Effects of Boundary Conditions and Friction on Static Buckling of Pipe in a Horizontal Well

IADC/SPE 111511Effects of Boundary Conditions and Friction on Static Buckling of Pipe in a Horizontal WellGuohua Gao, SPE, Chevron Corporation, and Stefan Miska, SPE, University of TulsaCopyright 2008, IADC/SPE Drilling ConferenceThis paper was prepared for presentation at the 2008 IADC/SPE Drilling Conference held in Orlando, Florida, U.S.A., 4–6 March 2008.This paper was selected for presentation by an IADC/SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the International Association of Drilling Contractors or the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the International Association of Drilling Contractors or the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the International Association of Drilling Contractors or the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of IADC/SPE copyright.AbstractA comprehensive buckling model, a group of fourth order non-linear ordinary differential equations, was derived by applying the principle of virtual work. Lateral friction force is included in this model. The equations were normalized to make the solutions independent of the wellbore size, type of pipe and mud. The critical sinusoidal buckling load of tubing with different boundary conditions typically seen in drilling and well completion applications was analyzed based on the analytical solution of the linearized buckling equation. The results show that the effect of the boundary conditions can be neglected when the dimensionless length of tubing is greater than π5. The authors further investigated the effects of friction on sinusoidal buckling by applying the principle of virtual work. The critical conditions for initiating sinusoidal buckling were determined by a group of three non-linear equations. A perturbation solution of these non-linear equations was obtained. It was found that the critical loads for sinusoidal buckling will increase by 30% to 70% for friction coefficients between 0.1 to 0.3. The authors also conducted an experimental study. The experimental results, including both data obtained by the authors and results published by other researchers, support the proposed model.IntroductionVarious pipes, including drill pipe, casing, tubing, coiled tubing, and sucker rods, are widely used in drilling, well completions, formation stimulation, water injection, and the pumping of wells. During drilling, completion, production, or stimulation operations, the drilling pipe or tubing may be subjected to some degree of axial compression, or the pressure inside the pipe may exceed the external pressure. In both cases, the pipe may lose its stability and buckle into a sinusoidal or helical shape. Consequently, stability and post-buckling analysis of pipe in various kinds of wellbores attracts intense interest from the petroleum industry. J.C. Cunha (2004) and R.F. Mitchell (2006(b)) presented an extensive review on the subject of buckling of tubulars inside wellbores.A. Lubinski et al. (1950, 1962) investigated the stability and helical buckling of drill string and tubing in vertical wells. P.R. Paslay and D.B. Bogy (1964) studied the stability of a circular, laterally constrained rod with two pinned ends in an inclined well by using the energy method. For a short rod, their critical load approaches that of Euler’s prediction when the rod is subjected to an axial compressive load without any lateral constraint except at both pin ends. For a long rod, theirformula can be simplified (R. Dawson and P.R. Paslay, 1984) as follows:c crs r EIqF 2=. Y. Chen, et al. (1990) derived thecritical load for helical buckling of a long pipe in a horizontal well with the energy method, ccrh r EIqF 22=. R.F. Mitchell (1988) established the equation describing the post-buckling configuration of a pipe constrained in an inclined well and the equation to calculate the normal contact force between the pipe and the wellbore0sin sin 62222244=++⎟⎠⎞⎜⎝⎛−θαθθθθcEIr q dx d EI F dx d dx d dx d , (1) θαθθθθθcos sin 432433222cc c EIr q dxd EIr F dx d dx d dx d dx d EIr N +⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛−+⎟⎟⎠⎞⎜⎜⎝⎛=. (2)2 IADC/SPE 111511Mitchell further investigated helical buckling, the axial friction force acting on the helically buckled pipes, and the effects ofwell deviation on helical buckling. He then obtained the analytical solutions for these equations (R. F. Mitchell, 1986, 1988, 1995, 1997, 1999, 2002, 2006(b)). A. Kyllingstad (1995) investigated the effects of wellbore curvature on the buckling of pipe in a curved well with the energy method. G. Gao (1996) derived the buckling equations of a pipe constrained in a three dimensional curved well by applying the static equilibrium equations. G. Gao also obtained the perturbations solutions for both sinusoidal buckling and helical buckling. S. Miska et al. (1995, 1996) and W. Qiu et al. (1998) analyzed the helical buckling of pipe subjected to axial load and torque, and the buckling of pipe in curved wells. Most of the buckling models proposed thus far do not consider the effects of boundary constraints and ignore the effects of lateral friction.P.V.R. Suryanarayana and R.C. McCann (1994, 1995) conducted an experimental study on the effects of friction on the buckling of a rod constrained in a horizontal well. Their experimental results show that friction has significant impacts on the critical load of both sinusoidal buckling and helical buckling. However, they did not establish a theoretical model to explain their observations. D. J. Miles and C. R. Calladine (1999) investigated the lateral thermal buckling of pipelines on the sea bed, and their results also show that lateral friction is an important factor in lateral buckling of pipeline. R. F. Mitchell (2006(a)) investigated the effects of friction on the initial buckling of tubing based on the assumption that the pipe rolls on the borehole wall.In summary, the effect of boundary conditions and friction on the buckling of pipe constrained in a wellbore has not been well understood and investigated. It is not known how many errors are introduced by ignoring the type of constraints on both ends of a pipe without a sound theoretical analysis. Also, a frequently asked question is: how long does a pipe need to be in order to neglect the effect of boundary counditions? At the same time, one can ask the question: when must the end conditions be considered in the calculation of the critical buckling force? It is also not known how and why lateral friction affects the initiation of buckling and post-buckling behavior of a pipe constrained in wellbores.The main objective of this paper is to establish a theoretical model that can provide satisfactory answers and solutions to the above issues, and to conduct experiments to verify the proposed theoretical model. While we will focus our discussions on a pipe in a horizontal wellbore, the methodology can be applied to inclined and curved wellbores as well.Theoretical Model Basic Assumptions1. The wellbore is a horizontal cylinder.2. The pipe is in continuous contact with the borehole.3. The clearance between the pipe and the well bore,c r , is small and the pipe’s deformation is in the elastic range.4. The effect of torque is negligible.5. The pipe slides against the borehole when it buckles. The direction of the sliding friction force does not reverse.6. The heat generated by friction dissipates into the surrounding environment and does not pass into the pipe.Analysis of Sliding Friction ForceWhen a pipe is subjected to an axial compressive load, it moves along the axial direction. As the axial load increases and exceeds its critical buckling load, the pipe also moves laterally. The spatial position of a point on the axis of a buckled pipe can be completely determined by its axial, radial, and angular displacements: )(x u , )(x r , and )(x θ. The positive angulardisplacement (0)(>x θ) is defined as shown in Fig.1(a). Let i x v x v r r )()(11= and q x v x v rr )()(22=, represent the velocity alongthe axial and lateral directions at a point x , where k j q r r rθθsin cos += (as shown in Fig. 1(a)), and i r , j r and k r are unitvectors along the x (axial), y (horizontal) and z (vertical) directions. The total velocity is q x v i x v x v rr r )()()(21+=, and thetotal sliding friction force is in the opposite direction of )(x v r. Let f denote the total sliding friction coefficient, while )(1x f and )(2x f represent the axial and lateral components of the sliding friction coefficient at the location x . The following identities hold: 22221)()(f x f x f =+, and)()()()(2121x v x v x f x f =. Thus, we have )()()()(222111x v x v fx v x f += and)()()()(222122x v x v fx v x f +=.We now consider a pipe that does not rotate and assume that the pipe slides on the wellbore instead of rolling on it. As shown in Fig. 1, the pipe slides upward to the right hand side for 0)(>x θ, and the lateral friction force q N f F f rr 22−= isopposite to k j q r r rθθsin cos += and points downward. However, if the pipe slides upward to the left hand side for 0)(<x θ, thelateral friction force is in the same direction as q r , i.e.,q N f F f rr 22=, as shown in Fig. 1(b). The lateral friction force is not aIADC/SPE 111511 3continuous function of )(x θ. Such a discontinuity of the friction force as a function of )(x θ brings about some difficulty in obtaining the analytical solution for the buckling equations, and we must use the energy method to obtain a solution.(a) (b)Fig. 1 Direction of lateral friction force for 0)(>x θ (a) and 0)(<x θ (b) .Static Buckling Equations and Their NormalizationA detailed analysis of the forces acting on a pipe constrained in a horizontal well, the net work (W ) done by these forces, and the elastic deformation energy (U ) are given in Appendix A and Appendix B. The difference between the elastic deformation energy and the net work done by all external forces,W U −=Π, represents the total energy of the system. In the case of continuous contact between the pipe and borehole, the total energy is∫∫∫∫−−−+⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛=ΠL c Fb L x c L c dx qr W dx Nd f r dx dx d dx d r EI 00)(02042222)1(cos )(sign 2θϑθθθθ, (3) where ∫∫∫−=L b x u b L L u Fb x s u s N s f L u F W b b 0)(01)(0)d (d )()()(d , ds ds d r x u x c b ∫⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛=02221)(θ, and ⎩⎨⎧<−>=0for ;10for ;1)(sign θθθ. According to the principle of virtual work, the net work done by all external forces with respect to any small admissible virtual displacement of the pipe must be transformed into elastic defromation energy when the buckled pipe takes its new equilibrium configuration, i.e., 0=Πδ holds for any small admissible virtual displacement. Consequently, the buckling equations of a pipe constrained in a horizontal well are derived considering the effects of lateral friction as seen in Appendix A. The normal contact force isc c c EIr q dxd EIr F dx d dx d dx d dx d EIr N θθθθθθcos 432433222+⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛−+⎟⎟⎠⎞⎜⎜⎝⎛=. (4) The axial force satisfiesN f dxdF1=. (5) The buckling equation of a pipe in a horizontal well is0sin )(sign 6222244=++⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛−cc EIr q EIr N f dxd EI F dx d dx d dx d dx d θθθθθθ. (6) The three unknowns, axial force )(x F , angular displacement )(x θ, and normal contact force )(x N , can be solved usingEqns 4-6 and the proper boundary conditions. When friction is ignored (0=f ), Eqns. 4 and 6 become exactly the same as Eqns. 1 and 2 in a horizontal well derived by R.F. Mitchell (1988).Since for virtual displacements the forces remain constant, Eqn 3 takes form:4 IADC/SPE 111511∫∫∫∫∫−−⎟⎠⎞⎜⎝⎛−+⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛=ΠL c L c L x c L c dx qr dx dx d x F r dx Nd f r dx dx d dx d r EI 00220)(02042222)1(cos )(21)(sign 2θθϑθθθθ. (7) By letting 25.0⎟⎟⎠⎞⎜⎜⎝⎛=c EIrqμ(1/m), and introducing the dimensionless distance,x μς=, dimensionless normal contactforce qN n =, dimensionless axial load EIq r Fc 2=β, and dimensionless total energy Lqr c Π=Ω2, Eqns. 4 through 7 can be normalized asθςθβςθςθςθςθcos 2432433222+⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛−+⎟⎟⎠⎞⎜⎜⎝⎛=d d d d d d d d d d n , (8) n f r d d c 121μςβ=, (9) 0sin )(sign 26222244=++⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛−θθςθβςςθςθςθn f d d d d d d d d d d , (10) ςϑϑθθςςςθβςθςθςθςςd d n f d d d d d d d L LL L⎥⎥⎦⎤⎢⎢⎣⎡+−+⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛=Ω∫∫∫020242220)()(sign cos 1221. (11)Effects of Boundary Conditions on Sinusoidal BucklingWe first ignore the effects of friction, i.e., assuming 0=f , so Eqn. 10 becomes:0sin 262222244=++⎟⎟⎠⎞⎜⎜⎝⎛−θςθβςθςθςθd d d d d d d d . (12) For a very small value of θ, Eqn. 12 can be linearized as022244=++θςθβςθd d d d . (13)The general solution of Eqn. 13 is)cos()sin()cos()sin()(24231211ςςςςςθp A p A p A p A +++=, (14) where 121−+=ββp ,122−−=ββp ,121=p p , and)(212221p p +=β. (15)The critical sinusoidal buckling load under different boundary conditions can be determined from the general solution given in Eqn. 14. For example, the boundary conditions for a pipe with two pin ends are:0022==ςςθd d , 0)0(=θ,022==Ld d ςςςθ, 0)(=L ςθ,where L L μς=is the dimensionless length of the pipe. Applying these boundary conditions into Eqn. 14yields 02=A ,04=A ,0)sin( )sin(2311=+L L p A p A ςς, and 0)sin( )sin(22231211=+L L p p A p p A ςς. The necessary conditionfor Eqn. 14 to have a non-trivial solution is for0sin sin sin sin 22212121=LL LL p p p p ςp p ςςς, which yields 0 )sin(1=L p ς or 0)sin(2=L p ς,i.e.,L k p ςπ=1,πςk p L =2, (k =1, 2,…). Substituting Lk p ςπ=1,πςk p L =2 into Eqn. 15 gives the critical load for sinusoidal buckling⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛=2221πςςπβk k L L crs , ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛=42221ππL EIr q k k L EI F c crs . (16) Eqn. 16 is exactly the same formula as that derived by Paslay et al (1964) with the energy method. The integer k represents the number of half-sinusoidal waves of a buckled pipe and is chosen as such that minimizes crs β in Eqn. 16.IADC/SPE 111511 5In a similar manner, the critical sinusoidal buckling conditions for a pipe with a pinned end and a fixed end, or a pipe with two fixed ends can be obtained.The boundary conditions for a fixed end and a pinned end are 0022==ςςθd d , 0)0(=θ,0==Ld d ςςςθ, and 0)(=L ςθ.Similarly, we then have 02=A ,04=A , andcos cos sin sin 221121=L L L L p p p p ςp p ςςς.The boundary conditions for two fixed ends are00==ςςθd d , 0)0(=θ,0==Ld d ςςςθ, and 0)(=L ςθ. The necessarycondition for Eqn. 14 to have a non-trivial solution is0sin cos sin cos cos sin cos sin 0 0 1 0 1 0 22221111221121=−−LL L L L L L L p p p p p p p p ςp ςp p p p p ςςςςςς. (17)Fig. 2 shows the results of dimensionless critical loads (crs β) under different boundary conditions. The abscissa in Fig. 2 represents the dimensionless length,L ς, divided by a factor of π. The critical loads,crs β, approach 1 as L ς approaches infinity. Based on the results shown in Fig. 2, we can approximately apply 1=crs β as the critical sinusoidal buckling condition, i.e., the effects of boundary conditions at both ends of the pipe can be neglected for a long pipe such that πς5>L . For a short pipe,we can see from Fig. 2 that crs β approaches infinity as L ς approaches zero. However, the limit of 2⎟⎠⎞⎜⎝⎛=πςβL crs crs Bor crs crs B B L 00,lim →=ς, exists as L ς approaches zero as seen in Fig. 3. For example, in the case of a pipe with two pin ends, let1=k in Eqn.17, and we then have 21lim 00,==→crs crs B B L ς. The critical load approaches20,22222⎟⎠⎞⎜⎝⎛→⎟⎠⎞⎜⎝⎛==L EIB L EI B EI F crs crs crs crs ππμβ. (18)Fig. 2 Critical load vs.πςL . Fig. 3 2⎟⎠⎞⎜⎝⎛=πςβL crs crs B vs.πςL.Eqn. 18 is the Euler’s type formula for critical load with different end conditions. For a pipe pinned at both ends, 5.00,=crs B . Similarly, we obtained the limit of crs B for a pipe fixed at both ends (20,=crs B ), and a pipe with one end fixed and the other end pinned (10,=crs B ). As shown in Fig. 3, when the dimensionless length of a pipe is less than π5.0, Euler’s critical formula gives a good approximation. In the following sections, we will focus on the buckling analysis of a long pipe with a dimensionless length that exceeds π5.Effects of Friction on Sinusoidal BucklingFor sinusoidal buckling of a long pipe, we can assume that the configuration of the pipe takes the form of6 IADC/SPE 111511ςςθp a sin )(=. (19)Using Eqn. 19 in Eqn. 11 gives the the dimensionless total energy of a sinusoidally buckled pipe in a horizontal well (seeAppendix B)())(922941611214312142242222222224a o a a p a p a f a a a p a a p s +−+++⎟⎠⎞⎜⎝⎛−+−⎟⎠⎞⎜⎝⎛+=Ωβπβ. (20)According to the principle of virtual work, the variation of total energy equals zero for any possible variation of a δ when the pipe is under its new equilibrium configuration. The value of a with respect to a given axial load β is such that 0=Ωs δholds for all possible variations of a δ, i.e.,0=∂Ω∂as . In the case of 02=f (no friction), we havea a p a p a s ⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛−+−⎟⎠⎞⎜⎝⎛+=∂Ω∂22248112231β. (21) Obviously, 00=a is a solution of 0=∂Ω∂a s . The second derivative of s Ω is⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛−+−⎟⎠⎞⎜⎝⎛+=∂Ω∂2224228312291a p a p a β, 122402+−=⎟⎟⎠⎞⎜⎜⎝⎛∂Ω∂=p p a a β. (22) The solution of 0022=⎟⎟⎠⎞⎜⎜⎝⎛∂Ω∂=a a is⎟⎟⎠⎞⎜⎜⎝⎛+=22*121)(p p p β. (23) When )(*p ββ<, 0022>⎟⎟⎠⎞⎜⎜⎝⎛∂Ω∂=a s a , 0=a is the only minimizer of s Ωand is the only solution for 0=∂Ω∂as. As a result, the original straight line shape is a stable configuration, and the pipe will keep its original straight line shape when )(*p ββ<.On the other hand, when )(*p ββ>, 002<⎟⎟⎠⎞⎜⎜⎝⎛∂Ω∂=a s a , 0=a becomes a maximizer of s Ω. However, Eqn. 21 has two othersolutions, 112)12(84422,1−−−±=p p p pa βand )12(2422222,1−−=⎟⎟⎠⎞⎜⎜⎝⎛∂Ω∂=p p p a a a β. When )(*p ββ>, 02,12>⎟⎟⎠⎞⎜⎜⎝⎛∂Ω∂=a a s a , 1a a = and 12a a a −== are two minimizers of s Ω. Thus, when )(*p ββ>, the original straight line configuration (0=a ) becomesunstable. Although it is still an equilibrium configuration (because 00=⎟⎟⎠⎞⎜⎜⎝⎛∂Ω∂=a s a still holds), any small disturbance will make the pipe turn from a straight line into a sinusoidal shape.The critical conditions for sinusoidal buckling can be determined by both 0=∂Ω∂as and 022=∂Ω∂as . Let an integer k denotethe number of half-sinusoidal waves of a buckled pipe with limited length L ς. Then, k and the parameter p satisfy therelationship of Lk p ςπ=, and Eqn. 23 becomes exactly the same equation as Eqn. 16, which gives the critical load of sinusoidalbuckling for a pipe with two pin ends. It is noticed that the assumed sinusoidal configuration of Eqn. 19 satisfies the boundaryconditions of a pipe with two pin ends, i.e., 0=θ and 022=ςθd d hold at both ends of the pipe (0=ς and p k L πςς==). Obviously, we should select such an integer for k or such a value for p that minimizes )(*p β. For a long pipe, we have12=crs p and the critical condition for sinuaoidal buckling is 1=crs β.In summary, the above discussion clearly shows that the critical condition of sinusoidal buckling can be determined by0=∂Ω∂a sand 022=∂Ω∂as with crs p p = minimizing the critical load. For example, given 02=f and 1=crs p ,s Ω is now a function of a and β. Fig. 4 illustrates the plots of s Ωvs a with different values of β. The dimensionless total energy s Ωhas only oneminimizer which is at 0=a for 1=≤crs ββ. However, when crs ββ>, there are three stationary points, a maximizer at 0=a , and two minimizers which are located at 11142,1−±=βa . Fig. 5 shows the plots of s Ω for 3.02=f . Similarly, for a given frictionIADC/SPE 111511 7coefficient 2f , there is a critical value of )(2f crs β beyond which the plot of s Ω has more than one minimizer. When )(2f crs ββ≤, s Ω has only one minimizer (0=a ), which means that the principle of virtual work holds only at 0=a and that the original straight line shape of the pipe is stable. When )(2f crs ββ>, the plot of s Ω has two more minimizers, which implies that in addition to 0=a , there are other possible solutions of a that also satisfy the principle of virtual work, i.e., the pipe has different equilibrium configurations other than the straight line shape.Fig. 4 Plot of s Ω for 02=f . Fig. 5 Plot of s Ω for 3.02=f .The critical conditions, which occur when there is more than one minimizer of s Ω and sinusoidal buckling is initiated, aredetermined by 0=∂Ω∂a sand 022=∂Ω∂a s . A value of crs p that minimizes crs βis selected. The non-linear equations can then be solved numerically. As shown by the pink colored curve in Fig. 5, at the critical values of crs β, crs a and 2crsp , the total energy, 0>Ωcrs , is a small positive value but not zero for 02>f . crs Ω is the extra energy required to disturb the pipe when sinusoidal buckling is initiated. If an extra lateral disturbing energy of crs Ω, such as a lateral impaction or vibration, is applied to a pipe, the pipe will restore its original straight line shape when crs ββ<, but the pipe will turn into a sinusoidal wave-like shape when crs ββ≥. The critical values, crs β, crs a , and 2crs p depend on the lateral friction coefficient 2f . Figs. 6 through 8 show how crs β,crs a , and 2crsp change as the friction coefficient increases.Fig. 6 Plot of )(2f a crs vs. 2f . Fig. 7 Plot of )(2f p crs vs. 2f .The open circles in Figs. 6, 7, and 8 represent the results of the numerical solution. A perturbation slolution is also derived as seen in Appendix C. In terms of 2f , the results of the purterbation solution can be expressed as322)(193.01f p crs +=, 2312371.0)(774.0f f a crs −=. (25)⎟⎟⎠⎞⎜⎜⎝⎛+++⎟⎠⎞⎜⎝⎛−=crs crs crs crs crs crs a f a p a p πβ2222288112123121. (26)8 IADC/SPE 111511The solid curves in Figs. 6 through 8 show the results of the perturbation solution. The results predicted with thepurterbation solution agree well with those of the numerical solution.Fig. 8 Plot of )(2f crs β vs. 2f .Here, we consider an example of 3 ½” pipe in a 6 ¾” horizontal borehole. The critcal buckling load is ccrscrs r EIqF β2=. In this example, )m (041272.0)in ( 625.1==c r , )N/m (101.2211×=E , )m (1081.1)in (35.4464−×==I , )N/m (0.206=q , and)KN (56.43=cr EIq. In the case of no friction (02=f ), Eqns 25 and 26 yield 1=crs p , 0=crs a , and 1=crs β, thus )KN (11.87=crs F . When the friction coefficient 3.02=f is applied, we have 0865.1)3.0(193.0132=×+=crs p ,4068.03.0371.0)3.0(774.031=×−×=crs a , 671.1=crs β, and )KN (6.145=crs F .Experimental Study Experimental FacilityThe buckling experimental facility shown in Fig 9 includes a transparent plastic tubing (used to act as a well bore), a pipe, a loading/rotating assembly, two load cells at both ends, and a displacement transducer. The main characteristics of the experimental facility are listed in Table. 1. The configuration of the tested pipe can be observed through the transparent tubing. The axial loads at both ends and the displacement at the loading end are measured with the load cells and displacement transducer simultaneously. The measured data is transferred to a PC and then recorded automatically for further analysis.Table 1 Main Characteristics of Experimental FacilityTotal length of the facility (m) 27.7 Length of the tested pipe (m) 26.2 Transparent tubing ID (cm)5.08 Pipe OD (cm) 1.27 Pipe ID (cm) 1.092 Pipe weight (N/m)2.539 Young’s Modulus of the pipe (MPa) 200 Maximum axial displacement (cm)15.24 Load cell capacity (N)4448.2Experimental Observations and MeasurementsExperimental observations show that a pipe subjected to axial compressive load in a horizontal well experiences three different configurations: stable shape—original straight line, sinusoidal wave-like shape, and helical shape.When the axial load is small, the pipe keeps its original straight configuration and lies down straight on the lower side of the well. When the compressive load exceeds the first critical load, the critical load for sinusoidal buckling, the straight configuration of the pipe becomes unstable and then sinusoidal buckling occurs. In this case, the pipe takes the configuration of a 3D snake, or a shape similar to a sinusoidal wave whose amplitude depends on the dimensionless load β. When the axial load exceeds the second critical load, the critical load of helical buckling, the pipe buckles into a helical shape.To measure the friction coefficient, we first move the pipe along the wellbore very slowly. During this process, the measured load at the loading end (or top end) is almost a constant. This load is the total sliding friction force along the wholeIADC/SPE 111511 9pipe, which is about 4.5 (lbf) or 20.0 (N). As seen in Fig. 10, the curves with two different colors show the measured results for two different cycles of loading and unloading. Note that the top load gage reads 2 (lbf) when no axial load is applied during the unloading process. All the top load measurements were corrected by this shifting. The weight of the pipe is about 66.5 (N). The estimated sliding friction coefficient is about 0.30.Fig. 11 shows the measured dimensionless axial loads at both the top end (pink curve) and bottom end (blue curve) vs. the axial displacement measured at the top end. Note that the top end is the end where the axial force is loaded and unloaded, or where axial displacement is permissible. Furthermore, the bottom end is the constrained end, where axial displacement is not permissible. During the experimental process, the axial load increased very slowly. The measured top end load was greater than the measured bottom end load. The difference between the loads measured at both ends,Bottom Top βββ−=Δ, represents the dimensionless axial friction force. Fig. 12 shows the plot of βΔ vs. Top β. At the same time, Fig. 11 clearly shows that the measured dimensionless critical sinusoidal buckling load is about 1.61 (see the arrow A in Fig. 11). When the top load is less than the critical load (force), the top load is almost a linear function of the measured axial displacement, which indicates the elastic axial compression of the pipe and experimental system. However, when the top end load reaches the critical load, the axial displacement increases about 0.1 inches without increasing the axial load, which we interpret as the initiation of sinusoidal buckling.Fig. 9 Schematic of experimental facility. Fig. 10 Measured axial friction.Fig. 11 Measured axial loads at both ends. Fig. 12 Axial friction force vs. axial load at top end.Fig. 12 shows that the axial friction force is almost constant when the top load is less than the critical load. The small increase of friction force is due to the imperfection of the pipe (it is not a perfect straight line). Fig. 12 also clearly shows a sudden reduction of axial friction force when buckling is initiated as indicated by the downward peak denoted by the arrow A in Fig. 12. One explanation of this reduction of axial friction force is that the pipe slides laterally against the wellbore when buckling initiates and releases part of the axial friction force. When the dimensionless top load exceeds 1.61, the axial friction force increases dramatically. This is also a clear indication that the dimensionless critical load for sinusoidal buckling is 1.61. This holds true only for a buckled pipe where the normal contact force, and thus, the axial friction force, increases as the axial compressive load increases. When the dimensionless top end load reaches about 2.2 (see the arrow B), another sudden increase of axial displacement is observed at the top end when axial load is no longer increased, as seen in Fig. 11. Fig. 12 also shows that a sudden reduction of axial friction load takes place when the dimensionless top end load reaches 2.2 (at the arrow B). We take this value of 2.2 as the measured dimensionless critical load to initiate helical buckling. The analysis and discussion of post buckling behavior is beyond the scope of this paper.。