Simple Frictional Analysis of Helical Buckling of Tubing

Transcription

1 Simple Frictional Analysis of Helical Buckling of Tubing R.F. Mitchell, SE, Enertech Engineering & Research Summary. revious analyses of helical buckling of tubing have not considered frictional forces. In this paper, differential equations are derived and solved for two simplified cases of interest: downward motion of the tubing-e.g., when buckling occurs during the landing of the tubing-and upward motion of the tubing-e.g., when buckling occurs as a result of thermal and differential pressure loading subsequent to landing. While somewhat more complicated than the conventional frictionless buckling equations, these solutions are still suitable for hand calculations. These solutions, however, do not represent general solutions to buckling with friction. Load reversals and lateral frictional forces add complications that would require computer analysis. Several examples are examined to evaluate the relative importance of friction, which has a significant impact on tubing length change for loaded cases. For instance, the choice of a conservative value for the friction coefficient may allow the solution of a difficult seal-design problem by reducing a large predicted length change. Friction also has an important effect on set-down loads. Frictionless buckling calculations do not give conservative results for this problem. Introduction The buckling behavior of well tubing has an important impact on well design and production operations. Lubinski et al. I first analyzed tubing buckling comprehensively. Hammeriindl 2-4 applied the same basic buckling model to more complicated situations, including combination strings and intermediate packers. The mechanical basis for this buckling model consists of the following features. 1. Slender-beam theory is used to relate bending moment to curvature. The tubing must remain elastic for the analysis to remain valid. 2. The tubing is assumed to buckle into a helical shape. This assumption is reasonable for a vertical wellbore but might not be valid for a deviated wellbore. 3. The principle of virtual work is used to relate the buckling load to the pitch of the helix. 4. Certain conditions on bending moment at the packer are implied by the formulation. It has been shown that these boundary conditions influence the solution of the buckling problem Friction between the buckled tubing and the constraining casin~ is neglected. Recent work showed that the virtual work anal~sis is not necessary to derive useful approximate solutions to buckling equations that satisfy Item 4. These results are summarized in Appendix A. The friction assumption is often mentioned in the buckling literature, and the importance of friction is often stated. In Fig. 14 of Ref. 2, for instance, the 50% deviation of the measure<! buckling length change from the predicted length change is attributed to friction. In this paper, the buckling model is modified to include the effects of friction for two special cases. The elementary theory of friction is discussed and the history-dependence of friction forces is described. Two simple load histories are de- Copyright 1986 Society of etroleum Engineers SE Drilling Engineering. December 1986 scribed that give analytical solutions: tubing loaded at the packer arid tubing slacked off at the surface. Sample problems based on the cases presented by Lubinski et al. I are calculated to illustrate the importance of friction. The actual technical development of the buckling equations is presented in three appendices. In Appendix A, the buckling-force/pitch relation and the contact forces between the buckled tubing and the constraining casing are determined by an approximate solution to the slenderbeam equations. In Appendix B, the differential equations governing the buckling force when friction is present are derived and solved for the cases of loading and landing. In Appendix C, the buckling and "piston-effect" length changes are determined for these special friction cases. Friction The main features of frictional forces are illustrated in Fig. 1. The normal force, F n, presses Body A against Surface B, and a tangential force, F T, is applied to A. A friction force, F, is generated at the interface of A and B. The main properties of the friction force are that (1) if there is no relative motion between A and B, then the friction force is exactly equal and opposite to the applied tangential force, F T, and (2) if A slides on B, then the friction force will have a direction opposite to the velocity of A. The friction force in roperty 1 is limited by a value called the static friction force, which is proportional to the normal force, F n. The friction force in roperty 2 is called the kinetic friction force and is also proportional to F n. The constants of proportionality are called the static and kinetic coefficients of friction, respectively. These coefficients are properties of the interface between A and B. Typical values of these coefficients for metal! metal and metal/nonmetal interfaces under various conditions are given in Table

2 777 ~ 777 B F A Fig. 1-Frlctlon between Block A and Surface B. I t II H t A t l=-f' ----,. F=F, i I.. R 8 A ----,. ----,. F=Fr F=F T STE F, + F\ STE A 8 To illustrate some of the difficulties introduced by friction forces, a simple sliding block problem is described in Fig. 2. Two sliding blocks, A and C, have identical normal forces and friction coefficients. Blocks A and C are connected with a spring, and a force, F T, is applied to Block A. Assume for Fig. 2a that the static friction force is equal to FT' In Fig. 2b, F T is increased by Fr. Block A slides, loading Block C with Fr. When the force on Block A is reduced to F T again, the spring between Blocks A and C still carries load Fh a friction force equal to Fr is developed by Block C, and the friction force on Block A is not equal to FT' Notice, however, that the external load on the block system, F T, is the same in Fig. 2a and c. This example shows that the load sequence-i.e., the load history-determines the final state of a system with friction. It is clear that proper application of friction to buckling requires that the direction, magnitude, and order of applied loads be specified. In the interest of simplicity, only two loading cases are considered in this paper: loading of an unbuckled tube and landing a tube with nonzero slack-off. These two load cases are illustrated in Fig. 3. In Case 1, a buckling force, F f (called the "fictitious" force in Ref. 1), is applied to the bottom of the tubing. The friction force and the tubing weight per unit length, W, are applied in the opposite direction to Ff. In Case 2, the landed tubing is "slacked off' at the surface load, Fs. The friction force opposes the slack-off, but the tubing weight is applied in the same direction. The details of the analysis of these two load cases are described in Appendices Band C., The friction force in this analysis is assumed to point in the z-coordinate direction. This may not be consistent with Friction roperty 2 because the incremental displacements of the tubing for a load variation are not vertical only. Lateral friction forces are beyond the scope of the Lubinski et ai. buckling model, but recent developments 7 may open the way to a more complete analysis. This analysis neglects these lateral forces. ~ I_F. R II A ----,. -:>" F =F', F F. ~ STE C Fig. 2-Load history-dependence of frictional forces: (a) Step A, (b) Step B, and (c) Step C. Summary of Buckling Results With Friction The results of the buckling analysis with friction for Cases 1 and 2 are summarized. The results are compared with the equivalent results of Ref. 1 to illustrate the effect of friction qualitatively. The complete details of the analysis may be found in the Appendices. Case 1. In this case, the tubing is buckled by force F IL) applied at the packer. The buckling force distribution, F Iz), for the friction case is given by FIZ)=.J; tan [,",WK(z-n)],... (1) K where the neutral point, n, is defined by n=l-tan- 1 [~ FIL)]/,",WK,... (2) and the parameter K is defined by rf K=-.... (3) 4EI TABLE 1-COEFFICIENTS OF FRICTION Materials Metal on metal Metal on nonmetals Surface Conditions Carefully cleaned Unlubricated Well lubricated Unlubricated Well lubricated f static 0.4 to to to to to 0.12 f kinetic 0.3 to to to to to

3 When Eq. 1 is evaluated at z=l, the applied for~, FIL), is recovered, and for z=n, the buckling force vanishes, as required by the definition of the neutral point. The equations from Ref. 1 equivalent to Eqs. 1 and 2 have the form Z=o Ft<z)=W(z-n),... (4) where the neutral point, n, is defined by n=l-ft<l)/w.... (5) CASE 1 LOADED CASE 2 LANDED Eq. 4 is equivalent to Eq. 62 in Ref. 1 when the coordinate system is defined as in Fig. 3. Eqs. 1 and 2 approach Eqs. 4 and 5 in the limit as f goes to zero. The length change as a result of buckling, t:.l2, calculated for the buckling force distribution in Eq. 1 gives r [K 2J t:.l2 = - 2f In 1 + w F t<l)... (6) for n greater than zero-i.e., for the neutral point within the string-and r W+KFt<L) 2 t:.l2 = - 2f In 1 W+KF t<o) (7) J J 1 1 l lw l when n is less than zero-i.e., for the neutral point above the string. The equivalents to Eqs. 6 and 7 for the nofriction case are the familiar results r2 t:.l2 = - --F 8EIW t<l) 2... (8) for the neutral point within the string, and r2 LW LW t:.l 2 =- 8E1Ft<L) 2 1 Ff(L) Ft<L) I.(9) for the neutral point above the string. As before, Bqs. 6 and 7 approach Eqs. 8 and 9, respectively, as f goes to zero. One of the interesting results of the friction analysis is that the buckling force is coupled to the actual tubing force through the friction. As a result, the piston-effect length change is affected by the buckling force. If the pistoneffect length change is L t:.l\ =--(Ma+Mab),... (10) EAs then M a is the change in the actual tubing force for the no-friction case and 1 K ~-~ Mab= 2KL In 11+ wft<l)21-ft<l)-l- (L-n)2 +W... (11) 2L. Z=L T Fig. 3-Load distributions for friction buckling cases. is the change in the actual tubing force caused by the buckling force and friction. Eq. 11 vanishes asfgoes to zero. Case 2. In this case, the tubing is buckled by the slackoff load, Fs. Because the tubing is lowered from the top rather than pushed up from the bottom, there are two important differences from Case 1: the friction forces point upward and the load boundary condition is at the surface. The buckling force for this case is given by Ft<Z)=~ tanh [ v'wk(z-n1)j,... (12) where the neutral point, n 1, is defined by nl =L- Fs.... (13) W A useful result is the force on the packer resulting from the slack-off force, F s: Ft<L)=~ Jf tanh( F s )....(14) 459

4 TABLE 2-SAMLE ROBLEM DATA Tubing Data 00, in. Weight, Ibmlft Length, ft ,000 Casing Data 00, in. Weight, Ibm/ft Length, ft ,000 acker bore 10, in. Young's modulus, psi oisson's ratio x Tubing Fluid Density, Ibm/gal Initial 7.30 Final Annulus Fluid Density, Ibm/gal Initial 7.30 Final 7.30 Tubing Surface ressure, psi Initial 0.0 Final 5,000 Annulus Surface ressure, psi Initial 0.0 Final 1,000 Qualitatively, Eq. 14 says that the force on the packer is always less than the slack-offforce when friction is present. The buckling length change for the landed case is The change in the actual force for piston-effect-lengthchange calculations is given by (L-n)~ (~K) (L-n) tanh - F -W--- L K W S 2L...,... (16) The no-friction forms ofeqs. 12 through 16 are the same as the no-friction equations given for Case 1. Eqs. 12 through 16 all reduce to the no-friction case in the limit as the friction coefficient goes to zero. Example Calculations The example cases considered here are identical to the cases'in Ref. 1 for convenience of comparison and discussion. The tubing, casing, fluids, and pressures for this case are summarized in Table 2. The last consideration in these sample calculations is the friction coefficient. From Table 1, a plausible range of f is 0.1 to 0.4. Interestingly, 10hancsik et al. 8 reported friction coefficients in the range 0.3 to 0.4 for rotating, slack-off, and pickup during drilling in wells with 70 to 99% of the hole cased. If the drillpipe-to-casing friction was dominant in each case, this would indicate friction coefficients in the upper range of Table 1. Case I-Squeeze Cementing. This problem is identical to Example 1 of Ref. 5. In this example, the tubing and annulus are originally full of 30 AI [0.88-g/cm3] crude. The fluid in the tubing is displaced with 15-lbm/gal [1797-kg/m3] cement. Finally, surface pressures of 5,000 and 1,000 psi [34.4 and 6.9 Ma] are applied to the tubing and annulus, respectively. The buckling-force distribution for this case with zero friction and with friction coefficients of 0.01 through 0.4 0r , ,500 t 5,000 Q. OJ Q 7,500 ~::~01 / //,=.05 1=.1 1 0,000 L ---"L--=:::::::==~~iiiiiI~'...J o 20,000 40,000 60,000 80,000 BUCKLING FORCE, LB Fig. 4-Buckllng force distribution for loaded case. 6 l~ 'Ie..iL2 \L1 (p'slont (buckhng) FRICTION COEFFICIENT Fig. 5-Tubing length change for loaded case. 460

5 .000 r ,000 8,000 9,000 20,000 BUCKLING FORCE, LB Fig. 6-Buckling force distribution for landed case. l W W IL iii " z :x: u :x: 2.0 l- " Z w... " z iii :> l- 4.0r , ~ is illustrated in Fig. 4. The no-friction buckling force has a straight-line distribution and is compressive from about 1,400 ft [427 m] to bottomhole. The tubing is buckled over that interval. For the friction cases, the buckling force decreases more rapidly near the packer and forms a curve parallel to the no-friction curve far from the packer. This means that friction is more important where the buckling force is large-near the packer. Note also that the spacing between curves decreases for higher friction coefficients. This means there is less sensitivity to friction coefficients in the 0.2-to-0.3 range than in the 0-to-0.1 range. The tubing-length change with friction is given in Fig. 5. Both the buckling and piston-effect length changes are given because both are affected by friction. Over the friction coefficient range of 0 to 0.4, the piston-effect length change varies by less than 50 % while the buckling length change varies by more than 50%. For a conservative friction coefficient of 0.2, the piston-effect length change varies by 38 %, the buckling length change varies by 78 %, and the overall length change varies by 53 %. Case 2-S1ack-Off. This case is identical to Example 2 of Ref. 1. In this example, the same tubing/casing combination is considered before application of the squeezecementing loads. Here we are assuming that the packer has limited downward motion so that we can slack off at the surface and load the packer. In the example, the slackoff load is 20,000 lbf [89 MN]. Fig. 6 shows the buckling-force distribution for the slack-off case. The important number is the force at 10,000 ft [3048 m]. For the no-friction case, this load is 20,000 lbf [89 kn], the same as the slack-off force. This set-down load reduces to 16,400 lbf [73 kn] for j=o.l, 14,000 lbf [62.3 kn] for j=0.2, and 11,200 lbf [49.8 kn] forj=o.4. This is clearly a case where neglecting friction is not conservative. The effect of friction on slack-off has been recognized by the petroleum industrye.g., figures in Ref. 9 relate slack-offtoset-down loads. Unfortunately, the analysis used to produce these charts is not available, nor are the friction coefficients. Fig. 7 shows the effect of friction on buckling and piston length changes. Interestingly, there is very little effect. One reason is that friction works against weight in the slack-off case, whereas friction works with weight in the..).l2 (bucklingl 0.0 o~---l------l----..l---...l l FRICTION COEFFICIENT Fig. 7-Tubing length change for landed case. loading case, as shown in Fig. 3. Thus the change in the load curve in Fig. 6 with friction is much less dramatic than the changes in Fig. 4. Finally, there is much less buckling in this slack-off case than in the loaded case. In the slack-off case, about 3,000 ft [914 m] of tubing is buckled, while in the loaded case, almost the entire length of tubing is buckled. Because the friction is coupled to the actual tubing force only over the buckled interval, the result is less sensitivity to friction. Conclusions Simple analytic solutions to two buckling-with-friction cases have been developed. While somewhat more complicated than the conventional frictionless buckling equations, they are still suitable for hand calculations. These s~lutio?s.do not represent general solutions to buckling wlth fnchon. Load reversals add complications that cannot be dealt with easily. The buckling-with-friction solutions are based on the assumptions used to develop the Lubinski et ai. 1 buckling analysis and are subject to the same considerations. Friction has a significant impact on tubing length change for loaded cases. The choice of a conservative value for the friction coefficient may allow the solution of a difficult seal-design problem by reducing a large predicted length change. Friction has an important effect on set-down loads. The results of frictionless buckling calculations are not conservative for this problem. Nomenclature Ai = area corresponding to tubing ID, ft2 [m 2 ] Ao = area corresponding to tubing OD, ft2 [m 2 ] Ap = packer bore cross-sectional area, ft2 [m 2 ] As = cross-sectional area of tubing wall, ft2 [m 2 ] 461

6 C,Cl,CO, C a = integration constants e = measure of error in Lubinski's buckling solution E = Young's modulus, psi [ka] f = friction coefficient F = force, lbf [N] Fa = actual force in the tubing, lbf [N]!!.Fa = change in actual tubing force (no friction), lbf [N]!!.F ab = change in actual tubing force caused by buckling with friction, lbf [N] Faj = final actual force in tubing, lbf [N] F ai = initial actual force in tubing, lbf [N] F j = buckling (fictitious) force, lbf [N] F Hl, F m = lateral force in tubing, lbf [N] F n = normal force, lbf [N] Fs = surface slack-off force, lbf [N] F T, FT = tangential force, lbf [N] I = moment of inertia, ft4 [m4] K = rji4ei, lbf-ft- l [(N'm)-l] L = length of tube, ft [m] tlll = piston effect length change, ft [m] tll2 = buckling length change, ft [m] n = neutral point, ft [m] t:.p i = change in pressure inside the tubing at the surface, psi [ka] t:.p 0 = change in pressure outside the tubing at the surface, psi [ka] = helix pitch, ft [m] r = radial clearance between tubing and casing, ft [m] ut.u2 = tubing centerline location, ft [m] W = buckling distributed load per length (W t + Wi - W o ), lbf/ft [N/m] Wi = weight of fluid in tubing per length, lbf/ft [N/m] Wn = tubing/casing contact load per length, lbf/ft W 0 [N/m] = weight of outside fluid displaced by tubing per length, lbf/ft [N/m] W t = weight of tubing per length, lbf/ft [N/m] z = axial coordinate, ft [m] o = angular coordinate in helix, degrees [rad] Superscript, = derivative with respect to coordinate z References 1. Lubinski, A. Althouse, W.S., and Logan, J.L.: "Helical Buckling of Tubing Sealed in ackers," JT (June 1962) ; Trans., AlME, Hammerlindl, D.J.: "Moment, Forces, and Stresses Associated with Combination Tubing Strings Sealed in ackers," JT (Feb. 1977) ; Trans., AlME, Hammeriindl, D.J.: "Basic Fluid and ressure Forces on Oilwell Tubulars," JT (Jan. 1980) Hammeriindl, D.J.: "acker-to-tubing Forces For Intermediate ackers," JT (March 1980) Mitchell, R.F.: "Buckling Behavior of Well Tubing: The acker Effect," SEJ (Oct. 1982) Sorenson, K.O.: "ost Buckling Behavior of a Circular Rod Constrained Within a Circular Cylinder," hd dissertation, Rice U., Houston (1984). 7. Mitchell, R.F.: "Numerical Analysis of Helical Buckling," paper SE presented at the 1986 SE Deep Drilling and roduction Symposium, Amarillo, April Johancsik, C.A., Friesen, D.B., and Dawson, R.: "Torque and Drag in Directional Wells-rediction and Measurement," JT (June 1984) Tech Facts, Baker ackers Div., Baker Oil Tools, Houston (1978) Appendix A-The Buckling Contact Force The geometry of the helix is described by the following equations: and Ul =r cos 0... (A-I) u2=r sin 0,... (A-2) where uland u2 are the tubing center-line locations in the x and y coordinate directions, 0 is the angular coordinate, and r is the tubing/casing radial clearance. Fig. A-I shows the tubing buckled inside the restraining casing with the coordinates and variables described above. The equations that describe a slender beam with axial load were derived in Ref. 5: E1ui ''''+(Fjuj)' -FHi =0, i=i,2,... (A-3) where EI is the bending stiffness of the tubing, F j is the buckling force, and F Hi is the lateral load exerted on the tubing. The prime denotes the derivative with respect to the z coordinate. Eq. A-3 was derived from the general equations that describe an elastic rod with large displacements by assuming that 0' is small compared to one. Most applications of helical buckling will easily satisfy that criterion. The buckling force, Fj' is given by Fj=FIL)-W(z-L),... (A-4) where F IL) is the applied buckling force and W is the effective weight per unit length of the tubing as given by Eq. 5 of Ref. 1. For tubing buckled within casing, the contact loads must be perpendicular to the casing wall: and FHl =-W n cos 0... (A-5) Fm = - Wn sin 0,... (A-6) where W n is the magnitude of the contact force and 0 is a function of z to be determined. When Eqs. A-I, A-2, and A-4 through A-6 are substituted into Eq. A-3, the following equations can be derived: and El[ -0"" +6(0,)20"] -F to" -FlO' =0...(A-7) El[ -40"'0' - 3(0,,)2 +(0')4] - F/0,)2 + W nlr=o.... (A-8)

7 Eq. A-7 is solved for (}(z), so Eq. A-8 can be used to evaluate W n' (}(z) is not considered a valid solution if a negative value of W n is produced because the contact force is considered to be compressive only. Eq. A-7 can be integrated to give 1 _()"' +2«(},)3 --F jj' +co =0,... (A-9) EI where 1 Co =(}"'(0)-2(}'(0)3 + -F jj'(o)..... (A-lO) EI y If (}'II and Co are neglected, Eq. A-9 reduces to an alge~aic equation for ()' with three roots: (}'=O, ()'= ±(Fi2El) 'h (A-ll) There are some attractive features of Eq. A-II. First, one of the solutions is for a nonbuckled tubing, (}=O. The two 'nonzero roots are equal with the opposite sign, which means that the tubing spirals to the right or left with the same pitch. We can define the pitch of a helix with variable () by =27r/()'..... (A-12) Substituting the nonzero roots ofeq. A-ll into Eq. A-12 and squaring results in Lubinski et al. 's formula: 2 87r 2 EI =--,... (A-13) F f except Ff and are explicitly variables in this derivation, not constants. At this point, we can go back to Eq. A-9 to test the assumption on (}"'. Eq. A-9 can be rewritten: 1 2(1 +e)«(}') 3 - -F jj' =0,... (A-I4) EI where e=[(}"'(o)_(}"']/ [2«(},)3]... (A-IS) The term e is a measure of error in the coefficient of the (},3 term in Eq. A-9. For e~ I, Eq. A-9 is approximately satisfied by Eq. A-II. When Eq. A-ll is differentiated, e can be evaluated: To give some feel for the magnitude of Eq. A-I6, the Example roblem Case I will be used to evaluate e along the tubing. For this example, if e=o.oi is chosen as an upper bound for application of Eq. A-ll, then Eq. A-16 gives the range of validity as 98.8% of the buckled length. It is clear from the form of Eq. A -16 that e decreases as axial force, Ff, increases. x Fig. A-1-Helical buckling geometry and coordinates. The impression at this point is that Eq. A-ll is a very accurate approximation to the helix pitch differential equation. R~member that to reach this conclusion, however, it was necessary to neglect both (}'" and the integration constant, Co. The validity of this assumption depends on the end constraints on the tubing. In Ref. 7, a cantilever end constraint at the packer gave a nonzero value for co; the value was small enough, however, that Eq. A-ll remains a good approximation. For ()' to be a valid solution for a buckled helix, the contact force, W n, as defined by Eq. A-8, must be greater than zero. Otherwise, the tubing cannot maintain contact with the casing. If we neglect higher-order derivatives, Eq. A-8 reduces to Substituting Eq. A-II into A-I7 results in rfj Wn = 4EI... (A-I8) Appendix B-Tubing Forces The next step in the analysis of buckling with friction is to determine the load distribution in the buckled tubing. Fig. B-1 illustrates an incremental force balance in the tubing. A length of tubing Z has contact force, F, and weight, W, applied. Weight W is given by W=Wt+Wi-W o,... (B-1) where Wt is the weight per unit length of tubing, Wi is the weight per unit length of the fluid in the tubing, and W 0 is the weight per unit length of annular fluid displaced 463

8 Force F #) is the applied buckling force that buckled the tubing. The neutral point, n, is defined as the point where FIn) =0. This is easily evaluated by Eq. B-S: n= -c/.jwk.... (B-lO) For z less than n, F Iz) is defined by Eq. B-5 with the solution Fig. B-1-Tublng force balance Including friction. by the tubing. The force balance corresponding to the loaded condition (Case 1, Fig. 3) gives Mf=(W+fWn)dz,... (B-2) where W n is greater than zero and f is the friction coefficient. Otherwise, M f = Wdz..... (B-3) Differential equations for the force in the tubing are obtained by taking the limit as z goes to zero: d dz Ff =W+fWn'... (B-4) where W n is greater than zero and Flz)=-W(n-z), z<n.... (B-ll) The solution to the landed case is similar except the hyperbolictangent, tanh, replaces the trigonometric tangent: FIZ)=~ tanh(.jwk z+cd.... (B-12) K The neutral point is defined by the surface boundary conditions because with friction the load at the bottom of the tubing is not known: nl d:l-fsiw,... (B-13) where F s is the slackoff load. Coefficient c 1 in Eq. B-12 can now be evaluated at the neutral point: Cl = -.JWK n..... (B-14) The force at the bottom of the tubing after slack-off equals d dz F f = W,... (B-5) FIL)=.j!f tanh ( ~ Fs).... (B-15) where W n is less than zero. The same equations hold for the landed case; however, the friction force is applied in the opposite direction, changing the sign of fw n' as illustrated in Fig. 3, Case 2. When Wn is eliminated from Eq. B-4 through use of Eq. A-II, the result is a first-order nonlinear differential equation for F / d 2 dz Ff=W+KFf,... (B-6) The actual force in the tubing, Fa' is needed to evaluate the length changes caused by piston effect. Because the same friction forces that affect the buckling force, F f, also affect the actual tubing force, Fa' the same force balance used for Eq. B-1 can be derived for the actual force: d 2 -Fa=Wt+KFf.... (B-16) dz Eq. B-16 can be greatly simplified by substituting Eq. B-6: where rf K=-.... (B-7) 4EI The solution to Eq. B-6 is given by FIZ)=~ tan(.jwk z+c),... (B-S) K where the undetermined coefficient, c, is evaluated at z=l: 464 c=tan -1 [~F(L)] -.JWK L.... (B-9) Eq. B-17 can be directly integrated to give Fa(z)=F Iz)+(Wt - W)z+ca..... (B-lS) The integration constant, c a' is evaluated at the bottom of the tubing: ca =Fa(L)-F IL)-(Wt - W)L,... (B-19) where Fa(L) is the value of the actual force at the bottom of the tubing. Eq. B-19 is interesting because it shows that friction couples the actual force in the tubing to the

9 buckling force. In the analysis of buckling without friction, the buckling force is independent of the actual tubing force, and further, the piston-effect length change is not affected by the buckling force at all. With friction, some piston-effect length change will be produced by the buckling force. Appendix C-Tubing Length Change With Friction Given the buckling force distribution (Eq. B-8 for the loaded condition and Eq. B-12 for the landed condition), the calculation of the tubing length change as a result of buckling is clear. Eqs. 61 and 63 of Ref. 1 imply the following equation for length change: rl r2 AL2 = - J 4EIF Iz)dz,... (C-l) n where F Iz) is interpreted in the sense of Eq. B-8. Subscript 2 denotes buckling length change. If the neutral point, n, is above the end of the string, then n is set to zero for the proper integration limit. To simplify the evaluation of Eq. C-l, a change of variables from z to F produces AL2 = - (f(l) ~ ( dz ) df..... (C-2) Ff(n) 4EI df Substituting Eq. B-4 into C-2 gives F/L) r2f AL = - df (C 3) 2 ) 4EI(W+KF2)',... - F/n) Eq. C-3 is easily integrated to give AL2 = -~ In[ W+KFIL)~ J... (C-4) 2/ W+KFln). For the landed condition, Eq. B-12 is substituted directly into Eq. C-l and integrated to give a buckling case with zero friction, the following equation, equivalent to Eq. 9 of Ref. 1, results: 1 rl L ALI = --J Ma(z)dz= --Ma,... (C-7) EAs 0 EAs where M a evaluated at the tubing bottom gives If Eq. C-6 is rewritten to factor out the normal pistoneffect and friction-dependent length changes, the result is L L ALI =---Ma---Mab,... (C-9) EAs EAs where M ab is the change in the friction part of the actual force in the tubing over the length L. This force change is determined from 1 L Mab=[,) [FIZ)-FIL)-W(Z-L)]dzlf n 1 L -[,) [Flz)-FIL)-W(Z-L)]dz! i'... (C-I0) n Note that the values of F Iz), F IL), n, and W may be different in the initial and final states. In the loaded and landed buckling cases considered here, the tubing is assumed to be in an unbuckled state initially. In these cases, the second term in Eq. C-9 equals zero. For the loaded case, M ab can be evaluated by substituting Eqs. B-8 and B-18 into Eq. C-lO and integrating: 1! W+KFIL) 2 I (L-n) M b=- In -FIL)-- a 2KL W+KFln) 2 L (L-n)2 +W- 2L... (C-ll) For the landed case, M ab can be evaluated by substitution of Eqs. B-12 and B-18 into Eq. C-lO: As stated earlier, the friction force couples the buckling force to the actual force in the tubing, as shown in Eq. B-18. This effect needs to be included in the evaluation of the piston-effect length changes. The effects of a distributed force are not included in the piston-effect formulas in the literature. 1-4 The following expression integrates the elastic strain in the pipe both before and after loading and calculates the difference for the net length change: 1 rl ALI = --J [Falz)-Fai(z)]dz,... (C-6) EAso where L is the length of the tubing and A s is the crosssectional area of the tubing. If Eq. C-6 is evaluated for _ (n-l) rw tanh ( fk F) _w(l-n)2 L '" K '" ~ s 2L... (C-12) SI Metric Conversion Factors ft x 3.048* E-Ol m in. x 2.54* E+OO cm lbf x E+OO N lbm/ft x E+OO kg/m Ibm/gal x E+02 kg/m3 psi x E+OO ka "'Conversion factor is exact. SEDE Original manuscript received in the Society of etroleum Engineers office Sept. 16, aper accepted for publication Aug. 7, Revised manuscript received June 26,1986. aper (SE 13064) first presented at the 1984 SE Annual Technical Can ference and Exhibition held in Houston, Sept

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