api-50-178 a study of the buckling of rotary drilling strings

A Study of the Buckling of Rotary Drilling Strisgst

ABSTRACT

The theory of elastic stability i s applied to the drilling string, and the critical conditinns for which the buckling occurs are investigated. 'rhe location of points at which the buckled pipe is tangent to the wall of the hole A d the force with which i t contacts the wall are determined. The location of points of maximum stresses and the value of these

s t re s se s are also calculated. The inclination of the bit and of the force on the bit are investigated. Measures for prevention of buckling and for rnini- mizing i t s effect are given. Essentially, these measures are carrying proper weight on the bit and/or the use of special drilling methods

T h e fatigue and failure of t h e drilling s t r ing have been a sub jec t of numerous investigations;@but, to the b e s t knowledge of the author, no mathematical approach t o the problem h a s been undertaken. In order to allow t h e conclusions of t h i s invest igat ion t o be eas i ly read by those who are not interested in mathematical developments, t h e derivation of formulas i s given in a n appendix.

Without weight on the bit, a drilling st;ing i s s t raight if the hole i s straight. With a sufficiently smal l weight on the bit, the s t r ing remains straight. A s the weight i s increased, a so-cal led cr i t ical value of weight i s reached for which the s t raight form of t h e s t r ing i s no longer s table . T h e drilling s t r ing buckles and contac t s the wall of the hole a t a point des igna ted a s the point of tangency. If the weight on the bit i s further increased, a new criti- c a l value i s reached a t which the drilling s t r ing buckles a second time. T h i s i s designated a s buckling of the second order. With s t i l l higher weights on t h e bit, buckling of t h e third and higher o rders occur.

At the point of tangency the drilling s t r ing rubs aga ins t the wall of the hole, and t h i s c a u s e s caving in cer tain formations. T h e rubbing effect becomes worse when the force between the buckled pipe and the wall of t h e hole increases. When the buckled s t r ing i s rotated, some reversing s t r e s s e s a r e developed. T h e s e s t r e s s e s increase with the diameter of t h e hole and resul t in fat igue fai lure of the string.

A s soon a s a drilling s t r ing buckles in a s t raight hole, t h e bit i s no longer vert ical and a perfectly vert ical hole cannot be drilled.

* Barnsdall Research Corp., Tulsa , s i n c e r e ~ o v e d t o Stano- l lnd 0 1 1 and G a s Company, T u l s a .

+presented at the sprlng meetlng, Mid-Cont~nent D l s t r ~ c t , Dl- v l s l o n of Productlon. Oklahoma City, ,March 1950.

@ ~ l b l ~ o g r a p h y 1 s at the end of the paper.

T h i s invest igat ion a t tempts t o find answers to the following ques t ions :

1. What a re the cr i t ical v a l u e s of weight on t h e bit a t which buckling occurs? 2. What i s the shape of the buckled s t r ing? 3. Where are the points of tangency and maximum fatigue s t r e s s located? 4. What i s the magnitude of the force between the buckled pipe and the wall of t h e hole? 5 . Under what c ircumstances a re t h e s t r e s s e s in a buckled s t r ing e x c e s s i v e ? 6. How grea t a re t h e incl inat ions of t h e bit and of the force on t h e bit? 7. How may buckling be avoided; or, eventual ly, how-to drill to minimize t h e bad effects when drilling with a buckled s tr ing?

Critical Conditions

A certain point of a drilling s t r ing i s usual ly des igna ted a s the "neutral point." In t h i s invest i - gation t h e neutral point i s defined by t h e following conditions: T h e weight i n mud of the portion of a drilling s t r ing below the neutral point i s equa l to t h e weight on t h e bit.@ E a c h va lue of weight on the bit corresponds to a value of the d i s tance between t h e bit and the neutral point. T h e cr i t ical val- u e s of th i s d i s tance depend upon the type o f pipe or dr i l l co l la r s and the specific gravity of the mud. It i s very convenient t o measure dep ths not in feet , but in a dimensionless unit in order t o obtain the r e s u l t s independent of the type of pipe, col lars , and mud.

I t w a s found that buckling of t h e first and second orders occurs when the neutral point i s located 1.94 and ,3.75 d~niens~onless units, respect ively above

@ A S shown In the appendix, t h l s polnt 1s different from that at whlch there 1s n e ~ t h e r compression nor tens lon, b e c a u s e such a polnt 1s displaced by the hydrostatic pressure and the pump pressure.

STUDY O F THE BUCKLING O F ROTARY DRILLING STRINGS 179

the b i t . @ ~ h e length in feet of one d imens ion less unit i s given by the followipg expression:

wherein: E i s Young's modulus for s t e e l , in pounds per square foot.

E = 29(106 ps i ) =4,176(106 Ib per s q ft)

wherein:

I i s the moment of iner t ia of the p ipe c r o s s s e c - tion, in feet4.

nz i s the length, in feet, of one d imens ion less unit. T h e va lues of nz for various t y p e s of drill pipe and drill co l la r s are plotted on F ig . 1.

p i s the weight in mud per unit of length of the drilling s t r ing, in pounds per foot.

T h e weight per foot of the dr i l l pipe must be increased to niake al lowance for tool ioints. T h e weight of a tool joint i s assumed to be equa l ly dis t r ibuted over a 30-ft joint of pipe. T h e weight, in pounds, of the length of a drilling s t r ing equa l to one dimensionless unit i s equa l to:

3 nip = d w (2)

By nlultiplying the va lues of m p by 1.94, the critical weights on the bit of the first order for various types of drill pipe and drill co l la r s were found and plotted on Fig. 2. Cr i t i ca l weights of t h e second order may be obiained by multiplying the readings of Fig. 2 3.'5 - 1.934. T h e following

b y m - conclusions may be drawn from th i s figure. 1. A drilling s t r ing coniprising drill pipe only buckles with a very small weight on the bit. A 4%-in. drill pipe can s t a n d no more than 1,400 Ib without buckling, and a t 3,000 Ib h a s already - buckled twice. 2. A drilling s t r ing of drill co l la r s only could s t a n d much more weight on the bit without buckling. However, the belief sometimes expressed that dr i l l col lars d o not buckle i s erroneous. Under normal drilling condit ions t h e drill co l la r s a re - general ly buckled a t l e a s t once and sometimes two or three times. F o r 6fi-in. drill co l la r s the cr i t ical weight of the first oraer i s between 7,700 and 8,700 Ib, and the critical weight for the second order i s between 15.000 and 17.000 Ib. No computation h a s been made for buckling of higher orders, but i s expected that a third buckling would occur a t a c r i t i ca l weight snlaller than 26,000 Ib. 3. T h e heavier the mud, the smaller are the criti- c a l weights on the bit. However, the influence of the specific gravity of mud on buckling i s not very significant.

or a complete understand~ng, see the appendix.

D R I L L I N G F L U I D SPECIFIC rfiVITY

I Fig. 1 - Length in F e e t of One Dimensionless U n i t

Fig. 3 i l lus t ra tes the general c a s e of buckling I of a combination s tr ing comprising 4%-in., 16.60-lb

API drill pipe and 6f4-in. drill collars. T h e weight on the bit i s plotted vs. the number of 30-ft drill collars. L i n e s 1, 2, and 3 correspond to cr i t ical weights of the first, second, and third order, respect ively. For any condition corresponding t o a point loca ted under l ine 1, e.g., no drill col lars and 1,000 Ib on the bit or 3 drill col lars and 6,000 Ib, the s t r ing i s straight. F o r any condition corresponding to a point located between l i n e s 1 and 2, e.g., 3 drill co l la r s and 8,000 Ib on the bit or 9 drill col lars and 14,000 lb, the drilling s t r ing i s buckled once. F o r any condition corresponding to a point loca ted between l i n e s 2 and 3, e.g., 3 drill col lars and 9,000 Ib on the bit or 12 drill co l la r s and 22,000 Ib, the drilling s t r ing i s buckled twice. T h e mathematical computation was not made up t o buckling of t h e third order. Line 3, corresponding to buckling of the third order, i s only approximated; consequently, i t i s not a l together correct.

Fig. 4 and 5 concern drilling s t r ings comprising 4%-in., 16.60-lb API dri l l pipe with 7-in. drill collars , and 3?4-in., 13.30-lb API drill pipe with 434-in. drill col lars .

180 ARTHUR

OR CRITICAL WEIGHTS OF

DRTLLJNG F L U I D SPECIFIC GRAVITY

Fig. 2 - Critical Weights on the Bit First Order

L i n e s 1, 2, and 3 of big. 3, 4, and s correspond to cr i t ical condit ions in a 12 Ib per ga l mud. T h e influence of drilling fluid densi ty i s negligible, except with heavy drill col lars in a heavy mud. A correction may be made by using Fig. 2. For in- s tance , for 6;i-in. drill col lars in a 16 Ib per g a l mud, Fig. 2 s h o w s tha t the horizontal portion of the curve 1 of Fig. 3 should be displaced downward from 8,400 to 8,000 Ib.

L e t u s ana lyze a l l t h e means which might be considered t o avoid the buckling of drill s t r ings, even though some of them d o not apply to present pract ice.

1. T h e s imples t means i s to carry a weight on the bit smaller than the cr i t ical weight of the first order; e.g., drill with s i x 65i-in. drill col lars and maintain a weight on the bit 6f only 8,000 Ib. Un- fortunately, such a weight i s too smal l to drill most formations economically. Nevertheless , inr some c a s e s i t should b e carried, espec ia l ly when particuIar c a r e i s indicated in order t o drill a s t raight hole o r t o avoid enlarging a cave. 2. Analyzing formula (2) i t may be s e e n tha t t h e c r i t i ca l va lue of weight on the bit might be in- c reased by increasing E, I , or p. I t i s obviously

impossible to modify Young's modulus E by choos- ing a mater ial other than s tee l . However, by us ing larger drill col lars , both the moment of iner t ia I and the weight per foot p increase . T h e cr i t ical weights vs. ou ts ide diameter of drill co l la r s were ca lcu la ted with formula (2) and plotted on Fig. 6, which shows tha t t h e c r i t i ca l value of the weight on the bit i n c r e a s e s by large amounts for smal l i n c r e a s e s in drill-collar s i z e . Consequently, a grea t improvement may be obtained by using larger dr i l l co l la r s with the same bit gage; that i s to say , by decreasing the play between the co l la r s and the hole. Using larger drill col lars and main- ta ining the conventional p lay requires drilling a larger hole and carrying more weight on the bit. In order t o es t imate the improvement which might be obtained in th i s l a t t e r c a s e , the curve of critical weights per square inch of bore hole h a s a l s o been drawn on Fig. 6. From inspect ion of the two curves i t i s s e e n that , in order to ach ieve a 100 percent improvement, 6;;-in. drill co l la r s must be replaced by 8 4 - i n . drill col lars if t h e bit gage i s maintained and by 11%-in. drill co l la r s if the clearance is maintained and a 1 4 k -in. hole drilled. T h e pract ical difficulties t o be encountered in t h i s l a s t form of solut ion a r e obvious.

On F ig .6 t h e ins ide diameter of the 6::-in. drill co l la r s i s assumed t o be 2!i in. F o r larger drill col lars , the ins ide diameter i s assumed t o be proportional t o the ou ts ide diameter. 3. Some new methods of rotary drilling require l e s s weieht on the bi t than the conventional meth-

h W R UF 30 W T DRILL CULlARb

Fig. 3 - Buckling Conditions for Drilling Strings Comprising 44-in. 16.60-lb API Drill Pipe and 64-in.(24-in. ID) rill Collars in 12Lb per Gal Mud

STUDY O F THE BUCKLING ROTARY DRILLING STRINGS 181

od. Drilling may be carried on with weigh ts smaller than the cr i t ical v a l u e s of the first order. T h i s h a s been accomplished in the rotary percuss ion drilling method described in a recen t paper.' T h e rotary percussion method w a s developed principal ly for u s e in very hard formations, but the author a l s o s u g g e s t s i t s u s e where conventional ly dr i l led h o l e s go crooked. T h e s t r a i g h t n e s s of ho les drilled by t h i s method and a l s o drilled with diamond b i t s i s undoubtedly due to the fact that the drilling s t r ing d o e s not buckle.

Fig. 4 - Buckling Conditions for Drilling Strings Comprising 44-in. 16.60-lb API Dr i l l Pipe and 7-in.

(3-in. ID) Dri l l 'Collars in 12 Lb per Gal Mud

Critical Length in Stack

T h e cr i t ical length of s t a n d s of pipe or col lars , ver t ical ly s tacked inside the derrick and supported c l o s e t o their top, i s equal t o 2.65 d imens ion less uni ts ( s e e Fig. 1); i.e., equal to:

107 ft for 3%-in., 13.30-lb API dr i l l pipe. 1 2 7 f t for 4k-in., 16.60-lb API drill pipe. 122 ft for 44-in., ( I t - i n . ID) drill collars.

144 f t for 64-in., ('24-in. ID) drill collars. 159 ft for ?-in. (3-in. ID) drill col lars .

T h i s conforms with the fac t t h a t the maximum

'~e fe r ences are at the end of the paper.

INURER OF 30 PMT D R I U . C O W

Fig. 5 - Buckling Conditions for Drilling Strings Comprising 3 4 .In. 13.30-1 b API Drill Pipe and 4 4 -in. (14-in. ID) Dri l l Collars in 12 Lb per Gal Mud l eng ths of drill pipe which can be s t a c k e d i n the derrick are "trebles." Longer s t a n d s must be supported near t h e middle.

Shape of the Buckled Drilling String

In th i s sec t ion we wil l consider how the s h a p e of the drilling s t r ing var ies a s the weight on the bi t i n c r e a s e s from zero to some high value.

T h e drilling s t r ing remains s t raight when the weight on the bit i s smal le r than the cr i t ical value. T h e s h a p e of t h e buckled pipe a t the cr i t ical value i s shown on Fig. 7 and 8. Curve 1 on both figures represen ts the a x i s of the drilling string. P o i n t 0 i s the bottom end, or the bi t .The vert ical d i s t a n c e s a re measured in dimensionless units. T h e length of one d imens ion less unit has been given in F ig . 1.

LRITIGL iiEICKIS CRITICAL WEIGHTS RI N U N = PER 511. IN. U P

(12 LB/CkL MUD) BORE HOLE

66,UX)

6 O . W

4 O , c c a

2 o . m

6 8 10 12 1L 16

O.D. OF DRILL W U R S , INCHES

Fig. 6 - Influence or Drill-collar Size on Buckling

18 2 ARTHUR LUBINSKI

IMMEDIATELY PRIOR TO SECOND BUCKLE CDNTACTlNt THE

Fig. 7 - Shape of Buckled Curves and Diagrams of Bending Moment Coefficient i

T h i s length d o e s not vary appreciably from one type of drilling s t r ing to another and i s usual ly be- t w e e n 4 0 and 65 ft. Therefore, for any order o f buckling, the shape i s very much the same regard less of whether the bottom part is a drill-pipe s t r ing or a drill-collar string. However, in the c a s e of drill pipe t h i s s h a p e ~ c o r r e s p o n d s t o a much smal le t weight on the bit than in the c a s e of drill co l la r s (Fig. 2).

Referring aga in to Fig. 8, ind ica tes the location of t h e neutral point (1.94 uni ts above the bit). TI ind ica tes the point of tangency.

A s the weight on the bit i n c r e a s e s between the critical va lues of the first and second order, t h e shape of the buckled s tr ing changes progressively between curves 1 and 2 of Fig. 8. T h e l a s t curve (neutral point N2 and the tangency point T 2 ) corre- s p o n d s to cr i t ical condit ions of the second order.

+ r d . + (r 1s the apparent radlus of the hole, t.e.. the rnaxlrnurn

possrble deflect~on )

Fig. 8 - Shape of Buckled Curves [Curve (1) - Critical, First Order; Curve (2) - Critical, Second Order; Curve (3) - Immediately Prior to Second Buckle Contacting the Wall of

the Hole.]

STUDY O F T H E BUCKLING O F ROTARY DRILLING STRINGS 18 3

Comparison of curves 1 and 2 of Fig. 8 shows that the portion of the string located close to the bit i s deflected more and more, while the portion located above the tangency point i s progressively straight- ened.

A s the weight on the bit increases above the critical value of the second order, a second buckle appears on the drilling string and grows rapidly for small increments of weight on the -bit. Curve 3 of Fig. 7 and 8 shows the shape of the drilling string at the very nloment when the second buckle contac ts the wall of the hole a t T3'.

The change of the shape of the buckled curves may also be seen on Fig. 9, in which the abscissa represents the distance between the bit and the neutral point, in dinlensionless units; this is , of course, proportional to the weight on bit. In order to help visualization, another abscissa sca le has been drawn to indicate the weight on the bit in the particular case of 6!i-in. OD 2!;-in. ID drill collars in 12 Ib per gal mud. The solid-line curve shows the distance between the bit and the lowest tangency point. Two ordinate sca le s correspond to this curve; one i s in dimensionless units and the other is in feet for the particular case of 6!:-in. drill collars. It may be desirable to determine the relative position of the point of tangency with respect to the neutral point. For this purpose, a dashed straight line, inclined a t 45 deg with respect to coordinate axes, has been drawn on Fig. 9. The ordinate of any point of this line indicates the distance from the bit to the neutral point, and the vertical distance between the solid and dashed lines represents the distance between the tangency and neutral points.

The following conclusions may be drawn from inspection of these two lines: 1. For weight on the bit equal to the critical value of the first order (1.94 du* 8,400 Ib**) the tangency point TI i s located 1.8 du or 104 ft above the bit; i.e., somewhat lower but close to the neutral point hi, which i s located 1.94 du or 113 ft above the bit. 2. As the weight on the bit increases between the critical values of the first and second order, the tangency point i s displaced downward by small amounts, while the neutral point i s displaced upward by much larger amounts. The tangency point i s progressively displaced from 1.8 du (104 ft) to 1.41 du (82 ft) above the bit. At the same time, the neutral point goes up from 1.94 du (113 ft) to

* Dimensionless unit. * * In thls case and in the following text the values in dimen-

s ~ o n l e s s units are general; and those in pounds or-feet per- t a n to the particular case of 6%-~n. O D 2 5 i n . ID drill collars in 12 Ib per gal mud.

LEIWT ON BIT IN POUNDS FOR .- DKILL COW I N u LB/CAL YUD --

Fig. 9 - Location of the First Tangency Point and Deflection of the Second Buckle

3.75 du (219 ft) above the bit. The distance between the tangency point and the neutral point increases from 0.14 du (8 ft) to 2.34 du (137 ft). 3. The downward displacement of the tangency point becomes much faster when the weight on the bit i s increased between the critical value of the second order and the weight for which the second buckle contacts the wall of the hole; i.e., when the weight on the bit increases from 3.75 du (16,200 1b) to 4.22 du (18,300 Ib). For such a small increase of weight on the:bit, the tangency point i s displaced from T2 (1.41 du, 82 ft) to T3 (1.12 du, 65 ft) above the bit.

The dotted curve (lower right-hand corner of Fig. 9) shows the deflection of the second buckle while the weight on the bit i s increased above the critical value of the second order. As previously mentioned, the second buckle grows very rapidly.

Curve 3 of Fig. 7 and 8 shows that the second buckle contacts the wall of the hole a t the point T3'located 4.20 du (245 ft) above the bit; i.e., under but very close to the neutral point located 5.22 du (346 ft) above the bit.

Curve 3 of Fig. 7 and 8 corresponds to the high- es t value of weight on the bit investigated in this study. I t i s very possible that, for greater orders of buckling, the highest buckle contacts the wall of the hole a t the neutral point; and that, a s the

184 ARTHUR LUBINSKI

Fig. 10.- Comparison of Electric Log Runs on Well in N. E. Lindsey Field, Oklahoma

STUDY O F T H E BUCKLING O F ROTARY DRILLING STRINGS 18 5

weight on the bit i s then increased, the point of tangency i s slightly displaced downward while the neutral point moves rapidly upward. Also, it seems logical to assume that an increase in weight on the bit distorts the highest buckle in contact with the wall of the hole, whereas the shape of the buckles below does not change very niuch.

Knowledge of the location of the tangency point may be useful when drilling in a hard formation below a soft and caving shale. Suppose that, in the caving shale, l e s s weight than the critical value of the first order had been carried in order to avoid caving. After having encountered the hard formation (which requires a large weight on bit) the weight should not be increased imn~ediately, if caving is st i l l to be avoided; for, if this i s done, the buckles of the string will contact the caving shale above. One and one-half dimensionless units or 87 ft of hole should be drilled in the hard formation before the weight on the bit i s increased. At that time the weight on the bit should be immediately rather than progressively increased, by a considerable amount, in order to obtain a tangency point in the hard formation. Fig. 9 shows that, with a weight on the bit very little larger than the criti- ca l value of the first order, the tangency point would be located 1.8 du (104 ft) above the bit; i.e., in the caving shale.. On the other hand, if the weight on the bit i s approxin~ately equal to the critical value of the second order (3.75 du or 16,200 IL), then the tangency point would be located at 1.41 du (82 ft), or in the hard non-caving formation. Consequently, drilling should be carried with a weight on the bit corresponding to approximately 3.75 du between the bit and the neutral point (16,200 Ib), but no more than 4.22 du (18,200 Ib), in order to keep the second buckle from contacting the caving shale above. Higher values of weight on the bit should not be carried before 4.2 du (245 f t ) of the hard formation are drilled, because a s seen in curve 3 of Fig.? and8 the second tangency point T3'is located 4.3 du above the bit.

In certain areas, e.g., Lindsey Field in CIklahoma, some fornlations have a tendency to cave and allow large boulders to drop into the hole which causes considerable trouble. With the thought that a special mud might solve the drilling difficulties of th is area, an API district study group was formed. After pro- longed investigation, the conclusion was reached that no mud could lift these boulders through the caves. A new approach to the problem i s needed, and the possibilities of special drilling methods must be considered. Use of weight on the bit, previously suggested, i s advised. Another possible way to avoid buckling i s by drilling with the rotary

percussion method previously analyzed. Evidence of the fact that the caves are not made by the bit, but by the drilling string rubbing against the wall of the hole, i s given by logs.

Fig. 10 represents a comparison of two electric log runs of a well located in the northeast Lindsey Field. In the shale, the top of which i s located at 9,964 ft, i.e., 48 ft above the bottom of the first run, the normal resistivity curve shows a smaller apparent resistivity in the second run than in the first run. Inasnluch a s the mud resistivity remained almost the same and no invasion occurred in the shale, the, only reason for the change i s a cave created between the two runs. The s ize of the cave nlay be estimated by using the departure curves. This cave was evidently made by drill collars rubbing against the wall of the hole, and possibly might have been avoided by using better drilling procedures. Force Applied by the Buckled Drilling String on

the Wall of the Hole

The larger the hole, the greater i s the force applied by the .buckled drilling string on the wall of the holq. In the forn~ulas, however, i t i s more practical to use not the hole s ize , but what will be termed "the apparent radius of the hole" which i s the maximum possible deflection of the string in a hole of a certain s ize .

The apparent radius of the hole r may be calculated with the following formulas for drill collars and drill pipe, respectively:

r = %(D-D,I (3)

r = % ( 0 - 4 ) (4)

wherein:

D i s the diameter of the hole. D, i s the outside diameter of the drill collars.

I), i s the outside diameter of the tool joints.

It will be proved in the appendix that the force applied by the buckled drilling string on the wall of the hole nlay be expressed a s follows:

F = fpr ( 5 )

wherein: p i s the weight in mud per unit length of the string, in pounds per foot; r i s the apparent radius of the hole,expressed in feet in order to usr a consistent system of units; and f i s a coetficient which depends on the distance in dimensionless units, between the bit and the neutral point. This distance i s , of course, proportional to the weight on the Lit.

The variation of f vs. the distance in dimensionless units between the bit and the neutral point i s shown in Fig. 11.

186 ARTHUR LUBINSKI

DISTANCE x;! BETWEEN BIT AND IEbTRHL WINT I N DIME?ISIUNLESS IINTTS

Fig. 11 - Coefficient f for Calculating Force Applied by Buckled Drilling String on the Wall of the Hole

When the weight on the bit is progressively increased and the distance between the bit and the neutral point reaches the critical value of 1.94 du, the drilling string buckles; but, a s seen in Fig. 11, the coefficient f and, consequently, the force F are

equal to zero. T h i s fact may be visualized by means of the following consideration. When buckling i s on the verge of occurring, the sl ightest increase in rigidity of the pipe would prevent it. Similarly, if it occurs, the sl ightest force can stop it and the reaction of the wall of the hole on the buckled string i s nil.

As the weight on the bit Leconies greater than the critical value, i.e., a s the distance between the bit and the neutral point increases above 1.94 du, the coefficient f and consequently, the force F increase too. The greatest value of this distance studied i s 4.22 du for which the second buckle contacts the wall of the hole. Fig. 11 shows that the corresponding value of f i s 2.7.

In order to show the order of magnitude of the - force F in the latter case, substitute f = 2.7 into equation (S), which becomes:

The values of the force F given by this expression have been plotted on Fig. 12 for various s i z e s of drill pipe and drill collars in a 12 1b per gal mud. This illustration shows that the force F i s not very great-a few hundred pounds a t the most. For that reason, in most formations the hole remains on

FORCE F LB 900

ACTUAL DIAMETER OF THE HOLE, INCHES

Fig. 12 - Force F Applied by the First Buckle on the Wall of the Hole for the Smallest Weight on Bi t at Which the Second Buckle Contacts the Wall of the Hole (12 L b per Gal Mud)

STUDY O F THE BUCKLING O F ROTARY DRILLING STRINGS 18 7

gage in s p i t e of the buckled drilling s t r ings. How- ever, if the formation h a s a tendency t o cave, t h e rate a t which t h e cave grows a c c e l e r a t e s b e c a u s e the force involved i n c r e a s e s with t h e diameter of the cave.

It should be borne in mind that t h e v a l u e s of the force F given in Fig. 1 2 correspond to the weight on the bit for which the second buckle con tac t s the wall o f t h e hole, which i s e q u a l t o about 18,000 Ib for 6?i-in. drill col lars . Fig. 11 s h o w s t h a t for va lues of weight on the bit beyond the limit of the diagram-higher va lues a t which the s t r ing i s buckled many times-the probable, trend i s toward a large increase of the coefficient f and, consequent- '

ly, of the force F. Bending Bloments and Stresses

When {he drilling s t r ing buckles , each c ross s e c - tion becomes sub jec ted t o a bending moment gen- erating a tension s t r e s s on one s i d e and a compression s t r e s s on the other. A s the drilling s t r ing rotates t h e s e s t r e s s e s reverse; and, consequently, they c a u s e a fatigue of t h e s teel . Sometimes they may be a reason for failures.

T h e following expression for the bending moment h1 will be proved i n the appendix:

M = ipmr (6) wherein: i i s a coefficient which wil l be explained further; and p, m, and r have the same meaning as previously s e t forth.,

F o r any given s i z e of drill pipe or dr i l l col lars , the weight per foot p and the length of one dimen- s i o n l e s s unit m are constant , and formula (6) shows that the bending moment W i n c r e a s e s with t h e apparent rad ius of the hole, which i s obvious. If t h e s i z e of the hole i s a l s o constant , then the bending moment hi depends only upon the coefficient i and i s proportional t o L .

T h e variation along the drilling s t r ing of the coefficient i for two different buckling condit ions i s shown in Fig. 7. A s previously explained, curves 1 and 3 indicate the shape of buckled curves for cr i t ical condit ions of the first order and for the smal les t weight on the bit a t which the second buckle con tac t s t h e wall of the hole, respect ively. Curves 1A and 3A are the corresponding diagrams of the bending-moment coefficient i .

Curve 1 A s h o w s tha t on a drilling s t r ing subjected t o buckling of the first order there a re two points (designated a s Mi and Mi') a t which the bending moment i s maximum. T h e largest of these two maxima occurs a t the point MI, which i s c loser to the bit than A!1'.

Curve 3A s h o w s tha t on a drilling s t r ing s u b jected to buckling of the second order there a r e

three points (designated as M3, M3/, and '14") a t which. t h e bending moment is maximum. A s for buckling of t h e first order, the bending moment i s the l a rges t a t point M3, which i s neares t t o the bit , medium a t ,M3', and the smal les t a t point M3", which is fa r thes t from the bit.

A discont inui ty is s e e n on curve 3A a t depth T3, a t which the s t r ing contac t s the wall of the hole which r e a c t s and appl ies a lateral load on t h e string. There, is n o discontinuity of the bending- moment coefficient i a t the depth T3', a t which the second buckle con tac t s the wall of the hole, be- c a u s e there i s n o Ia te ra l load appl ied t o the s t r ing a t th i s point. I t should be kept in mind tha t curve 3 corresponds to the s m a l l e s t weight on the bit a t which the second buckle con tac t s the wall of the hole; and, consequently, no force i s 'applied on the wall, exac t ly a s in the c a s e of the point TI of curve 1 for cr i t ical condit ions of the first order.

Fig. 7 s h o w s the bending-moment coefficient for two extreme c a s e s s tudied in t h i s paper. In order t o ana lyze the conditions between t h e s e c a s e s , Fig. 13 h a s been &awn in which the ab- s c i s s a represen ts the d i s tance in dimensionless un i t s between the b i t and the neutral point, which i s proportional to the weight on the bit. Dashed l i n e s represent the d i s tance between the bi t and the two lowest points on t h e string, a t which t h e bending moment i s maximum. I t i s s e e n that, as t h e weight on the bit inc reases , a l l the points of maximum bending moment move downward. Between the extreme condit ions of the diagram the lowest point of maximum bending moment moves from 14 to 114; i.e., from 1 t o 0.75 du above the bit or, in the c a s e of 6:A-in. drill col lars , from 5 8 t o 44 f t above t h e bit. T h e second point of maximum bending moment moves frornMif to M3'; i.e., from 4.3 t o 3.35 du above the bit, or from 250 t o 195 ft in the c a s e of 6x411. drill collars.

Fig. 13 s h o w s not only where the points of maximum bending moment a r e located, but. a l s o which a re the corresponding va lues of the maximum bending-moment coefficients if and i2. Coefficient if corresponds t o the point of maximum bending moment which i s n e a r e s t the bit, and i2 corresponds to the point above.

T h e following conclusions may be drawn from inspect ion of curves if and i2 of Fig. 13: 1. Coefficients if and i2 and, consequently, the maximum bending moments, increase with weight on the bit. T h i s increase becomes very sharp for weights on the bit above the cr i t ical value of the second order, a t which the second buckle i s growing. Between the extreme condit ions of the diagram the largest maximuni bending-moment coefficient


DISTANCE xp BETVE!ZN B I T AND NEUTRAL POINT I N DIMENbIONLESS UNITS

Fig. 13 - Diagram of Bending Moment Coefficients i (Solid Lines) and Location of Maximum Bending

Moments (Dashed Lines)

increases by a factor of about 2.5. 2. Between the critical conditions of the first and second orders, iq i s very small compared to r l ; il becomes appreciable only after the second buckle has grown.

In order to calculate the magnitude of s t r e s ses generated by the bending moments due to buckling, consider the condition a t which the second buckle contacts the wall of the hole. The corresponding value of the nlaxirnum bending-moment coefficient if for the first buckle, a s shown in Fig. 7 or 13, i s equal to 1.84. Substituting this value into (6 ) , this formula becomes:

M = 1.84pmr (7) The weight in mud per foot of drilling string, p,

may be easily calculated for any s ize of drill pipe or drill collars. The length m, in feet, of 1 du i s given by Fig. 1. Consequently, bending moments !il, in foot-pounds, may be calculated for various values of the apparent radius of the hole r, in feet.

The s t r e s ses may be calculated, when the bending moments are known, with the following well- known formula: kl v

0 - - I

(8)

where~n:

o i s the stress, in pounds per square inch. iM i s the bending moment, in inch-pounds. I i s the moment of inertia, in inches4. 2, i s the outside radius of the drilling string, in

inches.

Maximum s t r e s ses have been calculated with formulas (7) and (8) vs. r . They change l i t t le with mud density and s ize of drill pipe or drill collars. Those corresponding to &;-in. drill collars in a 12 Ib per gal mud are plotted against the hole diameter on Fig. 14, which shows that the diameter of a cave must be very large tocause dangerous s t resses in a drilling string subjected to the second order of buckling. Given a cave diameter equal to 100 in., the s t ress i s equal to about 16,000 psi. Some wide caves occur a s shown on Fig. 15, which represents a portion o f a caliper log of a well located in north- eas t Catherine Field, El l i s County, Kansas. The actual diameter of large caves i s not known because the surveying caliper does not extend beyond a certain limit. Neither are the maximum safe s t r e s ses known. The tensile strength of drill-pipe s tee l and drill-collar steel is very high, but the metal i s subjected to reversing s t resses in the order of magnitude of 100,000 reversings a day. A s t ress which i s normally safe for s tee l may cause failure after a certain number of reversings.' There i s a s t r e s s for which the number of reversings may be practically infinite without any failure. Un- fortunately, the value of this safe s t ress i s not known; such t e s t s have never been made. It seems logical to assume that a drilling-string s tee l can withstand a continuously reversing s t r e s s of a t least 20,000 psi without failure. Fig. 14 shows that such a s t ress would be reached with cave diameters equal to about 120 in. Such holes are probably not very common.

It should be kept in mind that Fig. 14 concerns the condition when the second buckle contacts the wall of the liole. On the other hand, Fig. 13 shows that the bending moments increase with the weight on the bit, and undoubtedly the s t r e s ses would be much larger for high orders of buckling. I f the weight on the bit i s no larger than the weight of

STUDY O F T H E BUCKLING O~ ROTARY DFULLING STRINGS 189

drill collars, the order of buckling i s low (first, second, or third); and, therefore, even in caved holes the s t r e s ses in s tee l are low. If the same weight i s carried with too few drill collars, the order of buckling becomes high and the s tee l may be subjected to reversing s t resses of 20,000 psi or more, even in caves of smaller diameters. This i s the reason for the occurrence of fatigue failures when the number of drill collars i s too small. The conclusion i s that it i s always safer to drill with a sufficient number of drill collars and adequate weight on the bit s o that the order of buckling remains low. Fig. 3, 4 and 5 give useful information in this connection.

The foregoing analysis o f s t r e s ses i s based upon the assumption that the cave extends sufficiently deep for a buckled curve, a s shown on Fig. 7 and

0 20 LO 60 80 100

HOLE DIAL6ElER. INCHES

Fig. 14 - Maximum Bending Stresses for 64-in. Dri l l Collars (24-in. ID) in 12 L b per Gal Mud for the Smallest Weight on Bi t at Which the Second

Buckle Contacts the Wall of the Hole

Fig. 15 - Caliper Log of Well in N. E. Catherine

Field, Kansas

8, to be located inside the cave. Consequently, the depth of the cave must be larger than 3 dimension- l e s s units, or, generally, more than 160 ft. In Fig. 15 the cave extends from 1,530 ft down to 1,850 ft, or 320 ft. However, the large cave i s interrupted a t 1,710 ft where the diameter of the hole i s only 21 in. At this depth the drilling string contacts the wall of the hole; and, for this reason, the maximum deflection probably does not exceed 15 in. which corresponds to a diameter of 30 in. Without the streak a t 1,710 ft the drilling string would more or l e s s follow the contour of the cave.

Fatigue failures of drilling strings may occur not only because of buckling in caves, but a lso because of the bending due to crooked holes and key seats, which are outside of the scope of the present investigation. ,

tnciination of the Bit and of the Force on the S i t

Suppose a drilling string buckles in a straight

190 ARTHUR LUBINSKI

DISTANCE xp BETWEEN B I T AND NEUTRAL P O I N T IN D I E N S I C N L E S S U N I T S

Fig. 16 - Inclination of Bit Coefficient t, and Inclination of Force on Bi t Coefficient n

hole. In that c a s e t h e s t r ing i s no longer vert ical a t i t s lower end, and t h e bit s t a r t s drilling an inclined hole.

Actual ly i t i s not t h e inclination of the bit, but the incl inat ion of the force on the bit which i s the c a u s e of the hole going crooked, and the two in- c l ina t ions a r e not equal. I t will be proved in the appendix hereto tha t t h e s e incl inat ions a re given by the following formulas:

. . . wherein:

r and m are a s previously defined. t and n are coefficients which depend upon the

d i s t a n c e between t h e bit and t h e neutral point; and a and /3 designate the incl inat ions of the bit and of the force on the bit, respect ively. 3 y incl inat ion we mean the s l o p e with respec t t o the vert ical , which i s , of course, equa l t o the angle expressed in rad ians because a and /3 are small.

The length rn of one d i n ~ e n s i o n l e s s unit i s con- s tan t for a given s i z e of drill pipe or drill co l la r s in a given mud. T h e apparent rad ius of the hole r d o e s not change if we assume a hole of cons tan t diameter. Then the incl inat ions vary only with the coefficients t and n. T h e s e coefficients have been plotted in Fig. 1 6 v s . the dis tance, in dimension- l e s s uni ts , between the bit and the neutral point, which i s proportional t o the weight on the bit.

T h i s figure s h o w s that t and, consequently, t h e inclination a of the bit inc rease with the weight on the bit. Fig. 1 6 d o e s not extend beyond the be- ginning of the second order of buckling, but t h e trend of the curve t ind ica tes clearly that the inclination of the bit k e e p s increasing with the weight on the bit for higher orders of buckling.

From an inspect ion of curve 11 of Fig. 16 i t i s s e e n that: 1. T h e inclination of tlie force on the bit i s smaller than the inclination of the bit. 2. T h e inclination of the force on the bit a l s o in- c r e a s e s between the cr i t ical condit ions of the first and second order. 3. T h e inclination of the force on the bit r e a c h e s a maximum when the second buckle i s growing on the drilling s t r ing; and, for more weight on the bit, the inclination of the force on the bit de- c reases . T h i s surpris ing fact may be e a s i l y explained. F o r the first order of buckling, the s ing le buckle i s located entirely on one s i d e of the a x i s of the hole, and the unbalance of the sys tem gen- e r a t e s a horizontal component of the force on the bit. F o r the second order of buckling, the two buckles are in opposi te direct ions and part ia l ly compensate each other. I t i s quite poss ib le that for some value of the weight on the bit within the range of the second order of buckling, the inclination of the force on the bi t i s nil.

T h e conclusion i s that in order t o drill a s traight hole it i s best t o carry l e s s weight on the bit than the cr i t ical value of the first order a t which the drilling string buckles . However, if s u c h weight is not sufficient, i t i s advisable to avoid the first order of b u c k l i n g a n d t o carry a weight on the bit c lose to the cr i t ical value of the third order. Ac- cording t o th i s rule , for 6'4-in. drill col lars Fig. 2 and 3 show that the recommended weights on t h e bit are either under 8,000 Ib or about 20,000 or

STUDY O F T H E BUCKLING

22,000 Ib. In order to ca lcu la te the magnitude of t h e incli-

nation of the bi t and of the force on the bit caused by buckling, consider, for example, 6!&-in. drill col lars in a 1 2 Ib per gal mud sub jec ted t o the smal les t weight on the bit for which the second buckle con tac t s the wallof the hole. Fig. 1 6 s h o w s that the corresponding va lues of the inclination coetTicients are: t = 1.52 and n = 0.52. Fig. 1 shows that ni = 58 ft. Subst i tut ing t h e s e numerical va lues into equat ions (9) and ( l o ) , the incl inat ions are found a s funct ions of the apparent rad ius of the hole r . Converted t o degrees, these incl inat ions have been plotted on Fig. 1 7 a g a i n s t the hole diameter. I t must be well understood that a "diameter of the hole" equal t o 20 in., for example, means that there are some c a v e s in the hole and that the drilling s t r ing may take the same s h a p e in these caves a s i t would in a 20-in. hole. Fig. 1 7 s h o w s that for 6%-in. drill co l la r s in a SO-in. cave, subjected to the smal les t weight on the bit a t which the second buckle con tac t s t h e wall of the hole, the inclination of the bit i s equal t o 2 deg and the

0 20 60 80 1CQ 120

DIAMETER OF HOLE, INCliES

Fig. 17 - Inclination of Bit and Inclination of Force on Bit for 64-in. O D ( Z ~ - i n . ID) Drill Collars in 12 Lb per Gal Mud for the Smallest Weight on Bit at Which the Second Buckle Contacts the Wall

of the Hole

F ROTARY DFULLING STRINGS 191

I:

Fig. 18- Kel ly Displacements vs. Weight on the Bit

inclination o f t h e force on the b i t i s equal to deg. Drilling crooked ho les i s caused not only by

buckling of drilling s t r ings, but i s a l s o a resul t of drilling in a dipping formation. T h i s l a t t e r factor i s outs ide the s c o p e of th i s investigation.

Length of Buckled Curves

Consider that the kelly i s displaced while no drilling i s in progress. Such a c a s e occurs when drilling in a very hard formation, or when the rotary tab le i s not turning. If the lielly posi t ions are plotted vs . the weight on the bit (Fig. 18), the diagram comprises a s t raight portion .4P and a curved portion PB. T h e curved portion corresponds to larger weights on the bit when the drilling s t r ing i s buckled.* L e t u s consider point E in the curved portion of the diagram. T h e corresponding ordinate C E i s equal to the sum of C D and DE. C D repre- s e n t s the displacement of the kelly due to e l a s t i c elongation of the drilling string, and DE represents the displacement of the kelly due to buckling.

I t will be proved in t h e appendix that the displacement AL due to buckling i s given by the following expression:

wliereirz: r and m are a s previously defined; and q i s a coefficient which depends upon the d i s tance , in d imens ion less units. between the bit and the neutral point, which i s proportional t o the weight on the bit. F o r a given s i z e of drill col lars or drill pipe and a given mud densi ty, the length rn of 1 du and the apparent radius of tlie hole r are con- s tan t , and AL depends only upon q. T h e coefficient q h a s been plot ted on Fig. 1 9 vs . the d i s tance , in dimensionless uni ts , between the bit and the neu- t ral point. Consider progressively increasing va lues of the weight on the bit. When the cr i t ical value of the first .order i s reached, q suddenly i n c r e a s e s from zero to 0.465. If the weight on the bit i s f u r ther increased, q becomes progressively larger; and the increase of q becomes extremely rapid beyond the c r i t i ca l condit ions of the second order for * D~agrams in which the welght on the elevators i s plotted in-

stead of the we~ght on the blt were applled to determine how much of a frozen or cemented column of pipe 1s free.3


0 1 2 3 L

DISTANCE x BETWEEN B I T AND NEUTRAL P O I N T ZN DIUENSIONLESS WITS

Fig. 19 - Kelly Displacement Coefficient

which the second buckle i s growing. This fact, which i s in perfect agreement with common sense , certainly also holds true. for higher orders of buckling. As the weight on the bit i s increased above any critical condition and a s long a s the new buckle i s growing, relatively large displacements of the kelly produce small increases in the weight on the bit. After the buckle contacts the wall of the hole, the same increments of the weight on the bit are produced by much smaller displacements of the kelly.

In order to calculate the magnitude ofthese facts and to realize in which circumstances they may be actually observed, consider the drilling string subjected to the smallest weight on the bit at which the second buckle contacts the wall of the hole and for which Fig. 19 indicates that q = 1.51. Sub- stituting this value into forniula (11) and replacing the length nr. of 1 du with the values corresponding

to various s i zes of drill pipe and drill collars in a 12 Ib per gal mud (Fig. 11, AL has been calculated vs. r and plotted in Fig. 20 vs. the hole diameter. This figure shows that for the secondorder of buckling AL i s equal to about 0.1 in. in a 20-in. hole, and cannot be observed. In other words, point E of Fig. 18 i s located close to P, e.g., a t El where the deflection of the curve PB withregard to i t s tangent PL, cannot yet be seen.

AL increases very rapidly with the hole diameter, and Fig. PO shows that, for 6:i-in. drill collars subjected to the second order of buckling in an 80-in. cave, AL i s equal to 3 in. and may easily be observed.

On the other hand, Fig. 19 shows that nL, also increases rapidly with the order of buckling. If the weight of drill collars i s a t least equal to the weight on the bit, the order of buckling i s generally low; and, therefore, the displacement of the kelly when no drilling i s in progress i s proportional to the increases of weight on the bit. A positive indication that something abnormal has happened

0 20 LO 60 80 100

HOLE D W T E R , LNCHES

Fig. 20 - Kelly Displacements for the Smallest Value of the Weight on B i t at Which the Second

Buckle Contacts the Wall of the Hole


i s given by the fact tha t the kel ly d i sp lacements a re not proportional to the i n c r e a s e s in t h e weight on the bit. One possibi l i ty i s a very wide cave; another tha t a tight spo t occurred somewhere in t h e drill pipe, which buckled above th i s spot , with a high order of buckling. Centrifugal Forces

Consider Fin. 23 which represen ts a c r o s s s e c - - t ion through a hole and a buckled, therefore eccen- t r ic , drilling string. When the drilling s t r ing i s ro-

PRESSURE DROP I N DRILL COLLARS

P. S. I./lIXX) FT

Fig. 21 - lnfluence of Drill-collar Pressure Drop on Buckling

ing s t r ing may behave l ike a flexible shaf t and ro- t a te about i t s own a x i s C. T h e motion which actu- a l ly o c c u r s i s that requiring the l e s s e r energy.

R'hile buckling, the drilling s t r ing w a s d i sp laced downward, and the work s p e n t by gravitational fo rces w a s s to red a s potent ial energy of e l a s t i c bending of the pipe. If thereafter the motion b occurs , the drilling s t r ing d o e s not go dovm any far- ther, and no more energy of bending i s spent . T h e only energy required i s that which i s spen t on friction a g a i n s t the wall of the hole and on v i scous forces i n the drilling fluid.

BROKEN LINES: 3-l/Zn A.P.I. DRILL PIPE

40 LM 1 2 0 I 6 0 200

PRESSURE DROP I N DRILL PIPE

P. S. I./1000 FT

Fig. 22 - Influence of Drill-pipe Pressure ~~o~ on Buckling

If the motion a occurs , the energy s p e n t on friction aga ins t the wall of the hole i s increased in the rat io of diameters of the hole and the pipe. Moreover, the res i s tance of . the mud t o a f a s t rotation of the pipe about the a x i s 0 of the hole must

Fig. 23 - Cross Section Through a Hole Showing Buckled Drilling String


be very high. Consequently, the normal motion of the buckled drilling string i s the motion 6; i.e., the rotation about i t s own axis C. Therefore, the re- sultant of centrifugal forces i s equal to zero; and their influence on buckling i s nil.

The foregoing holds true for an ideal pipe for which the flexural rigidity i s the same in all direc- tions. At the other extreme, imagine string which would have very small flexural rigidity in one direction and very large .in others, such a s a rec- tangular bar. It i s obvious that the bar would buckle in the plane of the smallest flexural rigidity and would rotate about the axis of the hole and not the axis of the bar. The actual case of a drilling string l i e s between the two extreme cases. The pipe i s not altogether symmetrical but the asymmetry i s much smaller than i n a bar.The string rotates about i t s axis unless the weight on the bit i s such that a new buckle i s just added, e.g., for a little more than 4 du (see Fig. 9) or about 18,000 Ib for 65i-in. drill collars. Drilling with a buckled string rotating about the axis of the hole should be avoided in caving formations. A good procedure for avoiding this condition i s to watch a rotary torque gage which would indicate a sharp increase. Another way i s to drill with weights on the bit which are not too close to critical values (see Fig. 2 through 5).

The normal motion of the buckled drilling string, i.e., rotation about i ts axis, i s the reason for the reversing, or rather rotating stresses, and for the inclination of the bit. Influence of Hydrostatic Pressure on Buckling

The influence of hydrostatic pressure on buckling i s a subject of controversial opinions. According to a general belief, the pipe buckles when under compression; and, i nasrnuch a s the hydrostatic pressure always gives a high compression to the lower part of the drilling string, an erroneous conclusion has sometimes been reached that the hydrostatic pressure contributes to the fatigue of the drilling string.4 These conclusions are part1 based on a theoretical investigation by Handleman Y which re!ates to the buckling of a beam submitted to pressure. Handleman's results are not applicable to the drilling string because, in the latter problem, - the weight of the string i s essential and cannot be neglected.

The only influence of the hydrostatic pressure on buckling resqlts from the changes in value of weight per foot p, a s indicated on Fig. 1 and 2. This influence i s smal1.1t will be proved in the appendix hereto that the increase of compression in the lower part of the drilling string has no effect on buckling and fatigue.

To avoid confusion, le t us point out that i t i s perfectly correct to say that buckling mainly concerns the portion of the drilling string located under the neutral point, provided the definition of the neutral point i s that adopted in this paper, viz.: "The neutral point divides the drilling string into two portions; the weight in mud of the upper portion being equal to the weight suspended from the elevators, and the weight in mud of the lower portion being equal to the weight on the bit."

According to the usual definition, which i s re- jected herein, the neutral point i s that at which there i s neither compression nor tension. Influence of Pump Pressure on Buckling

The pump pressure i s equal to the sum of pressure drops in the bit, in the drilling string, and in the annulus. The latter i s negligible. The first two will be considered in succession.

a. Pressure Drop in the Bit A s proved in the appendix, drop of pressure in

the bit i s of the same nature a s the hydrostatic pressure, but acting the opposite way. It decreases the compression in the lower portion of the drilling string, but has no effect on buckling and fatigue. 6. P r e s s w e Drop in the Drilling String

As explained previously, the buckling charac- terist ics (length of a dimensionless unit, critical weights, etc.) depend upon the loss of weight of steel in the drilling fluid. This loss should be considered somewhat higher than the actual ,one in order to make allowance for the drop of pressure in the drilling string. Fig. 21 and 22 show the values of k , the coefficient by which the weight per foot p must be multiplied, vs. the pressure drop in the drilling string. Fig. 21 concerns drill collars and Fig. 22 various types of drill pipe. On both figures lines for three densit ies of mud are drawn. Fig. 21 shows that the influence of the drop of pressure on buckling i s negligible for drill collars.

Fig. 22 shows that the situation i s different for drill pipe. For high rates of circulation, the critical weights are slightly decreased and the length of a dimensionless unit i s increased. Fig. 22 enables the proper use of formulas and curves involving p and m .

The values of pressure drop in drill pipe are a ~ a i l a b l e . ~

CONCLUSIONS The principal results of th is investigation may

be summed up a s follows: 1. The best drilling conditions occur when the drilling string i s straight; i.e., when the weight on the bit i s smaller than the critical value of the


first order. Unfortunately, such weights are usually not sufficient because the rate of drilling i s too low. New methods which might allow a satisfactory rate of penetration with a straight drilling string should be tried by the drilling industry.

2. Insofar as straight-hole drilling i s concerned, carrying weights on the bitwhich are slightly l e s s than the critical value of the third order i s better than using any smaller value of weight at which the string i s already buckled.

3. When the required weight on the bit i s large,

the use ofsufficient drill collars avoids high-order buckling which generates large bending moments and fatigue failures of the drill pipe.

When working out a drilling program for an area in which drilling i s difficult, a careful study should be made of the number of drill collars and the weight to be on the bit.

Rigs should be provided with good quality weight Their calibrations should be checked

from time to time. The use of rotary torque gages i s advised. A weight recorder i s useful in order to rheck whether the weight instructions are observed.

APPENDIX INTRODllCTION

In various treatises on elasticity, applied mathematics, or ~~~~~l functions,7~8~9,10 the only aria-

lyzed case of buckling with distributed weight concerns a vertical pole with the lower end fixed and the upper end free. There i s a reaction on only one end, and this reaction i s vertical.

~h~ case of the drilling string is much more plicated. Both ends can be considered a s hinged; and, consequently, there are reactions on the string at both ends, ~h~~~ reactions have vertical and horizontal components. hjoreover, there me reactions of the wall of the hole at the points where the buckled pipe contacts the wall.

Analytical Formulation of the problem

Let us begin with an assumption that the drilling string is a continuous pipe with no tool joints.* The external forces acting on the drilling string are represented on Fig. 24, in which: 1. The upward force IT$ i s the reaction of the elevators on the drilling string. 2. The upward force W2 i s the vertical component of the reaction of the bottom of the hole on the drilling string and i s commonly called "weight on the bit." 3. The force F2 i s the horizontal component of the reaction of the bottom of the hole on the drilling string. 4. The horizontal force 4 i s the reaction of the

-bushings on the drilling string. 5. The horizontal force F i s the reaction of the

of the 0' the drilling string i f the pipe i s buckled. '

Twoadditional forces not shown on Fig. 24 should be considered; viz., the weight of the pipe which i s a vertical downward force, and the buoyancy which i s a vertical upward force, both applied to the center of gravity of the drilling string.

'The generalization of the r e su l t s to combination strlngs of drlll pipe and drill collars wi l l be easy by the use of dl- mens lon l e s s unlts.

The influence of the viscous forces on the drilling string and of the "jet" force on the bit are neglected. They are small in comparison to the weight.

When IT$= 0, there i s no weight on the bit, the drilling string i s straight and this straight form of elast ic equilibrium i s stable; i.e., if a lateral force i s applied and a small deflection i s produced, this deflection disappears when. the lateral force i s re- "loved and the pipe becomes straight again- If E$ is increased, but maintained below a certain critical

the straight form remains If the critical of is reached, the form of the pipe becomes unstable; i.e., if a lateral force, however small, i s applied and a small deflection i s produced, this deflection does not disappear when the lateral force i s removed. On the contrary, the deflection increases until a bent form of stable

' equilibrium known as buckling i s reached. We shall choose, a s axes of coordinates, ,\ and

Y a s shown on Fig. 25. The ,T axis i s the axis of of the hole. The point of origin N i s the previously defined neutral point. I t i s well understood that the plane of the X and Y axes i s the plane of smallest flexural rigidity in which the buckling occurs. The A axis i s directed downward a s shown on Fig. 25. The function I'(S), representing the axis of the buckled string, can be represented by the following differential equation:

d2 Y hl = EI - dX2 (12)

where,n: ,)I is the bending moment; E is Young*s modulus of the steel; and I is the moment of inertia of the cross section.

Taking the derivative of both members of the (12), we obtain:

d3 Y A = E I -

dX3 (13)

wherein: [I i s the shearing force. We shall now determine the shearing force along

any cross section of the drilling string, such a s


Fig. 24 - Designation of External Forces Acting upon Drilling String

,MN ( see Fig. 24), and substitute in equation (13) - the value obtained. For this purpose, we shall determine all the known forces applied to the portion of the drilling string located below the section IMN. These forces are represented vectorially on Fig. 26 and are: W2 and F2 already defined; V or weight of the portion of the drilling string located under IMN; and the buoyancy.* If the portion of the drilling string under ,MN i s surrounded by the fluid; i.e., if we consider the portion above il!N a s non- existent, the buoyancy would be equal to the weight of the displaced fluid. & represents this upward force on Fig. 26. Actually, the hydrostatic pressure does not act on the section .MN. This part of the Luoyancy (& on Fig. 26) must be vectorially sub- tracted from Bl in order to obtain the true buoyancy.

Since the portion of the drilling string we are considering i s in equilibrium, the ve'ctorial sum of all the forces i s equal to zero. As shown on Fig. 27: A B i s the weight on the bit, or K$; BC i s the horizontal component of the reaction of the bottom of the hole, or F2; C D i s the weight IV of the part of the string located under MN; D E i s the buoyancy 13,; and E F is the buoyancy B2.

If the cross section ,MN i s taken higher on the - drilling string, we should consider a lso the force F, i.e., the reaction of the wall of the hole on the buckled pipe. Ilowever, let us consider first the theoretical- case of buckling which would occur outside the hole, in which case F = 0.

From the construction of Fig. 27, we rnay determine the force FA which represents the reaction of that part of the drilling string above hliV on the portion below. Th i s force has two components, viz., FG i s the shearingforce and C;A i s the compression or tension, according to i t s direction.

Consider a cross section ,MIN' somewhere above MR. Then the forces C D (weight of the portion located below ,M1N') and DE (buoyancy & or weight of displaced fluid) corresponding to the new cross section are larger than the forces corresponding to the cross section MR. However, the ratio of these two forces remains the same and i s eaual to the ratio of the densities of steel and mud. Therefore, instead of forces C D and DE, we shall consider the force CE a s being equal to the weight of the drilling string under the cross section, on condition that we multiply this weight Lv

As - % As

* I n us ing the term "buoyancy," it i s assumed that the drilling string i s subjected to a hydrostatic distribution of pressure. Actually, the c a s e i s more complicated; but, a s shown later, the dynamic and s tat ic distributions o f pressure lead t o the same buckling characteristics.

STUDY O F THE BUCKLING 0,F ROTARY DRILLING STRINGS 19 7

wnerein: As is t h e specific gravity of s t e e l , and -AM i s the specific gravity o f mud.

F o r the c r o s s sec t ion h!'N1, the force E F i s smaller than the corresponding force for the c r o s s sec t ion MN. However, E F i s perpendicular t o the

shearing force FG; and, consequently, t h e value of FG i s independent of E F . T h e final conclusion. then, i s tha t a s far a s formula (13), i.e., the shearing force, i s concerned, we need not consider the buoyancy a t all, if we consider the densi ty of the drilling-string material reduced in the rat io

As - 41 n,

The vectorial equation

gives, by projecting a l l the vectors on the a x i s ,lilAr:

A 3 s in a -BC c o s a - C E s in a -FG = O

Fig. 26 Fig. 25

198 ARTHUR LUBINSKI

Fig. 27 Then , the shearing force FG i s equal to:

FG = (AB - C E ) s in a- BC c o s a

Under ac tua l condit ions in the hole a i s very small ; therefore, w e may put c o s a = I and s i n a = tan a. Then t h e l a s t equation becomes: Shearing force A = F G = (AB-CE) tan a- BC (14)

L e t p designate the weight of the drilling s t r ing n - A

per unit of length multiplied by s M , a s previously explained. AS

L e t ,k and ,b designate respect ively t h e va lues of X for the two e n d s of the drilling string. Hence:

Then formula (14) may be written a s follows: Shearing force A = [Ri2 - p(X2- $)I tan a- (17)

Substituting equation (16) in equation (17) and replacing tan a - w i t h -dlYd,Y,..we obtain

and subs t i tu t ing t h i s l a s t expression in equat ion (13):

We have t h u s obtained the differential equation

of the buckled drilling string. B y properly choos ing t h e unit of length, the equat ion may be put in a simpler form. Let :

x = mx (19)

and

Y = my wherein: rn i s a constant which will be chosen later. Then:

Substituting express ions (19), (21), and (23) in equation (181, we obtain:

T h e value nL should be chosen s o that

L e t c be defined a s follows:

Subst i tut ing (25) and (26) into (24), we obtain

Substituting the express ions (22) and (25) into (12) we obtain: .-

wherein: i s the bending moment. L e t u s adopt a sys tem of cons i s ten t un i t s in

which: T h e unit of X and 1' is a foot. T h e uni t of R/i, .K$, F2, F, e tc . i s a pound. T h e unit of p i s a pound per foot. T h e unit of E i s a pound per square foot. T h e uni t of 1 i s a foot4. T h e unit of hi i s a foot-pound.

Formula (25) shows t h a t rn i s measured in feet. Then express ions (19), (201, (211, (281, and (26), show t h a t x, y, dy/dx, dZy/dx2, a n d c a r e dimension- l ess . Consequently, t h e a n a l y s i s made with those fac tors will be al together general and independent of the type of t h e drilling s t r ing and drilling fluid.

* T h i s expression e x p l a ~ n s formula (1) in the first part of this paper.


Solution of the Differential Equation

Let: d z =J dx

(29)

Substituting (29) into (27), the differential equation becomes

The variable z can be expressed in form of power series:

n = 0 and substituting (31) in equation (30), we obtain

Th i s expression i s a polynomial of powers of x. Expression (32) must be satisfied for any value of x; and, therefore, the coefficients of xO, xi, x2, x?, x4, etc. must a l l be equal to zero. We obtain thus the following expressions:

coefficient of x" = 2az + c = 0 coefficient of xl = a. t2(3a3) = 0 coefficient of x2 = al+3(4a4) = 0 .

coefficient of x3 = a2+4(5%) = 0 . . . . . . . . etc.

Consequently, q,, al, a*, a3, .a,,, etc. may a l l be expressed a s functions of w, a l , and c . Substitu- ting these expressions in equation (311, we find the general solution of the differential equation:

Putting in equation (33) ao=a , a1 = b , and expres- sing z by dy/dx we obtain equation (35) below. 3 y integrating and taking the derivative of (35), we obtain equations (34) and (36):

y = aS(x) + bT(x, + cU(x) t g (34)

In equations (34), (35), and (36) the following designations are made: g i s an integration constant.

The functions F(x), G(x), P(x), and Q(x) may be expressed in the form of Bessel functions of frac-

1 2 2 tional orders , - /J , /3 . and - /3.

Complete tables of Bessel functions of fractional orders have been made available very recently. l1

For the negative values of N, the corresponding Bessel functions of the second kind must be used. Functions s(x), T(x), U(x), li(x), and R(x) have been calculated by series, the convergence of which i s fairly satisfactory. The computation has been made for a range of x from -4 to t4.218 for

200 ARTHUR LUBINSKI

Fig. 28 - Functions F(x), P(x), and S(x)

the funct ions F(x) , q x ) , and ;I(%) and from -6 to 4.32 for the s i x other functions. T h e r e s u l t s are plotted on Fig. 2 8 t o 33. T h e number of decimal p l a c e s given by t h e s e figures i s not large enough

Fig. 29 - Functions of G(x), a x ) , and T(x)

for many calculat ions. When needed, funct ions have been computed for particular va lues of x with up t o s i x d ig i t s af ter the decimal point.

Inasmuch as the differential equation (27) i s of the third order, i t s general solut ion, equat ion (34), con ta ins 3 integration cons tan ts , viz., a, 6, and g. In addition t o the integration cons tan ts , the.parameter c i s unknown b e c a u s e F2 ( the horizontal component of the reaction of the bottom of the hole on t h e bit) i s a l s o unknown. T h i s parameter c a n be determined by imposing an addi t ional boundary condition.

Critical Conditions of the First Order L e t xi and r2 des igna te the v a l u e s of x for the

Fig. 30 - Functions H(x), R(x), and U(x)

upper and lower ends, respect ively, of t h e drilling string. L e t P i , Q , Rl, S 1 , etc. des igna te the v a l u e s of the funct ions P(x), W x ) , R(x), S(x), e tc . for X. = X I , and P2 , Q 2 , R 2 , S2, etc. des igna te respec- t ively the v a l u e s of the s a m e funct ions for r = x s .

At both e n d s of the drilling $tring t h e bending moment i s equal to zero (both the bush ings and the bit may be considered as hinged ends) . Therefore, formulas (28) and (36) give:

aP, + bQ, + cK, = 0 (50)

aP, t b Q 2 +cK2 =O (51)

For both k n d s y = 0; and , therefore, formula (34)

DIAENSIONLESS UNITS

Fig. 31 - Functions F(x), -P(x), and -S(x) gives:

asl + bTl t cU1 t g = O

as, t b T t cU2 t g = 0

By eliminating g between the two receding equations and rewriting equations (50) and (511, we get the following s e t of 3 equations in which a, 6, and c are unknown:

+ bQl t cK1 =O (50) {:" t bQ2 t cR, =O (5 1)

a(S1 - S2) t b(Tl -T2)+c(Ul - U,) =O (52)

value. of x2 corresponds to a stable equilibrium and has been drawn on Fig. 34. For the interpretation of this figure, i t should be remembered that xi (the ordinate) i s the distance from the neutral point to the top of the hole, and that x i s the distance from the neutral point to the b i t i is latter distance i s proportional to the weight on the bit U;. Fig. 34 shows that when the absolute value of xi i s small, i.e., when the hole i s very shallow, the pipe requires a larger weight on the bit in order to buckle. When the hole i s deeper, the critical value of the weight on the bit decreases and approaches asymp- totically to a certain value. This may a lso be explained without the previous mathematical analysis. If the upper end of the drilling string i s close to the neutral point, the bushings add some rigidity to the string. On the other hand, when the hole i s deep enough, buckling of a joint located under the neutral point i s independent of whether the portion of the string above this point i s 1,000 or 10,000 f t long.

Since the second n~en~bers of al l three equations of the se t are equal to zero, the solution of the se t has a physical meaning only if i t s determinant i s equal to zero.

Under actual drilling conditions;xl i s very large and x2 i s equal to i t s asymptotic limit. 'I'he curve of Fig. 34 has been plotted with calculated data between z1 = O and xl =-6. The extrapolation of

Pl Ql 1

p2 Q2 R, (Sl - S,) (TI - 3) (U1 U2)

= 0 (53)

Expression (53) i s the relation between xi and x2 which must be satisfied for the buckling to occur. By the trial-and-error method it was found that expression (53) may be represented by a series of curves. Only the curve pertaining to the s n ~ a l l e s t

2n2 ARTHUR LUBINSCI

t DIMENSIONLFSS UNITS

Fig. 32 - Functions -G(x), Q(x), and T(x)

this curve beyond xi = -6, shown a s dashed lines, seems to indicate that the asymptotic limit of x2 i s 1.88. On the other hand, for x1 =-6, x2 i s equal to 1.94. Consequently, we may assume with negligible error that x2 = 1.93 i s the critical condition of the first order.

For all practical purposes we may consider the point x1 as being not the upper extremity of the drilling string, but simply a point for which x1 =-6. The deflection and the bending moment are practically equal to zero a t this point.

Critical Lengths of Stands in Stack

The relations obtained previously may l e used to determine the critical length of stands of pipe stacked vertically in the derrick. Let us assume that a stand of pipe in stack i s vertical, has both ends hinged, and the whole weight i s supported by the lower end. These assumptions are close to the actual conditions.

Nhat i s the maximunl length for which the pipe remains straight and does not buckle? The answer

i s immediate: x1 = 0 and then we see on Fig. 34 that q = 2.65 a s used previously herein.

,

Critical Weights of Combination Drilling string

Consider first the case in which the,,length of one dimensionless unit rn i s the same for ..drill pipe and drill collars. For example, 47i-in: drill collars are many times heavier than &-in. drill pipe, but Fig. 1 shows that for both, m i s almost the same.

It is quite obvious that our previous analytical study concerning the single drilling string holds true in the present case, insofar a s the critical distances in dimensionless units between the bit and the neutral point are concerned. Two different cases must be considered according to the location of the neutral point, which may be in the drill pipe or in the drill collars. L et:

designate the critical weight of the first order. L, designate the total length of drill collars,.in

feet.

STUDY O F THE BUCKLING O F *ROTARY DRILLING STRINGS 20 3

t DIENSIONLESS UNITS

Fig. 33 - Functions -H(x), R(x), and U(x)

nz designate the length, in feet, of one dimension- l e s s unit of collars and pipe.

pc designate the weight per foot of the drill collars, in pounds per foot.

p,, designate the weight per foot of the drill pipe, in pounds per foot.

First case-the neutral point i s located in the drill pipe.

The critical weight on the bit i s equal to the weight of the drill collars plus the weight of a portion of drill pipe. The length of the collars, in dimensionless units, i s LJm, Therefore, for critical conditions of the first order, the lengthof that portion of the drill pipe located below the neutral point i s equal (in dimensionless units) to 1.94- LJm. The weights of these two portions of the drilling strings are Lp, and (1.94- L b ) m p p , respectively. Their sum i s equal to:

4 = LC ( ~ , - p ~ ) t 1.94mpp (54) '- Consequently, the critical weights are linear

functions of the total length of drill collars. These functions are represented on Fig. 3, 4, and 5 by the inclined portions of line 1.

Second case-the neutral point L S located in the drill collars.

The whole weight on the bit i s given by the drill collars. The critical values of weight on the bit cannot be increased by adding more drill collars. This case corresponds to the horizontal portions of line 1 on Fig. 3, 4, and 5 .

Consider now the case when length m of one dimensionless unit i s not the same for the pipe and the collars. Separate equations should be written for the pipe portion and collar portion of the string. The boundary conditions at the two ends of the string would be the same as for a single string. Three additional boundary conditions should express the fact that at the junction of pipe and collars the deflectior,~, the tangents to the ax i s of the string, and the bending moments are equal for these

204 ARTHUR

two portions of the string. Instead of th i s rigorous but very long method, a n

approximate approach h a s been used in t h i s study. Fig. 1 shows that t h e dimensionless un i t s of col- l a r s and pipe are -of approximately the same length. Consequently, i t i s believed that the method de- scr ibed previously for the c a s e of equal v a l u e s of m for pipe and co l la r s may be appl ied t o a c t u a l

DIiUZNSIONLF.SS UNITS

Fig. 34 - Critical Conditions of the First Order

c a s e s with a sufficiently c lose approximation for a l l pract ical purposes.

Consider , for ins tance , the broken l ine 1" of Fig. 3, which h a s been ca lcu la ted a s follows:

OA = cri t ical weight of pipe. DC = cri t ical weight of collars.

a = cri t ical length of collars. T h e l ine rigorously ca lcu la ted would a l s o p a s s

through point A. I t would approach asymptot ical ly the s t raight l ine BC . point of Tangency for Critical Conditions of Ule

First Order

Express ion (56) is t h e relation between t h e ab- s c i s s a ~ ~ of the point of t angency and the a b s c i s s a s xi and xg of the end of the s t r ing for c r i t i ca l condi- t ions.Aspreviously explained, z1 = -6 a n d x 2 = 1.94.

T h e value of x3 sa t i s fy ing the relat ion (56) w a s found by trial and error and i s equal t o x3 = 0.145. Equation Coefficients for Critical Conditions of the

First Order

We s h a l l determine t h e point of contact , t h e a b s c i s s a of which wil l be designated X3, in fee t , and x3, in d i n ~ e n s i o n l e s s units. Since for x = x3, dy/dx = 0, we have from the equation (35):

a& + bG3 + cH3 = 0 (55)

Equat ions (55), (50), and (51) form the following s e t :

I aF3 t bG3 t cH3 = 0 (55

aP, t bQ1 t cR1 = 0 (50)

aP, t bQ2 t cR2 = 0 (51)

which h a s poss ib le so lu t ions if t h e following determinant i s equal to zero.

In order to invest igate the shape of t h e buckled- s t r ing a x i s , the distribution of bending moments, etc., we must find the v a l u e s of a, 6, and c, but t h e s e t of equa t ions (501, (51), and (52) g i v e s indetermi- "

n a t e v a l u e s for t h e s e factors. Then , apparently, if we consider the pipe outs ide of t h e hole, t h e buckling once s tar ted would not s top. T h e pipe would bend more and more without reach ing any equilibrium. T h i s apparent absurdi ty i s due to the fact tha t formula (12) ho lds true only for small deflec- t i o n s for which the curvature i s equal t o d2y/d,y2. F o r large deflectionsfor which an equilibrium would be reached, a more complicated formula should be used. However, a l l t h i s i s without any pract ical

5 G3 H3

PI Q, R, P2 Q2 R2

*Lines 2 and 3 concerning the second and thud orders of buckl~ng have been constructed by analogous means.

= 0 (56)


meaning because ò he theore t ica l equilibrium of the buckled pipe would be reached far beyond t h e e l a s t i c limit of the s t e e l , and t h e pipe would not only become crooked but would break. Prac t ica l ly , such a condition never occurs because the buckling i s s topped when the p ipe .con tac t s the wall of the hole. T h i s should be taken into account in order t o remove the indeterminancy of equa t ions (50), (51), and (52). F o r t h i s purpose, l e t u s take into con- s iderat ion that a t t h e point of t a n g e i c y the deflection i s equal t o the apparent rad ius of the hole; i.e., for x = x3, Y = r, and, according t o equat ion (20), y = r /m. Therefore, equat ion (34) gives:

At the lower end of t h e s t r ing t h e deflection i s nil, i.e., for x = x2 = 1.94, y =O. Therefore, equation (34) g ives :

as2 + bT2 + c U 2 t g = 0 Eliminating g between the l a s t two equat ions and rewriting equat ions (50) and (51), we obtain the following s e t of equat ions:

r a(S3 - S1) + b(T3 -TI) + c(U3 - Ul) = (57)

= 0 (50)

+bQ, + c R 2 = 0 . (51)

In t h e s e equat ions xi = -6, x2 = 1.94, and x3 = 0.115. T h e solut ion of the s e t l e a d s to the va lues .of a(m/r), b(ni/r), and c(m/r) which a r e written in a tab le later herein. Points of Tangency for Weights on the Bit Above

Critical Conditions of the First Order

From equation (53), we have deduced that a buckled form of the first order is s t a b l e only for one va lue of the weight on t h e bit; i.e., a value corresponding t o x2 ~ 1 . 9 4 . We s h a l l now inves t iga te what happened af ter buckling of the first order occurred and t h e weight on the bit i s gradually further increased. Apparently there should be no buckling a t all. Such a conclusion is erroneous because the differential equat ion (181, which w a s the s tar t ing point for a l l of our deduct ions, d o e s not hold true any more for the whole length of the drilling string. F o r the upper portion of the drilling s t r ing ( that which i s loca ted above the point o f tangency, s e e Fig. 241, we must take into consideration t h e force F which i s the react ion of the wall o f t h e hole aga ins t the pipe. Equat ion (18) is replaced by:

Express ion (26) and the differential equation (30) a r e rep laced by:

and

respect ively. The integration of the differential equa t ions (30)

and (60) g i v e s the s a m e kind of genera l solut ion, viz., (34), (35), and (36). However, not only c, but a l s o the integration c o n s t a n t s a, 6, and g become different for the lower and upper portions of the drilling string.

Cons tan ts corresponding respec t ive ly to the ,upper and lower portions of the drilling s t r ing will be des igna ted by subscr ip t s 1 and 2, respect ively. Then the equa t ions corresponding to the upper por- t ion are:

y = alS(x) + blT(x) + clU(x) + gl (61)

and t h o s e corresponding to the lower port ionare:

y = a2S(x) + b2T(x) + c2U(x) + g2 (65)

F2 C2 = -

Pn' (68)

L e t xi correspond, as previously, to the upper end of the drilling string; or, rather, t o the point where XI =-6. L e t x2 correspond t o the lower end of the drilling string, and x3 t o t h e point a t which the pipe i s tangent to the wall o f the hole. T h e three boundary conditions for the upper portion of the drilling s t r ing are a s follows: 1. T h e bending moment i s equal to zero a t the upper end of the drilling string, which l e a d s to equation (69). 2. dy/dx i s equa l to zero for x = x3, which l e a d s to equation (70). 3. F o r x = xi , y = O . 4. F o r x = x 3 , y = r / m

T h e s e l a s t two conditions, when introduced into the equation (611, give two express ions from which g h a s been eliminated giving finally equation (71).

206 ARTHUR LUBINSW

T h e boundary condit ions are the same for the lower portion of the drilling s t r ing a s they were for t h e upper part; therefore, no explanat ion i s neces- s a r y for equat ions (72), (73), and (74) following, which a re analogous t o equat ions (69), (70), and (71).

One addi t ional boundary condition e x p r e s s e s the fact that a t the point of tangency ( x = x 3 ) , the bending moments and, consequently, the v a l u e s of dZy/dx2 [ s e e express ion (28)] calculated with (63) and (67) must be equal , which l e a d s t o equation (75).

'alp, + b ~ Q ~ + c l R ~ = 0 (69)

% F3 + blG3 "IH3 = 0 (70) al(S3-S,) + bl(T-TI) + cl(U3-U,) = (71)

q P 2 + b2Q2 + c2R2 = 0 (72)

%F3 t b ; ~ , ' c2H, = 0 (73) a2(S3-S2) + b2(T-T2) + c2(U3-U2) = & (74)

a& + blQ3 t c1R3 - %a25 - b2Q3 - ~ 2 ~ 3 = 0 (75)

T h e foregoing s e t of s e v e n equat ions with s i x unknowns a l , b l , c i , a 2 , b 2 , and c2 h a s poss ib le so lu t ions only if t h e following condition i s satisfied:

Express ion (761, in which xi i s constant and equal t o -6, i s the relat ionship between x2 ( the d i s t a n c e from the neutral point to the bit) and x3 ( the d i s tance from the neutral point to the tangency point). T h e following va lues of x3 corresponding to various va lues of x2 have been found with equation (76) by t r ia l and error.

Table 1 x

2 x3 R ernarks

1.940 0.145 Crltical condition of the first order. 2.600 0.942 3.200 1.668 3.753 2.346 Critlcal condition of the second order. 4.000 2.672 4.218 3.098 Second buckle contacts wall of the hole.

T h e v a l u e s given in t h e first l ine of T a b l e 1 have been previously found. T h e y correspond t o the cr i t ical condit ions of the first order. T h e s p e c i a l meaning of t h e va lues of t h e fourth and the l a s t

l i n e s of Table 1 wil l be explained la te r herein. T h e d i s tance x 2 - x 3 between the bit a n d t h e

tangency point h a s been plotted vs. the d i s t a n c e x2 between the bit and the neutral point on Fig. 9, which w a s explained previously.

Equation Coefficients for Weights on the Bit Above Critical Conditions of the First Order

S e t s of numerical va lues of x2 and x3 from T a b l e 1 are subst i tuted into equa t ions (69) t o (74). T h e s e t of equat ions (69), (701, and (71) i s so lved for al(rn/r), bl(rn/r), and cl(rn/r). T h e s e t of equa t ions (721, (73), and (74) i s so lved for a2(rn/r), b2(n1/r), and c2(rn/r). T h e r e s u l t s t o three decimal p l a c e s a r e shown in Table 2. T h e r e s u l t s ac tua l ly used comprised s i x decimal p laces , necessary mainly when large negat ive v a l u e s of x are involved.

T h e first l ine of T a b l e 2 corresponds t o the criti- c a l condit ions of the first order for which al = a 2 , bi = b 2 , and cl = c 2 . T h e s e v a l u e s have been pre- v ~ o u s l y found.

T h e s p e c i a l meaning of the v a l u e s of the fourth and l a s t l i n e s of the t ab le above will be explained further herein.

Shape of Ule Buckled Drilling String

L e t u s define the deflection coefficient h by the following expression:

From equat ions (77) and (20) we obtain

Y = hr (78) %herein: h is equal t o unity for the deflection I' which i s equal to the apparent rad ius of the hole r.

At the upper end of the s t r ing the deflection is ni! and, consequently, equat ion (61) g ives :

0 = a,Sl tb lTl t c l U 1 t g , . Eliminating gl between th i s express ion and equa-

tion (61), we obtain for the portion of the s t r ing above the tangency point:

A similar equat ion for the portion of the s t r ing below the tangency point is :

Substituting in equa t ions (79) and (80) t h e v a l u e s of a l , b l , c l , a 2 , b 2 , and c 2 from T a b l e 2 and the numerical va lues of S1, T I , 4, SO, T2, and U2 from Fig. 28 through 33, e x p r e s s i o n s of the deflection coefficient h vs. x are obtained. h vs. x 2 - x h a s been plotted in Fig. 7 for x2 equal to 1.94 and 4.218, respect ively,* and in Fig. 8 for x2 equal t o

* T h e letter "h"has not been Indicated in Fig. 7 and 8 .

STUDY O F THE BUCKLING OF ROTARY DRILLING STRINGS 20'7

1.94, 3.753, and 4.218, ;espectively. These figures have been explained previously.

Deflections of the second buckle have been calculated with equation (79) and plotted in Fig. 9, previously explained. It has been found by trial and error that for x2 ~ 4 . 2 1 8 the deflection of the second buckle i s equal to the apparent radius of the hole.

Bending Moments

Let us define the bending moment coefficient i by the following expression:

By eliminating #y/dz? between equations (81) and (28), equation (6) has been obtained.

Substituting in equations (63) and (67) the values of a l , b , , c , , a 2 , b2, and c2 from Table 2, expressions of the deflection coefficient i vs. x are obtained; thus data plotted in Fig. 7 and 13 are found. These figures have been previously explained herein.

Critical Conditions of the Second Order

At the upper end of the string the deflection i s nil and, therefore, y = 0. Curves 1 and 3 of Fig. 7 show that, in the vicinity of the upper end, y i s positive for the first order of buckling and negative for the second order. Consequently, at the upper end dy/dx i s positive for the first order and negative for the second; and dy/dx is nil for the limiting condition, i.e., for the critical value of the second order. Similarly, d2y4dx2 i s nil a t the upper erid of the string and curves 1A and 3A of Fig. 7 show that, in the vicinity of the upper end, d2y/dx2 i s positive for the first and negative for the second order of buckling. Consequently, a t the upper end d3y/dz3 i s positive or negative for the first and second order, respectively, and d3y/dx3 is nil the critical condition of the second order. Sub- stituting dy/dx = d3y/dx3 = 0 into equation (27) and - .

replacing c with c , , we obtain c , = 0. It has been f o k d by trial and error that c; i s very close to zero for x2 = 3.753 (see Table 2) which i s the critical condition of the seconcl order.

Inclination of the Bit

Let us define the inclination a and the inclination coefficient t by the following expressions

i3y eliminating dl'/d,l and dy/dx between equations (82), (83), and (21), formula (9) has been obtained.

Substituting in equations (63) and (66) the values of a l , b l , e l , a 2 , b p , and c 2 from Table 2 , expressions for the inclination coefficient t vs. x are obtained. Values of these expressions for x = x 2 concern the inclination of the bit. They have been plotted vs. x2 on Fig. 16.

Length of Buckled Curves

The length L of a curve between two points of abscissas equal to ,Ii and .X2 i s equal to:

I n the case of drilling strings, I' i s small with regard to X; therefore, after expanding the square root in series, all the terms but the first two may be neglected. Thus we obtain:

A s S2- .XI i s the length of the projection of the drilling string on the axis of the hole, the increase A L of the length of this projection which i s due to buckling is equal to: .

Al

Substituting (19), (201, and (83) into equation (84), we obtain

Table 2

x2 a, (m/r) b, ( m h ) c1 ( m h ) a2(m/r) b2(m/r) c2(m/r) Remarks

1.940 t0.064 -0.406 t0.482 t0.064 -0.406 1-0.482 Critical condition, first order. 2.600 t0.260 -0.104 t0.313 -0.025 t0.495 t0.971 3.200 t0.308 t0.070 t0.165 -1.233 t1.849 t1.462 3.753 t0.278 t0.205 -0.002 -3.175 t2.853 t1.946 Critical condition, second order. 4.000 tO.224 t0.280 -0.124 -4.017 t3.007 t2.164 4.218 -0.009 t0.474 -0.512 -4.610 t2.905 +2.176 Second buckle contacts wall of hole.

208 ARTHUR LUBINSKI 1

Put: X 2

q = % J t 2 d x (86)

By el iminat ing the integral between equa t ions (85) and (86) formula (11) h a s been obtained.

It h a s been explained how t h a s been ca lcu la ted vs. x for var ious v a l u e s of x 2 . Simpson's method h a s been used i n the integration (86) and q plotted vs. x2 on Fig. 19.

Inclination of the Force on the Bit

Formula (68) g i v e s the horizontal component F2 of the react ion of the bottom of the hole on t h e string.

F2 = c2pm

Subst i tut ing (19) into (16), the vert ical component 5 i s found to be:

% = =Pm T h e rat io of these two components i s equal t o

the inclination f l of the force on the bit:

Put: m

C2 T - = n X2

Then: r ,6 = n m

which is formula (10) used previously. T h e v a l u e s of c2m/r have been previously cal-

culated for var ious va lues of x2 (Table 2). T h e n the incl inat ion of the force on the bit coefficient n i s calculated with equation (87) and plotted vs. x2 on Fig. 16.

Force Applied by the Buckled Drilling String on the Wall of the Hole

T h e reac t ion F of the wall of the hole on the buckled drilling s t r ing i s equal to:

Subst i tut ing equa t ions (64) and (68) into t h i s l a s t expression, we obtain:

F = p r ( c 2 ~ - c l ~ ) .

P ut: m m f = c ~ ~ - c ~ ~ (88)

Then: F = fpr

which i s formula (5) used previously. Values of c 2 m / r and ci m/r have been previously

ca lcu la ted for var ious v a l u e s of x2 (Table 2). T h e n the coefficient f i s calculated with equat ion (88) and plot ted vs . x2 in Fig. 11.

By inspect ing t h i s figure, we may s e e . that - fo r

weight on the bit smaller than t h e cr i t ical va lue (x* = 1.94) f i s negative which means tha t the p ipe should be pul led toward the wall i n order t o b e bent. F o r the cr i t ical va lue t h e pipe buckles , but i t c o n t a c t s the wal l with a force equa l to ze ro a t the point of contact.

Influence of Hydrostatic Pressure on Buckling Heferring again to Fig. 27, i t i s s e e n t h a t t h e

buoyancy i s equa l t o the vector sum of f o r c e s DE and E F . T h e force EF i s proportional to the hydro- s t a t i c pressure. A s E F i s perpendicular t o the shear ing force FG, the shearing force i s indepen- den t of EF but depends on DE. A s s t a t e d under "Analytical Formulation of the Problem," t h e force D E may be eliminated from our considerat ions by replacing t h e weight C D by the force CE s u c h tha t the rat io of C D and D E is equal t o the ra t io of d e n s i t i e s of s t e e l and mud.

Consequently, the shearing force and, therefore, a l l t h e buckling charac te r i s t i cs a re not dependent on the compression EF which r e s u l t s from the hydrostat ic pressure, but they do depend on t h e l o s s of weight DE of s t e e l i n the drilling fluid which is a l s o a r e s u l t of hydrostat ic pressure. Influence of Pump Pressure on Buckling

I'he effect of buoyancy on buckling of the dr i l l ing s t r ing w a s es tab l i shed with a n assumption of a hy- d ros ta t i c distribution of pressure in the hole. A s t h e mud i s actual ly circulating, a hydrodynanlic distribution of pressure should be considered. At a n y dep th the pressure i s larger inside the dr i l l p ipe than in the annulus. L e t u s inves t iga te the influence, if any, of the high inside pressure on the buckling of a drilling string.

Imagine a hydrostat ic sys tem b (Fig. 35) in which the dis t r ibut ion of pressure would be the same as in the hydrodynamic sys tem a. T h e sys tem a com- pr i ses the pump P, the drilling s t r ing D , t h e bi t nozz les N, and t h e annulus A. T h e equiva len t hy. d ros ta t i c sys tem b comprises a drilling s t r ing D not communicating on bottom with the annulus A. T h e hydrostat ic pressure i n c r e a s e s with depth. In the hydrodynamic system t h i s increase i s smal le r in- s i d e the drilling s t r ing and larger in the annulus - -

because of the l o s s of head due t o v i s c o u s or turbulent forces. Therefore, the dr i l l ing s t r ing of the equivalent hydrostatic sys tem should be filled with a lighter drilling fluid; and, inversely, the annulus should be filled with a heavier one. Let:

- AM designate the dens i ty of 1.11e drilling fluid.

p des igna te the drop of p ressure due t o v i s c o u s OF turbulent fo rces per uni t of length of t h e drilling string.

STUDY O F T H E BUCKLING OF ROTARY DRILLING STRINGS 209

pl des igna te the drop of p ressure due t o v i s c o u s or turbulent fo rces per unit of length of the annulus.

T h e d e n s i t i e s of fluids i n the equivalent hydro s t a t i c system are:

AM - p in the drilling string, and

AM t p1 in the annulus.*

Moreover, t h e pump pressure of the hydrodynamic system should be replaced by a n equivalent head in the hydrostat ic sys tem as shown on F ig . 35.

Consider the forces act ing on a portion of the drilling s t r ing between t h e bit and any c r o s s s e c - tion iClN (Fig. 36). T h e reaction of the bottom of the hole on the bit, the shearing force, a n d the compression (or tension) in the c ross sec t ion hlN are s imilar t o those previously considered, but the weights and buoyancies are different.

T h e force A i s the weight in air of t h e portion of the s t r ing below h1N. T h e force R$ i s the buoyancy act ing on t h e portion of the s t r ing below ,MN sup- posedly completely surrounded by the fluid, i.e., severed from the portion above h l h . T h e force K$ i s then equal to the weight of the fluid of apparent densi ty Ahf + pl , and of the volume equal t o the volume of the portion of the s t r ing located be104 ,!IN, assuming that the s t r ing i s not hollow.

W3 i s the force which must be added vectorial ly t o F1 in order to obtain the true buoyancy, because the fluidpressure ac tua l ly d o e s not a c t o n the c r o s s sec t ion ,l!N.

In other words, the sun1 of vectors and IV3 i s the resul tant force ac t ing on the portion of the s t r ing below ICIN by the fluid.of the annulus.

Similarly, the resul tant force ac t ing on t h e portion of the s t r ing below MN by the fluid of the drilling s t r ing i s t h e sum of vectors IT$ and &. K$ i s equal to the weight o f t h e inside fluid located below d lN and of apparent dens i ty equa l t o AM -P . W3 i s the pressure inside the drilling s t r ing a t illA' multiplied by the inside cross-sect ional area.

By reasoning s imilar t o that followed previously when analyzing the influence of hydrostat ic pres- s u r e on buckling, we may s a y t h a t t h e compression (or tension) in the s t r ing i s a function of W3 and I&. T h e internal pressure i n c r e a s e s the t ens ion (or d e c r e a s e s the compression) of the drilling string, but the shear ing force in ,l!N and, consequently, the buckling, a re independent of If$ and ri because

, t h e s e fo rces a re perpendicular t o t h e c r o s s sec t ion IMN.

A s far a s buckling i s concerned, we must con- *These expressions and formula (90) following concern a sys-

tem of cons~s tent u n ~ t s , such as pounds per c u b ~ c inch for d e n s l t ~ e s and pounds per square inch per Inch for drops of pressure p a u n ~ t length.

W Fig. 36

Intluence of Dr~lling-string Pressure Drop on Buckling

2 10 ARTHUR

s ider only the fo rces W, R', and W2. L e t u s con- s ider U$ a s being equa l to the following sum:

W1 = W I 1 + Wlll

wherein: fi' i s the weight of the fluid of apparent dens i ty AM + p, and of the volume equal t o the volumeof the s t e e l below ;)IN; and &'I i s the weight of the s a m e fluid and of a volunle equal t o t h e internal volume of the portion of the s t r ing below MN.

T h e resul tant of ver t ical weights and buoyancies is then e q u a l to:

W - W l t W , = (W-W,')- (W,"-W,)

T h e expression in the first pa ren thes i s i s equal t o the weight of s t e e l in fluid of apparent dens i ty AM t pl . T h e express ion in the second ~ a r e n t h e s i s i s equal t o t h e difference of weights of the inside volume Llled with fluids of apparent d e n s i t i e s equa l t o AM t p1 and AM - p respect ively.

T h i s expression may be transformed a s follows:

8 ' -W, + W, = VsAsk (89)

wherein: K and A, a r e the volume and dens i ty of s tee l . a n d

whe'rein: D and d are the outs ide diameter and in- s i d e diameter of the drilling s t r ing, respect ively.

T h e meaning of the l a s t two express ions i s that in t h i s invest igat ion we should have considered t h e va lues of weight per foot of s t r ings multiplied by k; and, consequently, t h e s e va lues would have been somewhat smaller .

T h e va lues of drop of pressure p a n d p1 per unit of length of pipe and annulus are a ~ a i l a b l e . ~ T h e v a l u e s of k were then calculated for a few t y p e s of drill p ipe and drill co l la r s and for var ious r a t e s of mud flow, and plotted on Fig. 21 and 22. T h e s e figures were ana lyzed previously herein.

ACKNOWLEDGMENT

T h e author w i s h e s to thank t h e following: E. E. Huebotter, of Baroid S a l e s Divis ion, Na-

tional Lead Company, for having initiated th i s s tudy by h i s quest ions.

J. T. Hayward, P. A. Wolff, V. M. Williams, and T. I\!. P o t e e t of Barnsdal l Oil Company, and J. J. Arps and T. Gilmartin for helpful cr i t ic ism and advice.

Henry Schaefer and J. A. Muckleroy, of Stanolind O i l and G a s Company, for help and advice in re- v i s ing th i s paper following the API meeting a t which i t w a s presented.

L. B. Wilder, of Stanolind Oil and G a s Company,

for help in the numerous computations. Schlumberger Well Survey Corporation for useful

help. P h i l l i p s Petroleum Company, Carter O i l Company,

Magnolia Petroleum Company, and Anderson- Prichard Oil Corporation for furnishing l o g s a n d information.

BIBLIOGRAPHY

Mayborn, T. W: P ro longng Drill-pipe Life, D r r 11 r n g , 6 [dl, February (1945).

Texter, H. G: Corrosion Fatigue of-Drill P ipe , D r I 11 r n g , 5 [121, October (1944).

Crenshaw, Wm. H: Drill-pipe Failures, Dr r 11 r n g , 8 [5] March (1947).

Crenshaw, W. H; Bottoms, V. B; Wallace, C. N; and O'Dell, C. R: Drill-pipe Failures, Inspection, and Protection in the Permian Basin, Dr r 1 1 r n g a n d P r o - d u c t c o n P r a c t r c e , 247 (1948).

Thompson, A. W. and Texter, H. G: Field T e s t s on He- jected Drill P ipe , Dr r l l rng a n d P r o d u c t r o n p r a c t a c e , 8 7 (1948).

Graser, F . A: T h e Fundamental Mechanics of Directional Drilling, D r r 1 1 r n g a n d P r o d u c t r o n p r a c t r c e , '71 (1949).

Wilson, Gilbert hl: Locating Crit ical Point of Stuck P ipe or Casing, T h e 0 1 1 W e e k l y , 126 [I], June 2 (1947).

Lloyd, W. S: Surface Preparation and Drill-pipe Fat igue Failures, T h e o r 1 W e e k l y , 124 [2], Dec. 9 (1946).

Maln, Walter C: Detection of Incipient Drlll-pipe Fail- ures, D r r 1 1 r n g a n d P r o d u c t t o n P r a c t l e e , 89-102 ( 1949).

RlcMaster, Robert C: Prevention of Drill-string Fai lures in the Permian Basin, W o r l d obi (published in 2 parts) Part 1, 127 [131, April (1948); Part 2, 128 [I], May (1948).

Cavanagh, R. L: Non-destructive Tes t ing of Drill P ipe , T h e O r 1 W e e k l y , 125 [2], March 10 (1947).

Martrn, Phil ip W: Magnetic Device Measures S t r e s s on P ipe in Wells, T h e P e t r o l e u m E n g r n e e r , 21 [4], April (1949).

Main, Walter C: Controlled Vertlcal Drilling, W o r l d 0 1 1 , 128 [13], April (1949).

REFERENCES 1 Guarin, P. L; Arnold, H. E; Harpst, W. E; and Davis,

E . E: Rotary Percuss ion Drilling, Dr r 11 r n g a n d P r a - d u c t r o n P r a c t ~ c e , 112 (1949).

2 Grant, R. S. and Texter, H. G: C a u s e s and Prevention of Drill-pipe and Tool-joint Troubles, Dr r 11 r n g a n d P r o - d u c t r o n p r a c t r c e , 9 (1941); revised by H. G. Texter and S. C. Moore, W o r l d O r 1 (published in 5 arts) Part 1, 128 [6], 79, Oct. (1948); Par t 2, 128 [ I T , 124, Nov. (1948); Part 3, 128 [8], 100, Dec. (1948); Pa r t 4, 128 191, 90, 92, 96, 100, 102, 104, Jan. (1949); Part 5 , 128 [lo], 96, 97, 100-102, 104, Feb. (1949).

3 Wayward, John T : Methods of Determining How Much of a Frozen or Cemented Colunln of P ipe i s Free, Dr I 1 1 - r n g a n d P r o d u c t r o n P r a c t r c e , 16 (1935). 4 Hawkins, Murray F . and Lamont, Norman: The Analy-

s ~ s of Axlal S t r e s se s in Drill Stems, Dr I 1 1 r n g and P r o - d u c t r o n p r a c t r c e , 358 (1949).

Handleman, G. H : Buckling under Lpcally Hydrostatic P re s s l~ re , J. App. M e c h a n ~ c s , 13 [3], Sept. (1946).

STUDY O F T H E BUCKLING (

6 Nolley, J. P; Cannon, George E; and Ragland, Douglas:

T h e Rela t ion of Nozz le Fluid Velocity t o R a t e of p e n e - t rat lon with Drag-type Rotary Bi t s , D r r l l rng a n d P r o - d u c t t o n P r a c t r c e , 23 (1948).

' ~ i m o s h e n k o , ~ : T h e o r y o f ElastrcStabrIrtY,McGraw- Hill Book Co., Inc., New York, 1936.

8 Karman, T. V. and Blot , M. A: M a t h e n a t r c a l M e t h o d s rn E n g r n e e r r n g , McGraw-Hill Book Co., Inc., New York, 1940. 9

Bowman, F: I n t r o d u c t r o n t o B e s s e l F u n c t ~ o n s . Longman, Green & Co., New York, 10 Gray, Andrew and Mathews, G. B: A T r e a t r s e o n B e s -

sel F u n c t z o n s a n d T h e r r A p p l r c a t r o n s t o P h y s r c s ,

McMillan S; Co., Ltd., London, 1925. 11

T a b l e s o f R e s s e l F u n c t r o n s o f F r a c t r o n a l O r d e r s . prepared by t h e Computat ion Laboratory of the National Applied Mathematics Labora tor ies , National Bureau of Standards, Columbia U n i v e r s ~ t y P r e s s , New York, 1948, VoI. I; 1949, VoI. 1 1 .

DISCUSSION

E. E. Huebotter (Baroid S a l e s Divis ion, National Lead Company, Tulsa) : T h e paper by Rlr. Lubinski will undoubtedly s tand a s a milestone in a study which h a s been in progress for the p a s t 5 years , viz., invest igat ing the act ion of tha t portion of the rotary drilling s t r ing in compression while drilling. A recent invest igat ion undertaken in an ac t ive oi l field revealed that, a t t imes, a s much a s 30,000 Ib of the s h a l e formations penetrated were suspended in the drilling fluid e a c h day while drilling. Con- s ider ing the fac t that the bit in t h e s e i n s t a n c e s w a s not drilling in the s h a l e formations under consideration, the rapid ra te of suspension, for the m'ost part, can be reasonably attributed to the ac t ion of the compressed portion of the dr i l l ing s t r ing a g a i n s t the exposed s h a l e formations.

T h e mechanism by which the s h a l e formations penetrated are ground into suspens ion in the drilling fluids being used h a s been considered only by a simple approach in the pas t , with few refinements until now. Mr. Lubinsk i i s t o be congratulated for developing a most thorough approach to the theory underlying the buckling of drilling s t r ings in compression; thereby possibly more prec i se ly account- ing for their grinding action upon bore holes , and sugges t ing measures to minimize th i s action.

T h e high maintenance c o s t s of many of the present-day drilling fluids are a direct resu l t of the da i ly grinding of large amounts of sha le , e tc . from penetrated formations into suspens ion in the drill- fluid by drilling s t r ings rotated in compression. T h i s c a u s e s increased drilling-fluid weight, vis- cos i ty , ge l strength, and, frequently, filter l o s s e s beyond permissible limits. Such suspended s o l i d s must be diluted with water, quan t i t i es of which on

occas ion reach va lues exceeding 400 bbl in a s in- g le day. Average dai ly r a t e s of water addition of 100 t o 250 bbl t o maintain the suspended s o l i d s in a drilling fluid a t a workable value i s not a t a l l un- usual. E a c h barrel of water s o added t o t h e drilling- fluid sys tem ~ r o d u c e s a barrel or more of new drill- fluid which must be t reated t o maintain t h e proper- t i e s desired, thereby obviously gearing drilling- fluid maintenance c o s t s to dai ly make-up water ad- ditions. Inasmuch 9s the introduction of a certain quantity of s h a l e s o l i d s into the drilling-fluid s y s - tem requires a specific volume of water dilution to maintain a prescribed drilling-fluid weight, greater bit penetration per barrel of water added, reflected in lower mud-maintenance c o s t s per foot of drilled hole, can be real ized if the grinding act ion of the drilling s t r ing can be minimized.

,J. L. Holmquist (Spang-Chalfant Div. of the Na- t ional Supply Company, Ambridge, Pa.) (written)*: T h e paper by Arthur Lubinski i s a valuable and welcome addition t o the technical l i terature on drilling s t r ings.

A phase of the problem in which we are espec ia l - l y interested i s the s t r e s s in the drill p ipe when a considerable length of drill pipe i s in compression because of weight on the bit and high-order buckling occurs in consequence. An extension of the r e s u l t s t o higher buckling orders would be of considerable interest .

With regard t o the reverse bending s t r e s s which drill pipe can withstand without failure by fatigue, in the p resence of a corrosive environment such a s water-base drilling mud, t h i s s t r e s s can be of a surpris ingly low level . From various d a t a which have come to our attention, we judge that t h e maximum permissible s t r e s s i s likely t o l i e in the range of 10,000 t o 15,000 psi.

Mr. Lubinsk i h a s t reated the subject in a n ad- mirably rigorous and competent manner. We expect t o have many o c c a s i o n s t o refer t o t h i s paper i n the future.

Howard G. T e x t e r (Spang-Chalfant Div. of T h e National Supply Compan y, T u l s a ) (written): I am quite impressed with Rlr. Lubinski 's ca lcu la t ions of the cr i t ical l eng ths of s t a n d s of pipe or co l la r s ver t ical ly s tacked in a derrick. H i s figure of 127 ft for 4%-in. 16.60-lb drill pipe i s well borne out by s tudy of complaints which we sometimes receive when t reb les of Range 3, 4%-in. pipe are s tacked; and when a s ing le s tand g e t s away from the derrick man and s e t t l e s i tself into a buckle which perma-

'Presented by H o w a r d G. Texter, Spang-Chalfant Dlv. of T h e Katlonal Supply C o m p a n y , Tulsa.

212 ARTHUR LUBINSKI

nently bends one of the lengths. In t h e s e c a s e s the cr i t ical length h a s been exceeded by just a few feet inasmuch a s 3 l eng ths of Range 3 pipe, p l u s the tool joints, i s in t h e nature o f 1 3 0 ft. T h e com- plaint i s , of course, that the pipe i s "soft"; but

. the truth of t h e matter i s that the cr i t ical length h a s been exceeded because the derrick man failed to p lace the s tand s o that i t r es ted aga ins t the in- termedi a te supporting point.

I should l ike to emphasize R'lr. Lubinski 's conclusions; viz., "If the weight on the bit i s no larger than the weight of drill col lars , the order of buckling i s low; and, therefore, even in caved holes the s t r e s s e s in s t e e l are low. If t h e same weight i s carried with too few drill col lars , the order of buckling becomes high; and the s tee l niay be subjected to reversing s t r e s s e s o f 20,000 ps i or more, even in c a v e s of smaller diameters. T h i s i s t h e reason for the occurrence of fat igue fa i lu res when the number of drill co l la r s i s too snlall."

I would point out that th i s i s par t icular ly t rue where corrosion fat igue i s involved, a s in dr i l l ing the "sal t string9' ho les of West Texas . In corrosive media there can be no true endurance-limit s t r e s s figure and i t i s my opinion, from observation of field t roubles , that fatigue fai lures can and do occur when the s t r e s s e s from buckling are kept a s low a s 10,000 t o 15,000 psi , a s mentioned by Holmquist, or poss ib ly even lower. T h i s partly explains , then, the occurrence of corrosion-fatigue fai lures of dr i l l pipe even where there s e e m s to be an adequate number of drill col lars . I s a y "partly explains" because I s u s p e c t that bending s t r e s s e s from c a u s e s other than buckling son~et in les may be involved; e.g., whipping of the drill pipe from certain cr i t ical s p e e d s of rotation.

A s the author s t a t e s , i t i s qui te surpris ing that the inclination of the force on the bit i s smaller for buckling of the second order a s compared to buck- l ing of the first order. I, for one, would never have dreamed of such a thing. P o s s i b l y th i s exp la ins the more s u c c e s s f u l operation of some contractors whose p r a c t i c e s have not seemed in accord with the accept- ed relation of weight on the bit to s t ra igh tness of hole.

I am p leased t o note Rlr. Lubinski 's conclusion tha t hydrostat ic p ressure of the mud fluid in a well h a s no effect on buckling and fatigue. With that I am heart i ly in accord.

I do not s e e how anyone could poss ib ly take except ion to the very clear ly and succ inc t ly worded conclusions. T o me they seem almost self-evident af ter following rhrough the reasoning i n the body of t h e paper. I am very glad t o have such a complete a n a l y s i s of the fac tors involved in the buckling of rotary drilling s t r ings. I t will help me imnieasurably

in my never-ending s tudy of ac tua l drill-pipe and drill-collar failures.

W. S. Crake (Shell Oil Company, Houston) (written): At the ou tse t , the d i s c u s s e r wishes t o con- gratulate Rlr. Lubinski on t h i s paper, which deve lops the theory of the problem and a l s o s u g g e s t s some prac t ica l app l ica t ions of h i s work to rotary drilling.

Although those who have had to wrest le with the pract ical problem in the p a s t have Known tha t buckling took place in about t h e s h a p e s proved by the author, th i s is the first time t o the writer 's knowledge that a c lear picture h a s been presented; and, naturally, about a s many ques t ions a r i s e a s answers a re given.

In the first place, inasmuch a s good dri l l ing prac- t i c e requires , and t h i s paper p roves clear ly, tha t the neutral point must be contained within the drill- col lar s t r ing t o avoid s e r i o u s trouble, the d i s c u s s e r p roposes to neg lec t d i scuss ion on drill p ipe e x c e p t a s follows:

While the paper t r e a t s both the drill-pipe and drill-collar s e c t i o n s as uniform wall tubes,. what i s the effect of, first, interposing a tool joint of heavy sec t ion modulus (over 1 0 0 percent joint efficiency) every 3 0 ft on the dr i l l pipe? Second, what i s the effect on drill col lars , with a joint efficiency of about 65 percent and a lighter equivalent s e c t i o n nlodulus every 3 0 ft?

I t appears that a weak s e c t i o n a t drill-collar joints would tend to hold maximum buckling deflection a t the joint over quite a wide range of loading, and t h i s may be a c a u s e of premature fat igue of jo in t s if drilling i s carr ied on for long under t h e s e conditions. Therefore, i t appears to be des i rab le to determine loads which would c a u s e maximum deflection by second or third order buckling to fal l between drill-collar joints. I t i s rea l ized 'that other vibrational s t r e s s e s in the s t r ing, both vert ical and horizontal, would c a u s e ver t i ca l shif t ing of the point of maximum deflection. T h i s may be helped by s r e a d y feed ra tes , automatic bit-weight control, and s teady rotational speeds .

While t h e s t r e s s e s shown on Fig;. 14 for 6!4-in. drill c o l l a r s seem low, the in~por tan t quest ion i s 6 6 what are the pin s t r e s s e s on connect ions when first, second , third, and other order deflections are centered a t the made-up joint?" I t i s fe l t tha t some research on t h i s quest ion would follow, now that maximum deflection and use i s es tab l i shed by this paper for the various orders of buckling.

With regard to crooked hole, a number of inter- e s t i n g thoughts are provoked.

F i r s t , i t should be simple t o locate drill-collar supports , o r ' s t ab i l i ze rs , c l o s e enough t o the bit s o


tha t the collar between bi t and s tab i l i ze rs becomes a short column and almost unaffected in buckling by the l o a d s imposed. T h i s would follow surface ma- chine drilling which u s e s a s tab i l i ze r immediately above the bit, and often another a short d i s tance behind if the drill bar i s long and slender.

Second, as d i s c u s s e d in the paper, if the vert ical deflection plane of the drill collar a t the first buckle from the bottom ro ta tes with the pipe, t h e bi t trying t o go crooked a t a l l 360 deg of rotation i s hardly more likely to cause"crooked hole" than with light weight. If, however, the deflection plane s t a y s a t one point on the c i rc le and d o e s not rotate , the bi t i s perfect ly whipstocked usual ly in the natural direction the hole would tend t o deviate. T h i s follows the known method of making a hole s idetrack, by using a "limber joint" in~media te ly above the bit.

Pecu l ia r ly enough, truly aligned and perfect ly balanced col lars having a uniform r e s i s t a n c e to bending around the pipe a x i s would probably tend to c a u s e the deflection plane to s tand s t i l l and cause crooked hole; whereas out-of-balance or mis- aligned col lars would tend t o have the plane rotate because the natural "kink" of poor c o l l a r s a t the weakes t sec t ion would hold the posi t ion of the kink and make i t s plane rotate.

One final point i s tha t the invest igat ion h a s very significant bearing on t h e act ion of c a s i n g when "floating" into the hole, a l s o drill-stem t e s t i n g and tubing operat ions when running empty pipe. I t certainly exp la ins inabi l i ty t o ge t long, close-clear- a n c e too ls into pipe, the bottom end o f w h i c h i s pushed up in column loading. T h i s effect, t o a major or minor extent , may be s e r i o u s when se t t ing c a s i n g and bleeding off cas ing pressure a t the sur face before cement s e t h a s t aken place. T h i s act ion can sause se r ious kinks in c a s i n g s t r ings in washed-out s e c t i o n s if long columns of cement a re al lowed to 6 6 push up" on the float s h o e s and co l la r s without a n equa l internal pressure load holding the p ipe out of compression from bottom.

Xlr. Lubinski: T h e author a g r e e s with hlr. Holm- quist that the extension of the t o higher buckling orders would be of considerable interest . It should, however, be well understood that t h i s invest igat ion i s a s t a t i c s tudy of buckling in One

plane. T h e presence of s t r e s s e s due to torque, when the pipe i s rotated, probably brings lit t le change to the swing sub jec ted t o the first order buckling. On t h e other hand, i t i s very probable t h a t th i s change increases with the number of buck- l e s and that , for high order of buckling, the buckled curve resembles more a hel ix than a plane curve.

Consequently, another approach should be used to invest igate high orders of buckling of drilling strings. Probably some experimental approach to the problem would be the best . Theoret ical con- s iderat ions would be used only for extension of r e s u l t s from the model to the ac tua l s t r ings or from one s i z e of pipe to another.

T h e kformation given both by Mr. Texte r and Rlr. Holmquist, viz., that in corrosive media the reversing s t r e s s e s should be kept a s low a s 10,000 to 15,000 ps i , i s very valuable. Some o ther d a t a have s i n c e been published which confirm t h i s opinion.*

Mr. Crake r a i s e d the question of the influence of connect ions between joints of dr i l l pipe or drill co l la r s on t h e s t r e s s e s . T h e length of one connection i s shor t as compared t o a 30-ft joint of pipe or a drill collar. Consequently, the influence of a connect ion on the p ipe or on the drill col lar i s negligible a t some d i s tance frorn the connection. On the other hand, the s t r e s s e s in the immediate vicini ty of the connection or in the connection i tself are not known; these s t r e s s e s perhaps might be determined by p hoto-elastic s tudy of models.

I t would cer tainly be useful to drill with a weight on the bit for which large v a l u e s of bending moments occur somewhere in the middle portion of a drill col lar , and not near a connection between two drill collars. T h i s might be poss ib le only i f the portion of the s t r ing a t which the bendingmome!t i s large is much shorter than 3 0 ft , which, unfortunately, i s not the case . Inspect ion of curves 1A and 3A of Fig. 7 shows t h a t the lengths for which the bending moment i s larger than 7 5 percent of i t s maximum value a re a s follows:

Buckling of the first order (curves 1 and IA) : 1.2 diniensionless uni ts , or about 70 ft for 6!d-in. drill collars. Buckling of the second order (curves 3 and 3A): 1 dimensionless unit, or about 58 f t for 6!C-in. drill collars.

Consequently, the bending moment is necessar i ly tquite large in a t l e a s t two connect ions for buckling of the first and second order.

h t . Crake s u g g e s t s use of s tab i l i ze rs c l o s e enough to the bit s o that the col lar between the bit and s t a b i l i z e r s becomes a shor t column and almost unaffected in buckling by the loads imposed. In fact, this col lar would be pract ical ly s t raight , but the la teral forces on the s tab i l i ze r and on t h e bit would not be nil and, consequently, a s t raight hole would not be drilled. T h e author thinks tha t Rlr. Crake's suggestion of using s tab i l i ze rs should be

*performance equlrernents ot DTIII p lpe , D . A. Evans, Dr I l i lng, 11 [a, 23-25, 109, M ~ ~ C I I (1950).


modified. The stabil izers should be used not close to the bit, but a t the points of maximum deflection of the buckled drilling string. Thus, after stabi-

the stabilizer or on the bit. The author agrees with Mr. Crake that the diffi-

culty which i s sometimes encountered in floating an lizing, there would be no lateral forces either on empty casing into the hole i s due to buckling.

api-50-178 a study of the buckling of rotary drilling strings

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