Ibm/gal [1546 g/m3]. If the actual mud weight is lower than that, the higher buoyed weight of the pipe will increase the critical buckling load.

Unfavorable Approximations. The following approx­imations may allow the pipe to buckle even though it ap­pears stable.

1. The wellbore is assumed to be straight. A com­pressive load in an angle drop-off region will reduce the effective weight of the pipe there and, thereafter, lower the critical buckling load proportionally.

2. API Class I (new) drillpipe dimensions have been used in the analysis. Worn pipe will be less stiff, less heavy, and buckle more easily.

3. The effects of torque are neglected in the equations

30.000 Ibf

70 80

HOLE ANGLE (DEGREES I

Fig. 7-Critical buckling loads for 5-in. [13-cm) aluminum drillpipe.

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90

20

18

16

14

12

",\ 0°0' ,0,

10

6~0-L~'-~~--~--~--~50o---~60'--'7~0--~80'-~90

HOLE ANGLE !DEGREES)

Fig. 6-Critical buckling loads for 3.5-in. [9-cm), 13.3-lbm/ft [20-kg/m) drillpipe.

given here. Torque can be included in the stability analysis, but its effect is small for realistic values.

4. The analysis requires an estimate of hole enlarge­ment. If the actual hole is larger than expected, bending stresses will also be larger than planned.

5. Mud weights greater than 15.5 Ibm/gal [1546 g/m3] will lower the critical buckling load slightly.

Conclusions Drillpipe in high-angle holes can carry compressive axial loads without buckling. At hole angles above 50°, use of drill pipe in compression will permit a major reduction in BHA weight and a significant reduction in torque and drag on the drillstring.

The purpose of this paper is not to revolutionize drill string design but to provide an engineering basis for analyzing current practice and making improvements in the future.

Nomenclature

A = cross-sectional area of pipe, sq. in. [cm2 ]

E = Young's modulus, psi [kPa]

20,---,----r------.----,---..-,--,-----,----,------,

5000 Ibf

18

16

14

1;00 Ibf

12

10

80 90

HOLE ANGLE (DEGREESI

Fig. 8-Critical buckling loads for 4.S-in. [11-cm) aluminum drillpipe.

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F = axial compressive load, lbf [N] Ferit = critical axial load to initiate buckling, lbf

[N] g = gravitational force, lbf [N] f = moment of inertia, in.4 L = length of column, in. [cm] n = order of buckling r = radial clearance between pipe and hole, in.

[cm] Wb = weight on bit, lbf [N]

W BHA = buoyed weight of BHA, Ibm [kg] (J = hole angle, measured from vertical I' = Poisson's ratio p = weight per cubic inch, Ibm [kg]

Acknowledgments

We thank Exxon Production Research Co. for permis­sion to publish this paper. We also thank Arthur Lubin­ski for providing unpublished results and Trudy Atkins for typing the manuscript.

References

1. Lubinski, A.: "A Study of the Buckling Rotary Drilling Strings," Drill. and Prod. Prac., API, Dallas (1950) 178.

2. Lubinski, A., and Woods, H.B.: Drill. and Prod. Prac., API, Dallas (1953) 222.

3. Dellinger, T.B., Gravley, W., and Walraven, J.E.: "Preventing Buckling in Drill Strength," U.S. Patent No. 4,384,483 (1983).

4. Paslay, P.R. and Bogy, D.B.: "The Stability of a Circular Rod Laterally Constrained to be in Contact with an Inclined Circular Cylinder," J. Appl. Mech. (1964) 31,605-10.

5. Timoshenko, S.: Theory of Elastic Stability, McGraw-Hill Book Co. Inc., New York City (1936) 108.

APPENDIX

Derivation of Eq. 1 The derivation of Eq. 1 in this paper begins with Eq. 33 in Ref. 4, which is

(1-1')2 11"2 ( L4pAg ) Ferit = Ef-- n2 of . •

(1 +1')(1-21') L2 n211"4EIr

................... , ............... (A-I)

This equation applies to a rod or pipe in a horizontal hole only. To generalize this equation to inclined holes, multiply the weight-per-Iength factor, pAg, by sin (J. Also assume that Poisson's ratio, 1', is approximately 1/3 ,

in which case the leading coefficient in Eq. A-I becomes 1.0. These changes give

_ 11"2 (2 L 4 pAg sin (J) Ferit -Ef-

2- n + 2 4 •......•. (A-2)

L n 11" Efr

This equation simplifies to Eq. 2 (Euler buckling) when hole angle or pipe weight is zero or when radial clearance is infinite. In those cases the critical buckling load is determined by first-order buckling (n=I). Eq. A-2 does not predict the critical buckling length of 1.94 dimensionless units given by Lubinski. 1 This is because Lubinski considers the buckling of a critical column under its own weight, whereas Eq. A-2 is derived for pipe buckling under a constant, externally applied load.

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Eq. A-2 results from an eigenvalue problem where n is the number of buckles (or the order of buckling) that oc­cur in a pipe length L. Note that buckling of long lengths of pipe in inclined holes does not start with the first order and then proceed to higher orders at increased loads. The first buckle that appears will have some higher order (n> 1), depending on the pipe length. The buckle that first occurs is the one whose n gives the lowest value of Fcritas calculated by Eq. A-2.

For fairly long rods, the simplest way to proceed is to ignore the fact that n takes on only integral values and to treat it as a continuous variable. With that assumption the minimum value of F erit may be found by setting the derivative equal to zero.

a -Ferit =0, ............................ (A-3) an

which is

....... (A-4)

or

2L 4 pAg sin (J 2n- 0, ................... (A-5)

n 3 11"4Efr

which gives

n 2 = (L

4 pAg sin (J) 'h

1I"4EIr ................. (A-6)

If this value of n 2 is substituted back into Eq. A-2 then considerable simplification results (Eq. 1):

_ (EfPAg sin (J) 'h Ferit -2

r

This equation is the same as Eq. 1 given in the text. Ig­noring the eigenvalue nature of the problem has eliminated both nand L from the final formula. A few sample calculations made with the original Eq. A-I have shown that the approximate result is quite accurate for pipe lengths more than about 200 ft [61 m].

Note on Compression. Throughout this paper the "compressive load" being considered is the load that ex­ists above and beyond any isotropic compression caused by hydrostatic pressure.

SI Metric Conversion Factors

m. X 2.54* E+OO lbf x 4.448 222 E+OO

Ibm X 4.535 924 E-01 psi X 6.894757 E+oo

"Conversion factor is exact.

cm N kg kPa

JPT

Original manuscript received in the SOCiety of Petroleum Engineers office Dec. 8, 1982. Paper accepted for publication May 11, 1984. Revised manuscript received April 26, 1984. Paper (SPE 11167) first presented at the SPE 1982 Annual Technical Conference and Exhibition held in New Orleans Sept. 26-29.

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