Basic Calculations forPacker and Completion Analysis Albert R. McSpadden CTES, L.P. April 8, 2004 Introduction The design and installation of a well bore completion typically involves running tubing with a packer into the well to facilitate production or injection of fluids from or into the well.Once the packer is set, its position is fixed whereas the tubing may be allowed free, limited, or no movement.Over the service life of the completion, the well experiences changes in temperature, pressure, flow rate, etc. which cause tubing stress and strain.If the tubing is free to move at the packer, this strain translates into movement relative to the packer.Obviously, this movement should not exceed the seal bore length at the packer.If tubing movement is limited at the packer, stresses translate into force on the packer which must not exceed operating limits.Likewise, the total combined effect of all stresses may result in tubing failure due to burst, collapse, or material yield. A major aspect of down-hole tubular movement is buckling.Indeed, the analysis of tubing movement and packer forces is often taken to be synonymous with buckling analysis.Unlike the analysis of other influences on tube movement which is generally straight-forward, the analysis of tubular buckling is much less obvious.It is essential to understand that there are two types of buckling: Mechanical Buckling:buckling may result from applied mechanical compression on the tubing .For example, this may occur in response to weight slacked-off at surface onto the packer during or after installation.Or, changes in the down-hole conditions such as temperature may cause an increase in tubing length which cannot be accommodated by movement at the packer. And so buckling occurs. Hydraulic Buckling: it is often not understood that tubing installed in a packer can buckle simply due to difference in hydrostatic pressure inside the tubing versus the annulus.In this regard, buckling of production tubing in a packer differs from free hanging tubing such as drillstring or coiled tubing.For free hanging tubing, pressure changes inside the tubing and/or in the wellbore annulus have no net effect on buckling.Why does tubing in a packer behave differently?Because the packer functions as a down-hole pressure barrier and, in particular, it isolates the end of the tubing from the annular pressure effects above the packer.Surprisingly, this hydraulically related buckling often may exist even in the presence of tubing to packer tension. The following sections show how tubing strain and stress are calculated and how to model the resulting tubing movements and/or forces on the packer.It is important to understand how each different physical condition affects tubing stress and strain.In real life many different simultaneous physical factors affect the tubing.The net result of all the factors combined is not always obvious.The theory discussed is the basis for the numerical simulations in PACA, a part of the Cerberus modeling suite.Some analytical equations are provided as a means of double-checking simulation results in the simple case of a vertical well. 1. Basic Components of Tube Movement No Packer It is helpful first to analyze forces and stresses on tubing with no packer.A tubing string run into a vertical well without a packer will undergo change in length, i.e. stretch, simply due to elastic effects of hanging weight.Variation in temperature or fluid pressure with depth will cause additional stretch.We choose to define stretch as a length change relative to the nominal length of the tubing at the original surface conditions.Thus, a tubing string with nominal length at surface L will undergo some initial length change or stretch, L (Figure 1a). As different processes and operations are carried out during the life of the well, the tubing experiences more changes in temperature, pressure, fluid density etc. which result in a different amount of stretch L* with respect to the nominal length (Figure 1b).For the purposes of typical tubing stress and movement analysis, the focus of attention is change in stretch between different operational scenarios rather than the initial nominal length L itself. Usually the initial conditions correspond to installation.Thus, we define tubing movement L to be change in stretch: L =L* - L. L L* L L

Figure 1bFigure 1a There are four basic factors that result in stretch and hence tubing movement: 1) elastic stretch associated with axial forces in the tubing, 2) buckling resulting from compressive axial forces exceeding the buckling load of the string, 3) Poissons effect or ballooning caused by changes in fluid pressure in the tubing and annulus, and 4) thermal expansion or contraction due to changes in the temperature profile of the tubing.Thus, total tubing movement is given by 1 2 3L L L L L4 = + + + (1) where1 change in stretch due to elastic strain L = 2 change in stretch due to helical buckling L = 3 change in stretch due to Poisson's effect L = 4 change in stretch due to temperature change L = Movement from Elastic Strain L1According to Hookes Law for an elastic material, the strain or relative length change is proportional to the axial stress a imposed.This proportionality is valid as long as the stress remains within the elastic range of the tubing material.The constant of proportionality is Youngs modulus of elasticity E so that: aLE EL = = (2) Tubing axial stress a is total axial force FR applied to the cross-sectional area AS: RaSFA = (3) For a given section of tubing with uniform geometry and material composition, all of the parameters in the preceding equations may be assumed to be constant over the service life of the tubing except for axial force. Hence, change in stretch associated with elastic strain varies directly with change in axial force according to the following equation: *1 1 1RSF LL L LE A = = (4) There are various possible causes of variation in axial force in the tubing during the course of well operation.To analyze the nature of this variation, it is helpful to review the calculation of axial force in the tubing string itself.In the simplest case of an empty tubing string hanging in an empty vertical well, the only force acting on the tubing is hanging weight due to gravity which is distributed throughout the body of the tubing.Given the tubing weight per foot in air, wS , total axial force denoted by FR is given by R SF w L = (5) However, if the well and tubing are then filled with fluid, it is evident from Figure 2a that the bottom cross-sectional face of the tubing is now subject to hydrostatic pressure Po in the annulus at the bottom depth.Now, total axial force becomes ( )R S o iF w L A A P = oi(6) If the tubing is plugged at the bottom so that internal pressure Pi is maintained independent of the annulus, then another expression for total axial force at surface results (Figure 2b): R S o o iF w L P A PA = + (7) Likewise, the axial force at any distance X above the bottom is given by: , R X s o o i iF w X P A PA = + (8) In fact Eqns. 7 or 8 may be used in general even for open-ended tubing since in that case the internal and external annular pressures are equal at bottom. Since the tubing geometry and material density are assumed to be constant, only the fluid pressures are subject to change.Thus, change in total axial force is given by: R i i oF PA P Ao = (9) Hydrostatic pressure may vary due to change in surface pressure or due to change in fluid densities.Since hydrostatic pressure varies with depth according to fluid density, both Po and Pi may be expressed in terms of fluid densities o and i and pressures Po,X and Pi,X at some distance X above the tubing bottom: , o o X oP P X = + (10) , i i X iP P X = + (11) Combining Eqns (8), (10)and (11) results in the following for axial force at position X: , ,( )R X S o o i i o X o i X iF w A A X P A P,A = + + (12) Change in total axial force can be calculated in terms of density and surface pressures: , ,( )R i i o o i S i o SF A A L P A PoA = + (13) Thus, tube movement (or change in stretch) from elastic strain can be calculated by substituting FR as calculated in Eqn.(9) or Eqn.(13) into Eqn.(4). L PoX PiPoL Figure 2aFigure 2b Movement from Helical Buckling L2In general, a compressive axial force applied to the tubing will cause it to buckle if the force exceeds the buckling load.In the simple case of tubing hanging in an empty well, this means that mechanically applied force in excess of the buckling load will cause the tubing to buckle helically with a resulting decrease in depth, L2 (Figure 3a).In this case, we can say the tubing experiences mechanical buckling.In general we can say that mechanical buckling results from compressive reaction on the tubing.. If the tubing and annulus are filled with fluids, it seems reasonable to compare real axial force as calculated in Equation 12 to the buckling load and decide if the tubing is buckled at any given distance X above bottom.However, this assumption is incorrect. The presence of external and internal fluids changes the relationship between axial force and buckling.1,2Fluid pressure exerts a compressive force PoAo on the bottom face of the tubing which tends to induce buckling (Figure 3b). However, annular fluid pressure exerts lateral force which supports the tubing and resists buckling.It can be shown that curved tubing tends to stable equilibrium in the straight position when subject to lateral external pressures.In the simple hydrostatic case these lateral pressure forces exactly counter-balances the compressive force on the bottom (Figure 3c).Thus, compression due to pressure on bottom is neutralized as far as buckling is concerned.In general, the external lateral pressure forces enables the tubing to withstand a general compression up to a magnitudeof -PoAo without buckling.(see Appendix for rigorous discussion) Figure 3a F F =PoAo F =PoAo Figure 3bFigure 3c Po,X L2 x Similarly, internal pressure on the bottom (e.g. plugged tubing) relieves axial compression and which means that it acts to reduce buckling.However, at the same time internal lateral pressure forces on the inner tubing wall cause buckling.It can be shown that tubing subject to internal lateral pressure forces is in unstable equilibrium when straight, but given a perturbation buckles and thereby assumes a stable or neutral equilibrium.This effect is known as hydraulic buckling.In the simple hydrostatic case, it can be shown that hydrostatic pressure on the tubing walls is equivalent to an axial compression of magnitude -Pi Ai.So tension due to inner pressure on bottom is offset as far as buckling is concerned. So for freely suspended tubing, net fluid pressure acting axially on the tubing must be offset to correctly calculate buckling. This modified calculation of axial force is referred to as effective axial force FE and accounts for the influence of the lateral (i.e. non-axial) fluid pressure forces.Recall that the axial force on the tubing bottom as given by Eqn(7) is: ,0 R o oF A P Ai iP = + (14) So, effective force at the bottom by the tubing is converted from FR,0 thus: ,0 ,0 E R o o iF F A P AiP = + (15) The line of argument above applies to a cut away section of the tubing at any depth, not just the bottom end.This modified axial force at position X above bottom is the effective axial force, FE,X : , , , , E X R X o X o i X iF F P A P A = + (16) Combining Eqns. (12) and (14) results in the following complete expression for effective force: ,(E X S o o i iF w A A)X = + (17) The terms oAo and iAi equate to weight per linear foot of the annular and tubing fluids, wo and wi.If the tubing buoyant weight per foot, wB, is defined as B S ow w w wi= + ,(18) then a simplified expression results for FE,X : , E X BF w X = (19) Eqn. (19) implies that FE,0 =0 at bottom.The total effective force at the top of the tubing string is then: E BF w L = (20) Eqn. (20) says FE varies with change in fluid density but not with change in surface pressures.Hence, changes in pressure will not affect the buckled state of freely suspended tubing To determine buckling at a given depth, FE is compared to the critical helical buckling load.For tubing in a vertical well this is: 8BVHBcEIwFr= (21) where rc is the radial clearance between the tubing and inner casing diameter.The period or pitch of the helix, , varies with FE according to the following equation: 22 ,E VHBEEIF FF = . < (22) For a short section of length l above the packer in which axial force remains relatively constant, change in length is given by: 221 1crl = + (23) Since FE and vary continuously along the tubing string, proper calculation of length change requires a segmental summation or integration of Eqn.(23) along the entire tubing length.Comparing the change in length for two different buckling scenarios gives tube movement due to change in helical buckling. In the case of a vertical well where a mechanical force F is applied to the bottom of initially straight tubing, the following analytical solution for tube movement due to buckling has been derived:3 2 2*2 2 2, 8cVHBBF rL L L F FEIw = = < (24) In this case, the distance to the neutral point (i.e. the length of buckled tubing) is: BFnw= (25) Movement from Poissons Effect L3The presence of fluid pressure in the tubing and annulus results in another type of strain known as Poissons Effect, or ballooning.Fluid pressure acting laterally on the inner and outer tubing surface induces radial and hoop stresses, r and h.As a result the tubing expands or contracts in the cross-sectional plane.Associated with this cross-sectional expansion or contraction is a compensating axial strain, or length change. These stresses vary non-linearly across the wall thickness of tubing and are expressed by Lames equations for thick-walled cylinders.Integrating Lames equations across the tubing wall results in the total Poisson strain, 3: (32( )o o i io iP A PAA A E) = (26) where =0.3 is the accepted Poissons ratio for steel and where oP and iP are the average external and inner fluid pressures along the tubing length L.The tubing stretch from the Poisson effect, L3, is calculated from the preceding strain ratio thus: (3 32( )o o i io iL)L L P AA A EPA = = (27) Since Eqn (27) is based on average fluid pressures, then the calculation of L3 is accurate only if the pressures are relatively constant over the length L. Hence, a proper calculation of length change involves a numerical integration, applying Eqn.(27) segmentally.Once again, since the tubular material properties remain constant, Poisson stretch is only affected by changes in fluid pressure.Comparing the change in stretch for two different scenarios as before, the tube movement due to change in Poissons effect is then: (*3 3 32( )o o i io iL)L L L P A PAA A E = = (28) Movement from Temperature Change L4Often one of the most significant causes of length change in tubing during and after installation in a well is thermal expansion and contraction.In most any well the down-hole temperatures may vary greatly from the ambient temperature at surface.This difference between ambient surface temperature and the average down hole tubing temperature T along its length L results in a length change determined by the coefficient of thermal expansion for the tubing material, C: 4(SL C T T L = ) (29) where the value of C for typical tubular steel is 6.7 x 10-6 /oF.Note that this calculation is only accurate for tubing lengths L in which tubing temperature is relatively constant, and hence it is best to apply Eqn. 27 on a segmental basis along the tubing length. Tubing movement due to change in average tubing temperature between the initial scenario and a new operating scenario may be approximated thus: *4 4 4L L L C T L = = (30)2. Packers and Tube Movement So far it has been assumed that the tubing hangs freely in the well bore without a packer.Since production tubing is usually installed with a packer, analysis of movement and force must account for the packer.The packer functions as a barrier between upper and lower zones of the annulus.Because of this, the end conditions at the tubing bottom for real and effective force must be modified accordingly.The presence of a down-hole pressure differential at the packer has significant influence on buckling of the tubing, often with unexpected, non-intuitive results.In addition, the packer may prevent or restrict movement of the tubing bottom, and this determines the nature of resultant forces and stresses. Packer Pressure Force Obviously fluid pressure acts on all exposed areas of the tubing.Thus any time there is a change in tubing diameter such as with a crossover, a discontinuous change in axial force results.The packer seal bore diameter may differ from the tubing diameter, involving a crossover or expansion device.Several combinations of outer tubing area Ao and packer seal bore area Ap are shown in Figure 4(a-c). A oA oA oPo Pi Pi Pi PoPoA iA iA iA pA pA p Figure 4bFigure 4aFigure 4c Note that the tubing is typically open, so that annular pressure just below the packer is the same as tubing pressure Pi.The tubing bottom is isolated from the external annular pressure Po, which may now differ significantly from Pi.As a result, the end condition for tubing axial force is different from that in Eqn.(7).Now, the tubing is subject to a packer pressure force Fp which acts axially just above the packer and any associated crossover: ( ) (p o p o i p iF P A A P A A) = (31) Based on Eqn.(31) calculation of FR from Eqn.(9) which determines tube movement from elastic stretch(L1) according to Eqn.(4), is now given by the following: (32)( ) (R o p o i pF P A A P A A = )i,)iAReferring to Eqn.(13), this same result may be expressed in terms of change in surface pressures and change in fluid density: ,( )( ) ( )(R o o S p o i i S pF L P A A L P A = + + (33) Hydraulic Buckling:As with free hanging tubing, hydrostatic pressure in the tubing and annulus changes the analysis of tubing buckling.It is still necessary to differentiate between real force FR and effective force FE.Hence, effective force just above the packer FE,P is calculated from packer pressure force Fp by applying the conversion in Eqn. (15): , E P P o o i iF F P A PA = + (34) Eqns.(31) and (34) combined yield a simple equation for effective force at the packer, FE,P: ,( )E P o i pF P P A = (35) Change in effective force given at the packer is given by: ,(E P o i pF P P)A = (36) or equivalently by: ( ), , E P o i o S i S pF L P = + ,P A (37) For a vertical well, tube movement from change in fluid pressures or density is given by: 2 2,2( ), for 8E P ciBF rLEIw= > oP P (38) Thus, a sufficient increase in internal pressure Pi results in negative effective axial force which may cause straight tubing to buckle. The contrast in buckling behavior for tubing installed with and without a packer is sometimes a source of confusion. However, the same underlying physical principles are at work in either situation.In the case of tubing without a packer, there is a continuous pressure profile both inside and outside the tubing.This means the pressure effects on the bottom are dependent on the entire pressure profile, so that the pressures causing and preventing buckling cancel out.When a packer is installed and the pressure profile is discontinuous, those two pressure effects work independently.Then buckling may or may not occur, depending on which pressure effect predominates.(See Appendix discussion) As noted previously, buckling due to increased internal pressure is known as hydraulic buckling.It might be assumed that this is a localized effect at the packer itself.And so the effect may even seem intuitive in the case that the packer allows free movement.However, hydraulic buckling is a distributed effect such as the Poisson effect, and can result in buckled tubing even in situations where the packer prevents any upward motion of the tubing.In summary we can say that hydraulic buckling is pressure-induced buckling whichshortens the tubing and causes a tensile reaction between the tubing and the packer. Fixed vs. Free Movement at Packer The packer may allow free tubing movement or it may prevent or restrict such movement.In each case, this affects the development of forces and stress at the tubing/packer connection. Free tubing movement is considered first, and then these results are extended to the case of no movement and limited movement. 1.Free Tubing at Packer:The calculation of tubing movement in the case of free allowed movement is essentially the same as for free hanging tubing.Once the pressure effects at the packer are incorporated in change of real and effective force as using Eqns. (33) and (37), the basic tube movement components may calculated and summed as in Eqn. (1).Since the packer allows free tubing movement, the bottom of the tubing is assumed to move a distance L up or down, relative to the packer (see Figure 5). A practical consequence is the need to choose an appropriate packer seal bore length, LPSB, to accommodate the maximum expected tube movement (Figure 5c).If sufficient seal bore length is not provided, the tubing may pull out of the packer and communication between the upper and lower annular zones will result. L LPSBL PSBL >0 Figure 5aFigure 5bFigure 5c 2.Fixed Tubing at Packer:In the case that tubing is fixed at the packer, then changing well conditions cause tubing length changes which are not accommodated by movement relative to the packer..A resultant contact force is induced on the tubing by the packer.A net shortening of the tubing results in a downward tensile force exerted on the tubing by the packer; or equivalently, an upward force is exerted on the packer by the tubing.Likewise, a net lengthening of the tubing results in an upward compressive force exerted on the tubing by the packer, or equivalently, a downward force exerted on the packer by the tubing. A simple method may be applied to determine the magnitude of the resultant packer to tubing force: 1)Calculate tube movement L as if the packer allowed free motion (Figure 6a). 2)Apply a mechanical force FTP to the tubing: downward for shortened tubing, upward for lengthened tubing (Figure 6b). 3)Iteratively increase |FTP| until the tubing length is virtually restored to the packer position by a net virtual movement LTP =-L (Figure 6c). The force FTP thus determined is the resultant packer-to-tubing or tubing-to-packer force.The net effective force at the packer is now the sum of the packer pressure force and any induced packer to tubing force: ,0 , E E PF F FTP= + (39) FTPLTPL Figure 6aFigure 6bFigure 6c According to Eqn.(39), two distinct forces determine total tubing force at the packer.It is important to distinguish between these two forces and how they affect the tubing stretch and buckling.FE,P is an effective force which describes the pressure effects related to any hydraulic buckling.FTP is a real mechanical force which affects both elastic stretch and any mechanical buckling.Their resultant FE,0 is also an effective force which determines the resultant buckling of the tubing above the packer.The interaction between these forces and the determination of tubing force at the packer often leads to non-intuitive to results which are difficult to understand. Note that the specific components of tubing movement L1, L2, L3, and L4 may be reported as a part of the analysis before or after application of the force FTP.Note that in particular L1 and L1 will be affected by the final application of FTP.Obviously, in the latter case the movements correspond to the new final state of the tubing.However, the movements as calculated in Step 1) of the method given above may help to indicate the causes of change in the tubing state and of any induced tubing to packer force. 3.Limited Tubing at Packer:In many cases the tubing and packer are configured with a limitation on possible movement.A no-go may be provided in either direction or both, so that the tubing may move up and/or down at the packer in a limited range (see diagrammatic depiction in Figure 7).For the purposes of calculating tubing movement and resultant forces, the packer is assumed to be a free packer as long as resultant movements are within the limited range.When the tubing movement goes beyond the physical limit, then the analysis proceeds as in the case of a fixed packer.The induced force FTP is determined as for a fixed packer, which in this case brings the tubing back to the limiting no-go depth by means of a net virtual movement of LTP: TP no goL L L = + (40) L no-go >0 (limit up) L no-go (43) and SLbE A= (44) If the applied force F is tensile such that the tubing is being stretched, the length change is linearly proportional to the force F.However, if the force F is compressive in excess of the helical buckling load, then the length change vs. force relationship becomes non-linear.Given known tubing geometry and material properties so that coefficients a and b are constants, a graph of Eqn.(42) may be plotted as in Figure 8.Note that a simplifying approximation is made by assuming the buckling load is 0 in the vertical case. +F- F - LF Figure 8+LF Lubinskis method was put forward as a graphical technique, but it may be implemented by inverting and algebraically solving Eqn.(42) which is quadratic. Thus, to calculate FTP and FE,0 and to correctly determine tubing buckling f ited motion packers, proceed as follows: 1)Dete e the effective packer pressure force FE,P from Eqnsand (38). 2)Calculate theoretical tube movement LE,P by substituting FE,P F in Eqn.(42): ,(45) 3)Calculate LTP =-L from Eqn.(41) for total movement assuming a free packer. 4)Let L be the net resultant of LE,P and LTP : or fixed/limrmin .(37) for2, , E P E P E PL aF bF = +, E P TPL L L = + (46) 5)Calculate the FE,0 by solving the quadratic equation in Eqn.(42) with LF =L: 2b b 4a L ,02EFa + +)= (476)Solve for the restoring tubing-packer force FTP in Eqn.(39): , ,0 TP E P EF F F = (48) ote: although this method is straight-forward procedurally, the reasoning behind it may not be immincludeto Eqn. (42) is physically justified in the sense that a mould have to be applied to the tubing to move the neutraany hydbut inc buckling.Even though both LE,P L include hypothetical elastic strain, the differenceovement L effected by the packer.The actual restorthat mo he results of this method may sometimes be surprising.Fpressur sults in an xtremely compressive FE,P which hydraulically bucklepacker position can be greater inNediately obvious.LE,P is a hypothetical length change associated with FE,P, but d in it is the actualtubing movement due to hydraulic buckling.Substitution of FE,P in echanical force of magnitude FE,P w l point to the packer and remove raulic buckling.Likewise, L is a hypothetical length change associated with FE,0, luded it is the final actualtubing movement due to and between them is the restoring tube m ing force FTP is what accomplishes vement. T or example, suppose high internal e exists in tubing fixed at the packer.The high internal pressure ree s the tubing.The force required to move the neutral point to the magnitude than the tensile force FTP required to virtually restore the tubing to packer depth.In this case L (96) 23. Radial stress just above the packer on the innerEqns.(61) and (62): r o oP and outer tubing surfaces is given by , r i iP,= = (97) 24. Hoop stress just abov g surfaces is givenEqn 6e the packer on the inner and outer tubins.( 4) and (65): 2 2 2, 2 22 2, 2 2( ) 22 (i i o o oh io ii i o i oh oo iP r r P rr rP r P r rr r2)+ = +=(98) byEqn.(66) and r tubing surfaces: 25. The von Mises combined stress just above the packer is givencomputed by taking the maximum stress at the outer and inne2 2 21( ) ( ) ( ) 2VM a r a r a r = + + The packer-to-casing force is given by Eq(99) 26. n.(69): ( ) ) (PC TP i oF F P P AID pA (100)= + 7. ome example scenarios are presented below to illustraterovided above.In order to assist in verification of the con onsider a vertical well with completed with 7 OD, 3lb/f d w3.2revent motion of the tubing in either upward or downward dnumerical values are assumed:casing ID - 6.094, tubing OD - 2.875, tubing ID - 2.441. he tubing is assumed to possess the following material chathermal expansion coefficient, C: Possions ration, : 6.9 10-si 0.30The ided above: ro:1.438 ri:1.221 Ao:6.492 in2 iAs:1.812 in2 Ap:8.296 in2 cI:1.611 in4 w:6.50 lbs/ft For initial conditions at the time of installation of tubing and packer, we suppose that the well was dead with a column of 30 API crude equivalent.For the tubing and annulus we assume a coPo,Sare thu Temperature at surface is taken to be an ambient temperature of S =70 F and temperature at packer depth is TL =1 SceA cfluid is displaced with a 15 lb/gal slurry which is pumped with a pressure of Pi =5000 psi at surface.A surface pressure of Pi =1000 psi is maintained on the annulus during the peration. The pumping operation results in a cooling effec 65 F and TL =115 F. Note that effect of fluid flow anddrop is disregarded which in this case gives a more conservative result. Theo: i:0.0333 psi/ft b:7.680 lb/ft Pi,S:5000 psi Tavg:-20 F Pi: The resultinpacker is then FR =-30,712 lbs.Effective force at the packet packer are FE,P =FE,P =-66,302 lbs. Example Calculations Sand verify the analytical solutions p alculations, the scenarios are based examples formulated in the frequently cited paper by Lubinski. C 2 lb/ft casing.Suppose that 2.875 6.5 t workover tubing is to be installe ith a packer.The packer has a seal bore diameter of 5 and is installed at a depth of 10,000.The packer is assumed to be fixed, so as to p irections.The following exact T racteristics: 6 1/F Modulus of elasticity, E: 3.0 10 7 p following intermediate calculation values result from the inputs provA :4.680 in2 r :1.610 s nstant fluid density of 7.315 lbs/gal.Initial surface pressures in tubing and annulus are =0 psi and Pi,S =0 psi.Based on simple hydrostatics, the pressures at the packer depth s Po =Pi =3800 psi.T 50 F. nario A ement squeeze operation is performed though the tubing.In this case the tubing inner o t so the final temperatures are Ts =the resulting frictional pressure following intermediate calculation values result: 0 psi/ftPw12792 psio,S:1000 psiPo:4800 psi g real force at the packer is F=-37,598 lbs.The change in real force at the p r and change in effective force a Thus the basic components of tubing movement may be calculated as follows: L1 =-5.650 ft ngthening of the tubing of L=13.762 ft is effected by the packer which means a tensile ard force will be applied b ng FE89) where the coefficients of the force/length-10 and 0-4, we calculate a theoretical movement induc bing-to-packer force FTP =54,610 lbs. hat the tubing is completely in tension due to the tensile effect of the tubing to s s of stress just above the packer are as follows: VM,iasing forlbs. L2 =-3.835 ft L3 =-2.897 ft L4 =-1.380 ft Thus, if the packer were to allow free movement, the total tubing length change would have been L =-13.762 ft.Since L

(110) ed magnitude of FR is real force FR,X.The relationship between FE and FR is given byThis will be the case regardless of changes in the hydrostatic pressure profile if the combindensity of the tubing and internal fluid is greater than the density of the annular fluid.Any change in pressure affecting G orH will have a compensating effect on J or K.Using the vector FE allows us to reach this conclusion without knowing specific values of G and H.The magnitude of FE is the effective force FE,X defined in the main text, and likewise the E RF F M L = + + (111) Eqn.(111) is the equivalent vector equation which justifies Eqn.(16) in the main text, since||L|| =Pi,X Ai and ||M|| =Po,X Ao and L and M are oriented in opposing directions. If the tubing string is set in a packer, the end condition on the tubing bottom is different than r freely suspended string.As discussed in the main text, the tu a real force Fp given by Eqn(31).Once again, it is required tother the net effect of all forces acting on the tubing result in straightening or buckling of e tubing at some point O, a distance X above the packer (Figu pfo bing is subject at the packer todetermine in general wheth re 17a). The vector sum of all forces acting on the section of tubing below the position O is given by: R SF W G H F = + + +

(112) s with the freely suspended tubing, buckling or straight A ening at O is determined by the resultant moment at O and the magnitude and orientation of forces G and H is unknown: ( ) ( ) ( ) ( ) ( )O R O S O O O pM F M W M G M H M F = + + +

(113) nce again to avoid a direct calculation of the developypothetically cut-away the tubing at the cross-sectionl EO ed pressure forces G and H, we hat O.Also we ignore the packer and assume the tubing is plugged.This means that the tubing is now completely immersed in the annular fluid.Now as shown in Figure 17b, the tubing is subjected to several new hypothetical pressure forces.J and K are the hypothetical forces from annular and internal pressure acting on the tubing bottom.These forces are hypothetical since the bottom is in reality isolated from the annular fluid and it is not plugged.L and M are again hypotheticapressure forces acting at the cross-section at O due to the assumed cut-away. The new vector sum is again denoted by F : L ME SF W G H J K = + + + + + + (114) Reasoning as before, the resulting moment of FE about O associated is thus given by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) (O E O S O O O O O O)M F M W M G M H M J M K M L M M = + + + + + + (115) H Fp G FR FE Figure 17aFigure 17b G H WSWS O OJKL M Applying the principle of Archimedes to the immersed body of the tubing section below the position O, the we can again form the resultant forces B and Wi: B G J L = + + (116 iW H K M) = + + (117) Eqn(115) may be restated in the equivalent form ( ) ( ) ( ) ( )O E O S O O iM F M W M B M W = + + (118) or ( ) ( )O E O BM F M W = (119) We can now re-express G and H in terms of these theoretical force vectors: G B J L = (120) iH W K M = (121) If we substitute Eqns.(120) and (121) into Eqn.(113) and observe again that the moments ofL and M about O are nil, then we get that ( ) ( ) ( ) ( ) ( )O R O S O O i O pM F M W M B M W M F J K = + + +

(122) or ( ) ( ) ( ), O RMO E O E PF M F M F = +

(123) where , E P PF F J K = (124) defines a new hypothetical force vector acting at the packer. ine buckling, we needed to know the mirection of G and H which is impossible directly.However, Eqn.(123 a ng?le to apply the simple yet powerful Archimedes principle and ereby avoid calculation of important real forces which cannot namely the lateral pressure forces acting on the buckled tubing.Furthermore, the tubing will What is the significance of Eqns.(123) and (124)?Initially, to calculate the moment of realforces at O in Eqn.(113) and determ agnitude and d ) says that we can reach the same conclusion by using the effective force FE which is easily calculated andmodified packer force FE,P which includes the forces J and K which, although hypothetical, are easily determined in both magnitude and direction. Eqn.(124) is the equivalent vector equation which justifies Eqn.(34) in the main text and which is true in general for all combinations of packer seal bore diameter and tubing outer and inner diameters. Why make hypothetical assumptions that incorporate forces not really acting on the tubiBecause by doing so we are abth be determined directly, indeed actually buckle or straighten as if it were subject to these hypothetical forces.