SPE 15467

Basic Concepts in Static BHA Anaby U. Chandra, comxdtantSPE Member,formerlywith NLIndustriesInc.

ysis for Directional Drilling

Copyright 19S6, Society of Petroleum Engineers

This paper was prepared for presentation at the 61s1 Annual Technical Conference and Exhibition of the Society of Petroleum Enginaers held in NewOrleans, LA October 5-8, 1986.

This paper was setected for presentation by an SPE Program Committee following review of information contained in an abstract submit!ed by theauthor(s). Conlonts of the paper, aspresenled, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by theauthor(s). The material, as presented, does nol necessarily rellect any position of Ihe Society of Pelroleum Engineers, its officers, or members. Paperspresented at SPE meetings are subject to publication review by Editorial Committees of the Sociely of Petroleum Engineers. Permission 10 copy isreatricled to an abstract of not more than 300 words. Illustrations may not be copied, The abstract should contain conspicuous acknowledgment ofwhere and by whom the paper is presented. Write Publications Manager, SPE, PO. Box 833836, Richardson, TX 75083-3636. Telex, 720989 SPEDAL.

ABSTRACT Herein, a BHA (bottomhole assembly) is treated as abeam column subjected to its self-weight, buoyancy, and

Basic concepts related to the static analysis of two- weight-on-bit (WOB). For simplicity, a one stabilizer BHAand three-dimensional bottomhole assemblies are dis- in a straight but inclined borehole is discussed first. Then,cussed. Beginning with a straight one stabilizer assembly, further considerations of multiple stabilizers, boreholethe effects of multiple stabilizers, borehole curvature, curvature, torque, wall contact and of variations in collartorque, wall contact, etc. are introduced in steps. The cri- cross-section or material properties are introduced interia for defining the build, hold, drop and walk trends is steps. Both two- and three-dimensional BHAs are dis-clarified. Two methods for computing bending stiffness cussed. The difficulties experienced in the application ofand equivalent outer diameter of MWD collars with non- closed from methods, and the advantages of their numer-uniform cross-sectional properties are proposed, Also, ical counterparts (finite difference and finite element) insimple methods of estimating buckling loads for one and analyzing complex modern BHAs are highlighted. Then,two span assemblies are presented. using the equilibrium of forces at the drill bit and the for-

INTRODUCTIONmation, the criteria for defining the build, drop, hold andwalk trends is explained.

In the early days of drilling, the holes were shallow Since bending is the dominant mode of deformationand were supposedly drilled straight. The experience in a drillstring, the knowledge of its resistance to bending,with the Seminole fields in Oklahoma during the late i.e. bending stiffness, is important. This paper explainstwenties, made the industry realize that drilling does not the basic concepts of stiffness and presents two methodsnecessarily follow the intended trajector yl. Something for computing bending stiffness and equivalent diameterhappens downhole which makes the drillstring deviate of collars with non-uniform cross-section (e.g. MWD col-from its course, The efforts to understand the cause for Iars). Finally, simple methods for estimating critical buck-deviation of drillstring led to their mechanical analysis ling loads of bottom hole assemblies are discussed.using the concepts of structural mechanics. Many papershave been written on the subject29. However, the basic The scope of the paper is limited to the static analysisconcepts of mechanics, as related to directional drilling, of the drillstring. It does not discuss the effect ofhave not been put together in a systematic manner. This dynamics or of rock and bit characteri~lcs on the direc-paper attempts to fill such gap, tional tendencies of BHAs. Also, it emphasizes com-

bining the physical and mathematical concepts in aThe subject matter in this paper is presented on the simple fashion. No attempt is made to provide mathe-

Iines of standard textbooks on structural mechanics10-13. matical derivations; instead, the applicable expressionsare borrowed from easily accessible sources.

References and illustrations at end of paper.

THE TWO=DIMENSIONAL PROBLEM Equation (1) is a fourth order differential equation,and its general solution is,

When the BHA of interest and all loads acting on itare contained within one plane, we can refer to it as atwo-dimensional problem. For example, consider thestraight bottomhole assembly of Figure 1, which is =AsinEx+BcOsEx+ cxD+sinclined to the vertical at an angle 6. It consists of thedrill bit at A, the drill collar AB, a stabilizer at B, etc. In . . . . . . . . . . . . (2)engineering mechanics terminology, the drill bit and thestabilizers are termed as supports, and the length of drill where the coefficients A, B, C ancl D are obtained by

collar between two supports is termed as a span. applying appropriate boundary conditions. For the beamcolumn of Figure 2 with pinned supports, the boundary

For brevity, we first ccmsider only the span AB adja- conditions are,cent to the drillbit, The forces acting on this portion of the d2vBHA are weight-on-bit I? and the self-weight w (adjusted

=O, atx=Oandx= L. . ...(3)for buoyancy), as shown in Figure 1(a). The self-weight w v dx2can be decomposed into two components, the normalcomponent q = w sin8, and the axial component p =

12and the coet%cients are ,

w COS6. The normal component causes sagging or

[

qEI 1- COS j~- Lbending of the collar, i.e. it corresponds, to the beam A=action, The axial component corresponds to the column p2 sin ~~1 L 1 (4)action, and can cause buckling of the collar on reaching a

. . . . . . . . . . . . .

certain critical value. The behavior of the collar in thepresence of combined normal and axial loads is referred B=-D=~; c. LP2 2Pto as the beam column. We note that if the BHA is ver-tical, i.e. if 0 = 0, the normal component of the self- By substituting (4) in (2), one can find the displace-weight is zero, and the problem reduces to that of a ments along the drillstring. Then, using additional mathe-column. matics the reaction (side force) R at the bit, and the bit

rotation a can be computed. As explained in a subse-ln Figure 2, we consider the case of a Ctinpie beam quent section, the knowledge of R and/or x is needed in

column which represents the span AB of Figure 1. As a predicting the deviation tendency of a BHA.starting point, we assume that (1) the two ends do notcarry any bending moment, (2) the drill collar above the In some cases, it maybe justified to treat end B as astabilizer B is not effective, (3) there is no contact between fixed support; for example, in the case of a very long sta-the drill collar and the borehole wall, (4) the collar is bilizer or a packed assembly. Then, the expressions forstraight prior to the application of the loads (i.e. no initial coefficients A through D will change, resulting in differentcurvature), (5) the cross-sectional and material properties values of drillstring deflections, bit tilt and side force asof the collar are uniform, and (6) the axial component of compared to the pinned support condition. For a specialthe self-weight is negligible. We will discuss the implica- case of zero WOB, the results for assumed pinned andtions of relaxing each one of these assumptions later in fixed end conditions at B are compared in Figure 3.this section.

It is obvious that, with this closed-form approach,The normal deflection, v, of the beam column the analysis of even the most basic collar will require a

shown in Ft ure 2 is given by the following differentialP considerable amount of computation, A drilling engineer

equational1 , cannot conceivably perform such analysis in a routine

d4v d2v qfashion. His job will be even more difficult when working

EI +P= with real life BHAs where some (or all) of the six assump-dx4 dx2

l(1) tions mentioned earlier are invalid.

where, x is the distance alon~ the axis, P is the axial load, In the following para{ * -!e examine the impli-q is the norm> Distributed load, E is the modulus of elas- cations of relaxing each c assumptions one byticity of the material, and I is the moment of inertia of the one. Due to space limitat~. ussion is kept brief.collar section. The product EI is called the flexural rigidityof the collar cross-section. .

l In the absence of the axial ioad F! Equation (1) reduces tosimple beam bending equation.

SPE 15467 UmeshChandra 3

End Moments Contact Between the Drillstring and BoreholeWall

The bending moment at the bit is generally assumedto be zero in order to simplify the analytical effort. In To understand the problem of contact between thereality, the resistance offered by the rock, the mechanical drillstring and the borehole wall, let us consider the onedesign of the bit, and its cutting characteristics, all com- span BHA of Figure 5. If the available clearance betweenbine to generate a non-zero bending moment at the bit. the collar and the borehole wall (CLS) is more than theSuch bending moment will tend to limit the rotation at the maximum deflection 6, no contact occurs, Case (a), Ifbit. Although this moment can be explicitly accounted for CLS and 6 are equal, the two surfaces barely touch eachby adding a rotational spring of known stiffness, it is other resulting in a point contact with zero contact force,deemed advanta eous to combine it with the rock and bit

?3,14Case (b). However, if CLS is less than 6, the area of con-

anisotropy effects . tact as well as the contact force will have finite values,Case (c). The contact force (pressure) will not necessarily

In previous discussion, the bending moment at end be uniform over the surface. Also, the bit tilt and the sideB was assumed zero only for illustration purposes. The force will no longer be the same as in Case (a).presence of additional drill collar beyond stabilizer B (as inany multi-stabilizer BHA), the design of the stabilizer and Referring to Figure 5(d), Equation (1) can bethe flexibility of the rock will result in a non-zero moment rewritten as follows:at end B. Of these three, the first factor is addressed inthe following paragraphs and elsewhere in the paper, d4v d2v qwhereas, the other two factors are not discussed further. EI+P=c for AC, DB. . ( 5a)

dx4 dx2

Multi-Stabilizer Assemblies d4v + p d2vEI = q.q(x) force ... (5b)

In reality, a BHA consists of many stabilizers dividing dx4 dx2it into several spans, and more than one span must beconsidered in the analysis to accurately determine the The solution of the contact problem requires that firstdirectional tendency of the BHA. h-i mechanics termi- the deflection at each point of collar be computed withoutnology, such a BHA can be regarded as a beam which is regard to the borehole wall and compared against thecontinuous over several supports, i.e. a continuous available clearance. It further requires finding the locationbeam. The analysis of a multi-span BHA requires the and size of the contact region, and the magnitude andapplication of Equation (2) to each span, along with the distribution of the contact force. Such computation isuse of continuity requirements at intermediate supports tedious, and is best handled with the help of a computer(stabilizers) and boundary conditions at the end supports. using finite difference or finite element techniques57 15.The continuity conditions at the intermediate supportsare, (i) the deflection is zero, (ii) the slopes at the two Initial Curvaturesides of the stabilizer are equal, and (iii) the moments atthe two sides are equal and opposite. Obviously, the Equation (1) is meant for a beam column or a collarmanual computations will become very tedious for a whose axis is straight prior to the application of anymulti-stabilizer assembly, and the use of a computer-based technique will be profitable.

external loads. l [n order to study the effect of initial cur-vature, we consider the one span BHA of Figure 6. The

In Figure 4, we demonstrate the effect of stabilizerinitial shape of the collar can be represented by,

placement on the directional characteristic of a multi- 7rxvo= a kin . . . . . . . . . . . . . . . . . . . . . . . . . (6)stabilizer BHA. In the three cases shown, the total length, Lcollar size and the inclination from the vertical are the Then, it can be shown that the initial curvaj)re issame, i.e. tkte normal load q is the same. For each case, equivalent to an additional lateral load q given by ,the deflected shape of the drillstring and the side force atthe bit are shown in the figure. The importance of the lr2

cl ()=Pa~ sin~.............placement of first stabilizer above the bit is obvious. As we (7)L

will discuss later, the first assembly is dropping angle, thesecond holding, and the third is building the angle. In Figure 7(a), we consider the first span of a drop-

ping assembly in a borehole with positive curvature (i.e.its inclination from the vertical increases as we move

For the purpose of this discussion, the self-weight is treatedas an external load.

4 Basic Concepts in Static BHA Analysis for Directional Drilling SPE 1546

down the hole towards the bit). According to Equation(7), the effect of this curvature is the same as that of a lat-eral load q on a straight collar. This lateral load q addsup to the lateral component of the self-weight q, resultingin an increased dropping tendency of the assembly. How-ever, if the same assembly is used in a borehole of nega-tive curvature as shown in Figure 7(b), q would opposeq and the BHA will have a reduced dropping tendency.This concept can explain the behavior of the building,holding or dro ping assemblies in curved boreholes1?reported earlier .

CrossSectional and Material Properties

Consider a simple, one stabilizer BHA shown inFigure 8. It consists of two collars of different sizes and/ormaterials, and thus of different Et values. The analysis ofsuch a BHA would require writing Equation (2) for por-tions AB and BC separately, and using additional conti-nuity conditions of deflection and slope at point B.Obviously, for complex realistic BHAs where the EI ofcross-section changes very often, manual computation isnot practical and, again, the use of computerized tech-niques is warranted.

Distributed Axial Load

Equation (1) did not account for the axial compo-nent of self-weight of the collar shown as p = w COSOinFigure 1. The attempt to handle this term in a closed formadds to the computational complexity. However, thefinite element or f$ite difference method can easily dealwith this situation .

The application of the foregoing concepts to drill-string analysis requires some special considerations. Forexample, it is found convenient to (i) establish a straightreference axis passing through the bit to measure thenormal deflection v, (ii) replace the independent variablex in Equation (1) by a variable s measured along thecurved borehole axis, (iii) cut off the BHA of interest at acertain distance above the bit, and (iv) assume the WOBdirection to be coincident with either the borehole axis orthe deformed drillstring axis. The first three of these con-siderations are discussed in References 5, and 16,whereas the last one is explained in the section entitledCriteria for Defining Deviation Trends.

Often, a drilling engineer is interested in only com-paring the deviational tendencies of two or more BHAswithout a specific concern or prior knowledge of theborehole curvature or the WOB. It is then possible toignore these two factors and to analyze the BHAs ofinterest as straight continuous beams. In such cases, sev-eral standard techniques of structural analysis can beused to compute the tilt and the side force at the bit. Also,

the possibility of contact between the drill collar and theborehole wall can be examined by computing the max-imum deflection. Some of the popular techniques suit-able for hand computations are moment area orconjugate beam method for single span beams, andmoment distribution method for multi-span beams ormultiple stabilizer BHAslOo13. The moment distributionmethod has been further modified to account for a con-stant axial load such as the WOB17.

THE THREE-DIMENSIONAL PROBLEM

When the BHA of interest and all loads acting on itare not contained within a vertical plane, e ,g. when theazimuth angle is not constant, we refer to this as a three-dimensional problem. We again begin our discussion witha one stabilizer BHA (one span beam column) withstraight axis, as shown in Figure 9(a). It has loads ql andqz acting in two mutually perpendicular directions Y andZ, respectively. Direction Y lies in the vertical plane,whereas Z corresponcis with the azimuth,

In the absence of torque, the differential equationsrepresenting deflections v and w are12,

d4v d2vEI +P

~ dx4ql............(8)

dx2

d4w d2wEI +P= ,J...o,oms (9)

dx4 dx2 2

These equations are similar to Equation (1). Equa-tion (8) is independent of w, whereas (9) is independentof v, i.e. the two equations are uncoupled. However, iftorque is also present as shown in Figure 9(b), the equa-tions take the following form12:

d% d3w d2vEI -T +P

dx4 dx3 dx2=q~ (lo)

d4w d3v d2wand, EI +T +P= . . . . (11)

dx4 dx3 dx2 2

Thus, in the presence of torque, the two deflectionequations become coupled. The solution of such equa-tions is much more difficult. Additional complexities ariseif, similar to the two-dimensional case, some (or all) ofthe six assumptions discussed earlier are invalid; which isalways the case.

References 6, 8, and 9 have derived equations sim-ilar to (10) and (11) which are specialized for curved,three-dimensional driilstrings. Again, similar to the two-dimensional case, it is found convenient to (i) establish a

SPE 15467 Umest

straight reference axis passing through the bit to measurethe normal deflections v and w, (ii) replace the indepen-dent variable x in Equations (8) through (11) by a variable5 along the curved borehole axis, (iii) cut off the BHA ofinterest at a certain distance above the bit, and (iv)assume the WOB direction to coincide with either theborehole axis or the deformed drillstring axis.

Since the Z direction corresponds to the azimuth, theself-weight does not contrikmte to the term q7 in equa-tions (9) or (11). If we further assume that ttie contactbetween the drillstring and the borehole wall occurs onlywithin a vertical plane, qz becomes zero. Then, the onlysource responsible for the azimuth side force at the bit(indicative of walk tendency) is the initial curvature of theborehole in the azimuth plane. As discussed earlier, thiscurvature is equivalent to a lateral load similar to the self-weight. The assumption of contact being limited. to a ver-tical plane was made in DIDRIL in order to reduce thecomputational efforts15.

If we assume that the contact between the drillstringand the borehole wall does not necessarily occur in a ver-tical plane, the contact forces in the azimuth direction (D-irection) are not zero, and the computational effortsbecome much greater.

Equations (10) and (11) are meant for a straightbeam column where the torque T is constant along theaxis, Strictly speaking, such an assumption is not true fora curved beam column or collar. It is reasonable onlywhen the curvature is mild and the BHA length is small.On the other hand, a limited study indicates that torquehas practically no influence on the deviation tenden-cies15, If this result is confirmed by further analysis, onecan conclude that the inclusion of torque in Equations(10) and (11) is not warranted, i.e. Equations (8) and (9)are adequate to analyze a three-dimensional BHA. More-over, since these two equations are unco~pled, a two-dimensional program should be adequate to handle thethree-dimensional BHAs by analyzing them indepen-dently in inclination and azimuth planes. In this authorsopinion, the merit of a three-dimensional program liesnot so much in predicting the deviation trends but indetermining the actual direction of drilling especially byaccounting for the effects of rock and bit characteristics.

CRITERIA FOR DEFINING DEVIATIONTRENDS

Before we discuss the criteria for determining thedeviation trend (build, hold, drop and walk) of a BHA, itwould be appropriate to clarify the meaning of the termweight-on-bit (WOB) and its direction.

In Figure 10, we consider a dropping assembly in atwo-dimensional curved borehole. The bit is located at Aand the first stabilizer at B. The solid curved line indicates

handra 5

the borehole axis, and the broken line the deformed drill-string axis. The WOB can be defined as the net axialforce at the bit, which is equal to the axial component ofthe self-weight minus the sum of the axia) components ofall reactive forces at stabilizers and contact points and thepull at top end. Alternatively, we can limit the discussionto the drillstring between the bit and the neutral point.The axis in question is, in general, the centerline of theborehole5, which is also the axis of undeformed drill-string. Alternatively, the tangent to the deformed drill-string can be used as the W(?B direction, as is the casewith DIDRIL9115. Evidently, for a vertical borehole, thefirst assumption makes the WOB axis coincident with thevertical, whereas the second assumption does not. Wenote also that, these two different approaches wouldresult in somewhat different values of side force at the bit.

Now, we look at the forces at the bit and the forma-tion. The equilibrium of a dropping assembly in a straightbut inclined two-dimensional borehole is shown in Figure11(a). For brevity, we consider only one span of theassembly, with q as the normal component of the self-weight. The reaction at the bit, R, is normal to the bore-hole axis. If the WOB is assumed to act tangential to thedeformed drillstring it can be resolved in componentsparallel and perpendicular to the borehole axis, as shownin the figure.

The bit force system of Figure 11(a) is equivalent tothat shown in (b). AF is the axial force equal to WOB xcoscz, where m is the bit tilt. Also, SF is the net sideforce, and ii equais R - WOB x sin~. The forces acting onthe formation are equal but ~pposite to those acting onthe bit, as shown in (c!. These formation forces togetherwith the rock and bit characteristics are responsible for thecutting and removal of the rock, and further advance-ment of the drillbit.

The exact direction of the bit advance cannot be pre-dicted without considering the bit and rock characteris-tics. However, it is still possible to predict the tendency ofdrillstring deviation based on the knowledge of bit tilt,side force, or resultant force at the bit. Millheirn8 favorsthe use of the side force for this purpose. Accordingly, ifthe net side force on the formation points down as shownin Figure 11(c), the assembly is dropping. Whereas, if theside force points up, the assembly is building. Anassembly is considered holding if the side force is small,regardless of its direction. This is how the three BHAs ofFigure 4 were classified.

It is evident from Figure 1 l(c) that the use of resul-tant force direction as a criteria will give similar predictionof the directional tendency as the side force. However,the choice of the side force has two advantages. First,since the value of WOB is generally much larger than theside force, the latter has very fittle effect on the magni-tude or the direction of the resultant. If we compare twc

6 Basic Concepts in Static BHA Analysis for Directional Drilling SPE 15467

assemblies, one building and the other dropping, their bending stiffness (more appropriately, the bending stiff-side forces may be drastically different but the resultant ness coeffkient), kb, at end 1 or 2 can be defined as theforce will show little change. Thus, the side force is a moment applied at that end necessary to cause a unitmore sensitive indicator of the directional tendency than rotation (at the same end), with the rotation at the otherthe resultant. Secondly, the determination of resultant end held at zero (see Appendix). When the cross-force would require additional computation. For such sectional properties of the collar are uniform throughoutreasons, DIDRIL uses the side force on formation as the its length, Figure 12(a), kb at the two ends is the same,criteria for determining the deviation tendencies15. and is given by 1319,

In three-dimensional boreholes, the question of walk 4EItrend also needs to be addressed. Obviously, if tt,~ azi- kb =

.. . . . . . . . . . ........ . . ...+ (12)L

muthal component of the side force on the forrnatim ispointing to the right of the borehole (look ~g down When Eor I changes along the length, kbl is in gen-towards the bit), the BHA is walking to the right. Con eral not equal to kb2. To find kbl or kb2, the collar isversely, if this side force points to the left, the BHA is divided into several segments with constant EI withinwalking left. each segment, as shown in Figure 12(b). Then, if only

one or two changes in EI are involved, suitable expres-STIFFNESS AND EQUIVALENT DiAMETER sions can be derived using such techniques as moment

area method and the bending stiffness for each end canThe present day BHAs often include collars which still be hand computed. But, if several Et variations are

have non-uniform inner or outer diameter (ID or OD) involved, hand computations are impractical and the usealong their lengths, e.g. MWD collars. Obviously, the of finite element method becomes necessary.deviation characteristic of such a collar is not necessarilythe same as that of a standard 7- or 8-inch OD drill collar Method 2- Pure Bending Method: In this method,it replaces. eq~al and opposite moments are applied simultaneously

at the two ends of the collar which are free to rotate, asIn drilling industry, a collar with non-uniform cross- shown in Figure 13. Angle 4 is the included angle

section is often specified by its stiffness. But, the concept between the radii at ends 1 and 2, and is equal to the sumof stiffness (especially, the bending stiffness) is not well of the end rotations (31and Bz. The bending stiffness ofunderstood. The commonly used definition of stiffness as the collar can be defined as the moment needed to causethe product of E and I is misleading. In fact, the product an included angle of unity (e.g. one radian). Thus, for aEI is more appropriately called the flexural rigidity of collar of uniform cross-section as shown in Figure 13,one cross-section and it is not a property of the entirecollarlO. A proper definition of stiffness should account d~=p~ (13)for al; cross-sections with their respective EIs, lengths, M EIand relative iocations, since all these factors influence the and, kbl = kbz = kb = = . . . . . . . .. (14)overall bending behavior. A technical discussion of stiff- @Lness is given in the Appendix, and two methods for com- When the cross-sectional properties of a collarputing the bending stiffness are preser-ted in this section. change along its length, PI is in general not equal to @2.These two methods, although both correct, give vastly Also, for such a collar, the equivalent bending stiffnessdifferent answers and can lead to confusion if not prop- can be computed from the following expression,er}~ understood.

1 1 1 1Alternatively, a collar wiih non-uniform cross-section = + -t-

-- . . . . + . . . . . . . .. (15)kb kl kz k.

can be specified by its equivalent outer diameter (OD).Although, the value of equi~alent OD is computed via

L1 L2 Lnstiffness, it is practically the same regm!!ess of the = + +----+method used to compute the latter. Therefore, the use of EIII E212 EnInequivalent OD in specifying a non-uniform collar is notsubject to confusion. Also, knowing equivalent OD of the where, n is the number of different cross-sections in thecollar, one can further compute its equivalent moment of drill collar. This equation is similar to the case of severalinertia 1, or- flexural rigidity EL We elaborate on these axial springs connected in series.ideas in the following paragraphs.

Comparison of Methods 1 and 2 The stiffness coeffi-Computation of Bending Stiffness cient method is more accurate since it accounts for the

relative location of all cross-sections. However, for collarsMethod 1- Stiffness Coefficient Method: Consider with several variations in EI, it requires the use of a com-a drill collar or its segment shown in Figure 12(a). The puter, On the other hand, pure bending method is suit-

able for hand computations.BUCKLING LOADS

A comparison of Equations (12) and (14) shows thatfor a collar of uniform cross-section the value of bendingstiffness resulting from the stiffness coefficient method isfour times that resulting from the pure bending method.For a collar with non-uniform cross-section, this relation-ship is valid in an approximate sense.

A natural question arises at this stage, Which of thetwo methods is correct. In fact, both of them are correct.The primary difference in the two lies in the nature of theboundary conditions assumed at collar ends. A user mustunderstand the concepts of the two methods and thecontext of his application, because the stiffness is alwaysdependent upon the boundary conditions (or, in the ter-minology of the Appendix, on the selection of thedegrees of freedom). Indeed, there is no such thing asthe stiffness of a collar or a beam.

Computation of Equivalent OD

The equivalent OD relates to a hypothetical collarwhose length is equal to that of the MWD assembly orcollar it replaces, ID is an assumed value (nominal ID),and bending stiffness is equal to that of the replacedMWD collar as computed from one of the two methodsdiscussed above. When the first method is used to com-pute the bending stiffness, the equivalent OD, do, can becomputed by rewriting Equation (12) as follows,

[16kbL 11/4do= +di4 . . . . . . (16)

L rE J

where. di is the nominal ID of the collar. Since the kbvalues .3; the two ends of the collar are different, a deci-sion is needed to choose one of the two or their average.

In case of the pure bending method, the equivalentOD can be computed from,

[64kbL 1

1/4

do = ~E +dia . . . . . . . . . . .. (17)

If the material of the assembly is the samethroughout, i.e. if E = Ez = ------ = En, Equation (17)

tcan be further simpk led as,

[64 L

1

1/4

do = _ + di4r L1/ll + L2/12 + --- Ln/In

. . . . . . . (18)It is easy to see that Equations (16) and (17) give

almost identical results since kb in (16) is about four timesthat in (17). The specification of equivalent OD of theMWD collar in a vendors catalogue is very beneficialsince it eliminates the ambiguity associated with themethod chosen for ccmputing the bending stiffness.

Buckling of a BHA occurs when the weight-on-bitreaches a certain critical value and the normal deflectionof the drillstring suddenly becomes large. This criticalvalue of the WOB depends upon the BHA make-up, i.e.upon the size (OD, ID) of the drill collars, location of sta-bilizers and the elastic modulus of the collar material.Although, contact with the borehole wall prevents exces-sive deformation of the drillstring, its buckling has severaladverse effects, such as; (a) formation caving due to rub-bing of the drillstring against the borehole wall, (b) fatiguefailure of the drillstring, and (c) unintended deviation ofthe drillstring. Therefore, avoiding buckling is a goodobjective especially since many expensive MWD sensorsare used now-a-days in the BHAs.

Consider the first span of a BHA in a vertical bore-hole as shown in Figure 14(a). To begin with, we assumethat, (1) the bit and the stabilizer act as two simple sup-ports, (2) the collar above the stabilizer has no stiffness,(3) the self-weight is omitted, and (4) the drill collar is ini-tially strai ht. The critical buckling load for such a case is

#given byl 1220,

,,, = ~;.. . . . . . . . . . . . . . . . . . . . . . ., (19)

If the self-weight (adjusted for buoyancy) of thecollar is also included, Figure 14(b), the critical bucklingload can be approximated by 1112

pa= #EI PL___ . . . . . . . . . . . . . . . . . . ..(2o)

L2 2if the stabilizer at end B is very long, or the collar

above it is much stiffer than collar AB, the model shownin Figure 14(c) can be used and the buckling loadbecomes,

The situations described above can be considered astwo extremes for a collar located immediately above thebit. In general, the true condition lies in between the two,and the corresponding critical buckling load can bebounded by Equations (19) [rather (20)1 and (21).

If both ends of a collar could be considered as fixed,as shown in Figure 14(d), the corresponding bucklingload becomes,

pa = 4#EI. . .. (22)L2, . . . . . . . . . . . . . . . . . . . .

IObviously, a good estimate of the critical bucklingload of a collar will depend upon an engineers ability toproperly model the boundary conditions at its ends.

Above discussion was primarily meant for a BHA ina vertical borehole where no initial curvature or normalload was involved. For a BHA in an inclined or curvedborehole, the concept of critical buckling load has aslightly different meaning. In Figure 1, the BHA in aninclined hole was shown to have a normal load compo-nent, q, in addition to the axial component, p. Further-more, in Equation (7) an equivalence between the initialcurvature and normal load was established. Therefore,the two cases of inclined or curved borehole can be simi-larly treated under the general catagory of an imperfectcollar,

An imperfect collar is shown in Figure 15(a). It bendsimmediately upon the application of the axial compres-sive load, i.e. the WOB. If the imperfection VI is small!initially the lateral deflection V2 increases gradually withthe WOB. But, as the WOB approaches a certain criticalvaiue, V2 grows rapidly regardless of the value Of VItFigure 15(b). In its later stage, the load deflection curve ofan imperfect collar follows that of its straight counterpart

20 Thus, the critical buckling loadin an asymptotic sense .assumes a limiting value for the imperfect collar. Whenthe initial imperfection VI is relatively large (as in a highlycurved or inclined borehole), the lateral deflection of thecollar increases gradually but non-linearly, and there is nodiscernible knee in the curve.

Now, we consider the case of a BHA with two spansas shown in Figure 16(a). It can be shown that the criticalbuckling load for such a BHA lies between the two valuescalculated for separate spans as if each were a bar withhinged endslz,

i.e., Pcrl < Pcr < Pcr2. . . . . . . . . . . . . . . . . . (23)

or, Pcrl > Pcr > pcr2

#EII1 iT2E212where, Pcrl = and Pcr2 =

L12 L22

When the values of pcrl and pcrz differ onlY margin-ally, their average will yield good approximation for thecritical buckling load of the assembly, Per. When this is notthe case, we can use a different approach.

Consider, for example, the BHA shown in Figure16(b), with El = E2 = E, II = 12 = II and L1 = 2LZ =L. The correct value of the critical buckling load for thisassembly is 14.9 E1/L2. Also, using equations (19), (21)and (23), we obtain,

EI EI9.87 > Pcrl, it implies that span BC is much stiffer than ABmd we can replace span BC with a fixed support at B, asshown in Figure 16(c). We can then conclude that the;ritical buckling load, PCr, for the two span assembly ofFigure 16(b) should lie between 9.87 E1/L2 and 20.5 El/L2. The average of the two values gives a better approxi-mation to the exact buckling load of the assembly.

SUMMARY

In directional drilling, the ability to perform simplestatic analysis of bottomhole assemblies is important. Thepaper attempts to present some basic concepts of this dis-cipline in a systematic fashion. Herein, the BHA is treatedas a beam column which can be straight (but not neces-sarily vertical) or curved, and subjected to the self-weight,buoyancy, and weight-on-bit. Beginning with a one stabi-lizer BHA, the effects of additional stabilizers, boreholecurvature, torque, wall contact and of variations in cross-sectional properties are introduced in steps. Both two-and three-dimensional assemblies are discussed. The lim-itations of closed form methods and the advantages oftheir numerical counterparts are highlighted. The criteriafor defining the build, drop, hold and walk trends is clari-fied. Two methods for computing bending stiffness andequivalent outer diameter of collars with non-uniformcross-sections (e.g. MWD collars) are proposed. Finally,simple methods for estimating critical buckling loads ofone and two span assemblies are presented.

ACKNOWLEDGEMENTS

The subject matter presented in this paper was pre-pared when the author was employed with NL Industries,Inc. The opinions expressed here are of the author alone.Thanks are due to Carolyn McFarland and Mary Foutsfor their help in typing the paper.

DIDRIL i.s a trademark of NL Industries, Inc.

REFERENCES

1.

2.

3.

4.

Wilson, G. E.: How to Drill a Usable Hoie, Part l, Work! Oil(August 1960) .Lubinski, A.: A Study of the Buckling of Rotary DrillingString, Driliing and Production Practices, API (1950) 178-214.Lubinski, A. and Woods, H. B.: Factors Affecting the Angie ofInclination and Dog-Legging in Rotary Boreholes, DrillingandProduction Practices, API (1953) 222-250.Woods, H.B. and Lubinski, A.: Use of Stabilizers in Control-hng Hole Deviation, Drillingand Production Practices, API(1955) 165-182.

1 b 878 -------

5, Fischer, EJ.: Analysis of Drillstring in Curved Bore holes,paper SPE 5071 presented at the 1974 SPE Fall Meeting,Houston, Oct. 6-9.

6. Walker, B.H. and Friedman, M.B.: Three-Dimensional Forceand Deflection Analysis of a Variable Cross-Section DrillString, Transactions of the ASME, Journal of Pressure VesselTechnology (May 1977) 367-373.

7. Millheim. K., Jordan, S. and Ritter, C. J.: Bottom-HoleAssembly Analysis LMng the Fini:e-Element Method, JPT(Feb. 1978) 265-274.

8. Dunaevsky, VA. and Judzis, A.: Conservative and Noncon-servative Buckling of Drillpipe, paper SPE 11991 presented atthe 1983 SPE Annual Conference, San Francisco, Oct. 5-8.

9. Ho, H.-S.: General Formulations of Drillstring MechanicsUnder Large 3-D Deformations, NL Technology Systems,Houston, Texas (1985).

10. Popov, E. P.: Mechanics of Materials, Second Edition,Prentice-Hall, Inc., New Jersey (1976).

11. Tlmoshenko, S.: Strength of Materials, Part II, Third Edition,D. Van Nostrand Company, Inc., New York (1956),

12, Timoshenko, S.l? and Gere, J. M.: Theory of Elastic Sta-bility, Second Edition, McGraw-Hill Book Company, NewYork (1961).

13. Wang, C. K.: Statically Indeterminate Structures, McGraw-Hill Book Company, New York (1953).

14. Bradley, W.B.: Factors Affecting the Control of BoreholeAngle in Straight and Directional Wells, JPT (June 1975)679-688.

15, Rafie, S., Ho, H.-S. and Chandra, U.: Applications of a BHAAnalysis Program in Directional Drilling, paper lADC/SPE14765 presented at the 1986 lADC/SPE Drilling Conference,Dallas, Feb. 10-12.

16. Chandra, U. and Rafie, S.: Users Instruction Manual forDIDRIL-1,NL Technology Systems, Houston, Texas (1985).

17. James, B.W.: Principal Effects of Axial Load on Moment-Distribution Analysis of Rigid Structures, Technical Note No.534, National Advisory Committee for Aeronautics, Wash-ington (1935).

18. Millheim, K.: Here Are Basics of Bottom-Hole AssemblyMechanics, Directional Drilling - 3, Oil and Gas Journal (Dec.4, 1978) 98-106.

19. Przemieniecki, J. S.: Theory of Matrix Structural Analysis,McGraw-Hill Book Company, Inc., New York (1968).

20. Gerard, G.: Introduction to Structural Stability Theory,McGraw-Hill Bock Company, Inc., New York (1962).

APPENDIX - STIFFNESS MATRIX OF A DRILLCOLLAR

The stiffness of a collar or a beam column is a mathe-matical representation of its response to applied loads. Aload can be a force or a moment, and the response canbe displacement or rotation. Thus, a load could beapplied in axial, bending, torsion or shear modes. Theresponse of a collar to one type of load is different fromthat to other type, and hence its stiffness in one mode isdifferent from that in the other. Also, when the load isapplied in one mode, the collar may respond in othermodes as well. The stiffness matdx is a means of collec-tively describing the response in all individual as well ascoupled modes.

. .

Consider a collar segment (beam element) arbitrarilyoriented in a three-dimensional space, as shown in Figure17 (a), The behavior of such an element can, in general,be represented by a total of twelve degrees of freedom(DOF), six at each end as shown. Out of the six DOFlthree are translational and three rotational. The stiffnessmatrix [K] of such element is a [12 x 12] matrix consistingof diagonal as well as off-diagonal terms, kij. The forcedisplacement relationship for the element is given by,

where, {F }and {U }are vectors of force and displacement,respectively. The expended form of Equation (A. 1) canbe found in Reference 19.

For round collars, one can take advantage of sym-metry and only three DOF at each end are needed todescribe the behavior of the element. Of these three DOFtwo are translational and one rotational, as shown inFigure 17(b). The full stiffness matrix in such case is givenby,

12EIL3

6EIm

o

12EIF

6E1T

4EIL

o

6EI-7

2E1L

Symmerrtc

EA

-i12EI

oL3

6EI 4E1o-~y

r -1

u,

U*

U3{ >.

U4

U5

U6

(A.2)

6x1 6x6 6x 1

From Equation (A.2), or from the extended form of(A, 1), the definition of any stiffness coefficient, kij, can beeasily derived. It is the force (or moment) in the ith DOFnecessary to cause a unit displacement (or rotation) in thejth DOF with all other DOFZ held fixed.

Whenever bending is the predominant mode ofcollar behavior, for example, in directional drilling, we areconcerned with k~~ and k6G of Equation (A.2). Thisapproach is common in civil engineering structuresls. InFigure 12, these two stiffness coefficients were called kbland kb2, respectively. This forms the basis of the stiffnesscoefficient method discussed in the paper.

Pure bending method can be viewed as a specialcase of the above, where only the rotational DOF at eachend is retained to describe the behavior of a collar ele-ment.

xt%m

3$ m-G\

d

F-i SE 15467

V.

D-: +x

I \7Y,v 1

CURVED AXIS

q.e - 777Txrrr

7 \Y,v sTRAIGHT

Fig. 6lniNal curvature.

Fig. 8CoIlar with nonuniform El.

Z,wA ~ Bp-

fx

.J -pY,v

i

@

\ .\.q2-D Zlw

.

Y,v

N-N

(a) WITHOUT TORQUE

Y,v

N-N

(b) WITH TORQUE

Fig. 9Three-dimensional beam column.

(a) DROPPING AS.SSMSLY IN A POSITIVE CURVATURS HOLE

(b) DROPPING ASSENBLY IN A NEGATIVE CURVATURS HOLE

Fig. 7Effect of borehole curvature on a dropping lssembly.

ABIRSFERENCEAXIS -----n ~oMHom,/ AXIS< a - VERTICALA -. b- TANGENT TO++T zE=G BOREHOLE AXISc - TANGENT TODEFORMED DRILL-

b I STRING AXISa

Fig. 10Possible dlrecNonsof WOB.

WOB. cosa

/

I/

/

q ,

///.

/

Yt/

P

i~ STAS

\\I

E,I, L ;tIIIiZIBIT

tP

(a)

I&:;-%.( -l-x:+k \

b

1-E,I, L

+

bl = b2

(a) COLLAR WITH UNIF@RW EI

M=kbl~E1, 11,L1 E2,12, L2

4-t- -T- +

(b) COLLAR WITH NON-UNIFORM EI

Fig. 12SNffn.?ss coefficient method.

P-PL

[

\

iII

I

t

P(b) (c)

Fig. 14Concept of buckling.

P

P

(d)

Fig. 13Pure bending method.

P

i

+

\\\\I

1 V*

INITIALAXIS ~ (CIJRVED) ,/

A

Pcr

P

LINEARNON-LINEAR

\/

P*

} %~#RHT

= LARGE Vl

SMALL VI

tP

(a)

=V1+V2

(b)

Fig. 15-Buckling of imperfect collar.

E2,12, L2~ a~hp

EsPAN 1 L sPAN 2

--l

(a)

PP .l9Qm

L2

(b)

P A B_P

(c)

Fig. 16Buckllng of BHA with two spans.

P .205EJcr

L2

f

/

19Y!/3146 _ TAA!JSLATIONAL DOFY

A

~ ROTATIONAL DOF

x

z

(a) THREE-DIMENSIONAL ELEMENT

Y

t-x(b) TWO-DIMENSIONAL ELEMSNT

xYZ GLOBAL COORDINATESxyz LOCAL COORDINATES

Fig. 17-A beam @umn or collar element.