Copyright 2007, SPE/IADC Drilling Conference This paper was prepared for presentation at the 2007 SPE/IADC Drilling Conference held in Amsterdam, The Netherlands, 2022 February 2007. This paper was selected for presentation by an SPE/IADC Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers or International Association of Drilling Contractors and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the SPE, IADC, their officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers and International Association of Drilling Contractors 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 where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A., fax 1.972.952.9435.

Abstract A new type of tubing completion in use in the North Sea is a fixed packer with an expansion joint several joints upstream of the packer. The expansion joint installation may be either pinned closed, or sheared and spaced out after packer installation. The joint may or may not have a stop to prevent jump-out. If the joint is pinned close, then a shear rating must be specified for the shear pins. Additional information necessary for expansion joint analysis includes the joint stroke length, joint seal bore diameter, and installation space out, if sheared. There are distinct differences in the analysis of expansion joints compared with conventional packer installations. This paper details the tubing movement and stress calculations for both pinned and sheared expansion joints. The pinned joint is designed to fail at a specified tubing load, either compressive or tensile. After pin shear, the tubing has free movement until the joint closes, jumps-out, or is restrained by a stop. The sheared joint movement calculation is distinct from conventional tubing movement calculations because there are two piston loads, instead of only one in a free packer. The tubing below and above the joint may buckle. Bending as a result of buckling may cause binding and friction loads in the expansion joint. This paper presents several example cases that give insight into the potential benefits and problems of expansion joints in comparison with conventional tubing completions. Introduction Expansion joints have been in common use in steam injection wells to accommodate thermal expansion loads, for example, Leutwyler (1965) and for more modern practice, Brunings (2005). The use of expansion joints instead of packers or PBRs in North Sea well completions is a relatively new concept, for example, Gunnarsson (1994). The primary

motivation is the robustness and accessibility of the seals. With an expansion joint in the tubing string, you can pull the top 2/3 of the string with the seals in the upper (female) part of the joint. It is then relatively easy to replace these seals. The male polished stub remains in the well and, if needed, is easy to clean up. If its ruined, you only have to unscrew from the packer to get the final part of the tubing string, run a new assembly back in, and latch up again. Also, the expansion joint in the string goes in pinned. After you run it, you can shear the pins and then seat the tubing hanger with minimal space out issues. With a PBR, the seals are bigger, more costly, and you have to pull the entire string to get at them. Cleaning up inside a PBR is dodgy at best, and space out using pups in the string is difficult. What is special about the analysis of tubing movement with expansion joints? Tubing movement was first studied in detail in the classic paper by Lubinski (1962). This paper considered free and fixed packers in a vertical wellbore. Many of the calculations, such as ballooning and buckling, were tied to this vertical well assumption. First, we want to consider these effects in more generality. Second, the expansion joint introduces some new considerations. Expansion joints are run pinned. The pinned joint is designed to fail at a specified tubing load, either compressive or tensile. After pin shear, the tubing has free movement until the joint closes, jumps-out, or is restrained by a stop. The sheared joint movement calculation is distinct from conventional tubing movement calculations because there are two piston loads, instead of only one in a free packer. Both the tubing below and above the joint may buckle. Each of these conditions will be considered in the following analysis. Movement Calculations with Expansion Joints In order to focus on the special case of expansion joints, the basics of tubing movement calculations have been placed in three appendices. Appendix A summarizes tubing movement calculations, as in Lubinski (1962) and Hammerlindl (1977), but reformulated for more general load distributions and tubing configurations. Appendix B summarizes tubing buckling results necessary for tubing movement calculations. Appendix C summarizes the basics of curved wellbores, which are needed in Appendix B. There are four components of tubing movement identified by Lubinski:

1. L1 Hookes Law length change

SPE/IADC 105067

Tubing Buckling Analysis With Expansion Joints Robert F. Mitchell, Halliburton DEDS

HVan7Highlight

2 SPE/IADC 105067

2. L2 Buckling length change 3. L3 Pressure ballooning 4. L4 Thermal expansion

The Hookes Law component contains two effects: pressure boundary loads produce L1p, and, for lack of a better term, compatibility loads produce L1f. For fixed packers, we must find the load that satisfies the displacement compatibility conditions. In Lubinski (1962), that condition is L equal to zero. For more complex problems, such as the use of expansion joints, new compatibility conditions must be defined. There are four distinct configurations for expansion joint analysis:

1. Pinned joint 2. Spaced out joint 3. Closed joint 4. Restrained joint

In case 1, the tubing loads are carried by the shear pins, and we must know the magnitude of these loads to properly size, or evaluate the design. To determine these loads, we must know the bore of the expansion joint. The force balance needed to evaluate the shear pin force is illustrated in Figure 1. We are looking at the male end of the joint, and summing forces:

)AA(p)AA(pFF oboibiasp += . . . . . .(1) where Fsp is the total shear pin force, pi is the internal pressure, Ai is the tubing internal area, po is the annulus pressure, Ao is the total cross-sectional area of the pipe, and Ab is the bore area. How the force is distributed among the shear pins will have to be determined by the manufacturer. The buckling analysis of this scenario is simply the buckling analysis described by Lubinski (1962) or Hammerlindl (1977) for fixed packers, with the exception of using a more general buckling model (Appendix B). The second case has the tubing free to move within the expansion joint, so the compatibility loads vanish. The boundary condition pressure load for the male end can be determined from equation (1), with Fsp set to zero. The force equation for the female end is the same, assuming that the pressures do not vary much over the length of the joint, so that external pressure-area effects will cancel. There are now two tubing sections to analyze, the section upstream from the joint to the fixed packer, and the section downstream from the joint to packer or slips above. Each section must be analyzed separately, and the net length change compared to the motion available in the joint. If the motion is closing, and exceeds the space-out, then the joint will close and we must perform a type 3 calculation. If the motion is opening and exceeds the remaining stroke, then the joint will open, or be restrained by a stop. If restrained, we must perform a type 4 calculation. The third case has the joint fully closed, so that tubing loads are carried by the expansion joint. In this case, the displacement constraint is:

sodu LLL =+ . . . . . .(2)

where Lu is the length change of the upstream section, Ld is the length change of the downstream section , and Lso is the joint space-out. To close the expansion joint, the net tubing length change must be positive. The compatibility load, a compressive load applied to both strings, must produce displacements that satisfy this constraint. Finally, in case 4 we consider a joint that has a stop to prevent jump-out of the tubing. In this case, the displacement constraint is:

stsodu LLLL =+ . . . . . .(3) where Lst is the stroke of the expansion joint. To pull out, the net tubing length change must be negative (shorter), so the right hand side of equation (3) must also be negative. Finding the Compatibility Load In case 3, we must find the value of Fa that satisfies equation (2) or in case 4, we must solve equation (3). Once we have determined L1p, L2, and L3 for both upstream and downstream segments, and computed an initial value of L4, we can use equation (A-12) to compute an initial guess for Fa. You may need, at most, to double this estimate to bracket the solution. After the solution is bracketed, an efficient method for refining Fa, such as Ridders method (Press, 1992, pp 351-352) will give the required answer. Sample Calculations The following conditions were used for a sample calculation of axial forces on an expansion joint set above a fixed packer: - The wellbore is vertical - The packer is set mechanically - The expansion joint is set 95 meters above the packer. - The expansion joint is sheared and spaced out. (Free

movement, meaning that seal stem is not influencing the forces working on the female expansion joint)

- A 7 29 ppf tubing is used in the calculations. - Tubing length is 2550 meters - Annulus ID is 12.5 inches to 1750 meters, and 9.76 inches

to 2550 meters. For comparison, this same problem is posed without an expansion joint and with a PBR replacing the fixed packer, which would be the alternative packer installation. The load cases considered in the sample problem are shown in Figures 2-4. Figure 2 shows the tubing pressure for the five load cases

1. Running and Packer Setting 2. Tubing Pressure Test 3. Annulus Pressure Test 4. Injection 5. Production

In Figure 2, the Annulus Pressure Test has the same tubing pressure as the Running condition. Figure 2 shows the annulus pressure distribution, which only varies from the nominal hydrostatic pressure for the Annulus Pressure Test. Figure 4 shows the tubing temperatures for worst case injection and production, along with the undisturbed temperature which was used for the running, setting, and pressure test conditions.

SPE/IADC 105067 3

Figure 5 shows the tubing installation loads for the expansion joint and for the PBR. The expansion joint case shows lower axial forces. This is because the boundary condition in each case is pressure driven, but the pressure is evaluated and applied at different depths, the PBR case at the total tubing depth, the expansion joint 95 meters higher. When we look at the triaxail loads in Figure 6, we see that the triaxial loads are zero at the points of pressure application, at total depth for the PBR, and at the expansion joint location. Figure 7 shows the pressure test loads. Unsurprisingly, the loads are simply translated according to the applied pressures. The triaxial stresses, shown in Figure 8, are more interesting. First, internal pressure will cause buckling, which causes a discontinuity at 5742 ft (1750 m) for the tubing pressure cases. The discontinuity is caused by bending stresses developed by buckling, and the change in radial clearance because of the change in annulus diameter. The annulus pressure case does not have the discontinuity because there is no buckling, which is surpressed by the high external pressures. The trend with depth also changes because the impact of the increased pressure on the von Mises stress is reversed for internal and external pressures. Figure 9 shows the impact of injection and production loads on the tubing. The main change in loading is the different temperatures used for injection and production. Because neither the expansion joint nor the PBR close, the temperatures have no effect on the loads. Only the different pressure distributions cause differences in the load results, combined with the different initial conditions. The greater pressures in the injection case produce buckling stresses, as shown in Figure 10. Again, we see a discontinuity in the stresses at the change in annulus inside diameter. The production case does not buckle at this depth, so we see no discontinuity. Tables I and II show the various components of tubing movement for the expansion joint and the PBR, repectively. First, notice that the ballooning and thermal length changes are the same for both cases, because the loading is identical. Second, note that we have 362 feet of buckling when setting the expansion joint, but no buckling for the PBR. The weight of the string between the expansion joint and the fixed packer puts this section of the tubing in compression, resulting in buckling. We also saw this effect in Figure 6. The Hookes law terms, since they are driven by pressure boundary conditions, are different between the expansion joint and the PBR, because the pressures are taken at different depths. These differences in loading also drive the buckling length change terms. In addition, tubing weight buckles the string below the expansion joint. The buckled length terms differ for the same reason. Conclusions and Observations The classic methods of Lubinski (1962) and Hammerlindl (1977) are not general enough to model modern well completions, including the use of expansion joints. This paper outlines the length change equations that must be solved to determine the displacement compatability of the tubing string.

This consists, as in the original Lubinski (1962) paper, of four types of tubing length change:

1. L1 Hookes Law length change 2. L2 Buckling length change 3. L3 Pressure ballooning 4. L4 Thermal expansion

Here these quantities are expressed in a form suitable for efficient numerical calculations. The special boundary conditions appropriate to expansion joints are specified, and suitable numerical solution methods are given. A simple example shows the difference between the loads and stresses in expansion joints compared to a PBR. The most notable difference is the buckling which takes place in the section between the fixed packer and the expansion joint. This section will likely be buckled, exept in cases with high annular pressures. Overall, the loads and stresses do not vary greatly between these two cases, so a successful PBR completion could be converted to an expansion joint completion with confidence. Nomenclature Ab = area of the expansion joint bore, (in2) Ai = internal flow area of the pipe, (in2) Ao = total cross-sectional area of the pipe = Ai+Ap, (in2) Ap = cross-sectional area of the pipe, (in2) Apb = area of the packer bore, (in2) br

= binormal vector bz = z coordinate of the binormal vector I = moment of inertia, (in4) d0 = pipe outside diameter, (in) eB = buckling strain eH = ballooning strain eT = thermal strain E = Youngs elastic modulus, (psi) Em = Youngs elastic modulus of mth section, (psi) Fa = actual axial force in the pipe, (lbf) Fb = buckling force, (lbf) Fb0 = initial value of buckling force, (lbf) FL = critical lateral buckling force, (lbf) Fsp = total shear pin force, (lbf) g = acceleration of gravity, (ft/s2) Mb = bending moment, (in-lbf) Mt = axial torque, (lbf-in) nr

= normal vector nz = z coordinate of normal vector pi = internal pressure of the pipe, (psi) po = external pressure of the pipe, (psi) R = radius of curvature, (in) rv = position vector of wellbore center, (in) rc = radial clearance, (in) ) s = measured depth, (in) s1, s2 = measured depths, (in) tr

= tangent vector tz = z coordinate of the tangent vector u = axial displacement, (in) wc = contact force (lbf/in)

4 SPE/IADC 105067

cw = average contact force (lbf/ft) wbp = buoyant weight of the pipe (lbf/in) wp = weight of pipe in air, (lbf/ft) = coefficient of thermal expansion (F-1) Fa = compatibility force, (lbf) eB = change in buckling strain eH = change in ballooning strain eT = change in thermal strain Fjp= change in axial force due to area change at sj, (lbf) Fj= change in axial force over section j, (lbf) Ld = downstream total length change, (in) L1 = Hookes Law length change, (in) L1f = Hookes Law length change due to force, (in) L1p= Hookes Law length change due to pressure, (in) Lso= expansion joint space out, (in) Lst = expansion joint stroke, (in) Lu = upsteam total length change, (in) L2 = buckling length change, (in) L3 = ballooning length change, (in) L4 = thermal length change, (in) pi = change in internal pressure of the pipe, (psi) po = change in external pressure of the pipe, (psi) = wellbore curvature (in-1) f = dynamic friction coefficient = wellbore trajectory inclination angle i = density of fluid inside the pipe (lbm/in3) o = density of fluid outside the pipe (lbm/in3) = pipe angular deflection = wellbore trajectory azimuth angle = angle between tangent vectors in minimum curvature

calculation, (radians) = derivative with respect to s superscripts + = upstream value = downstream value References Brunings, C., Quijada, W., and Grisoni, J.C. 2005. New

Completion Developments for the Production of Heavy and Extra-Heavy Oil in Eastern Venezuela, SPE/PS-CIM/CHOA97914 presented at the InternationalThermal Operations and Heavy Oil Symposium, Calgary, Alberta, (1-3 November).

Gunnarsson, B., Tnnessen, S.H., and Stensland, J.F. 1994.

Evolution of the Snorre Field Downhole Completion Systems, SPE 28890 presented at the European Petroleum Conference, London, (25-27 October).

Hammerlindl, D. J. 1977. Movement, Forces, and Stresses

Associated with Combination Tubing Strings Sealed in Packers, JPT (February).

He, X. and Kyllingstad, A. 1995. Helical Buckling and Lock-

Up Conditions for Coiled Tubing in Curved Wells, SPEDC (March).

Leutwyler, K. and Bigelow, H. L. 1965. Temperature Effects

on Subsurface Equipment in Steam Injection Systems, JPT (January).

Lubinski, A., Althouse, W.S., and Logan, J.L. 1962. Helical

Buckling of Tubing Sealed in Packers, JPT (June). Mitchell, R. F. 1997. Effects of Well Deviation on Helical

Buckling, SPEDC (March).

Mitchell, R. F. 1999. Buckling Analysis in Deviated Wells: A Practical Method, SPEDC (March).

Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery,

B.P. 1992. Numerical Recipes in Fortran 77, Second Edition, Volume 1, New York City: Cambridge University Press.

Sawaryn, S.J. and Thorogood, J.L. 2003. A Compendium of

Directional Calculations Based on the Minimum Curvature Method, SPE 84246 presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, (October 5-8).

Timoshenko, S. P. and Gere, J.M. 1961. Theory of Elastic

Stability, New York City: McGraw-Hill Publishing Co.

Appendix A: Tubing Equilibrium and Length Change Calculations Tubular forces are determined by pressures, tubular weight, external mechanical forces (e.g. packer loads), and friction. The axial force varies with depth due to the tubular weight and friction: cfpa wcosw)s(F = . . . . . .(A-1) where Fa is the axial force (sign convention: tensile force positive), = d/ds, wp is the tubular weight per foot in air, is the angle of inclination of the wellbore with the vertical, f is the friction coefficient, and wc is the contact force between the tubing and casing. The friction is positive for incremental tubular movement upward, and negative for incremental tubular movement downward (such as landing the tubular). The contact force depends on the buoyant weight of the tubular, wellbore curvature, axial force, plus the effect of buckling. (Caution: in this development, s is measured from the surface. In some papers, s is measured from the bottom of the string.) The frictional force is not easy to calculate because it depends on the load and displacement history of the tubular string. Using elastic relationships, Equation (1) can be expressed:

[ ]{ } cfpBHTp wcosweee)s(uEA = . .(A-2) where: u = axial displacement E = Young's modulus Ap = tubular cross-sectional area

SPE/IADC 105067 5

eT = T , thermal strain eH = -2(piAi - poAo)/EAp = axial strain due to hoop stresses eB = the buckling "strain" in the sense of Lubinski

(1962). and is the coefficient of thermal expansion, T is the temperature, is Poissons ratio, pi is the internal pressure, po is the annulus pressure, Ai is the internal flow area, and Ao is the total cross-sectional area of the pipe. In this analysis we will not consider the frictional forces. We are interested in changes to equation (A-2) resulting from changes in well conditions:

[ ]{ } 0eee)s(uEA BHTp = . . . . . .(A-3) (3) The in equation (A-3) represent change of conditions from the initial state (running, packer setting and slackoff, or cemented conditions) to the final loaded state. If we refer back to equation (1), we can integrate equation (A-3): [ ]BHTpa eee)s(uEAF = . . . . . .(A-4) If we rearrange equation (4) and integrate:

+++= dseeeEAFu BHTpa . . . . . .(A-5) we get an equation for u, the total tubing length change. In the original Lubinski (1962) paper, four types of tubing length change effects were defined:

1. L1 Hookes Law length change 2. L2 Buckling length change 3. L3 Pressure ballooning 4. L4 Thermal expansion

We can identify these terms in equation (A-5):

[ ]

=

==

=

TdsL

dsEA

ApAp2L

dseL

dsEA

FL

4

p

iioo3

B2

p

a1

. . . . .(A-6)

If we assume no initial buckling, no area changes, a straight vertical wellbore, and incompressible fluids, then Lubinskis equations follow. Many modern wells will not satisfy these conditions, so we must consider equations (A-6) in more generality. The easiest terms to calculate are the ballooning and thermal expansion length changes. Given pressure and temperature distributions, the integrals in equation (A-6) need be evaluated only once, by any of several accurate numerical methods (see Press, 1992). A second pressure contribution can be seen in the Hookes law length change term. As Hammerlindl (1977) observed, a tubing string can be modeled as a series of cylinders connected one to another. At the points of connection there will be a change in internal and external area, see Figure (A-1), which will produce a force through fluid pressures:

)AA(p)AA(pF jijii

jo

joo

pj

++ = . . . . . .(A-7) where plus (+) indicates the area seen moving in the positive s direction from the connection at sj, and minus () indicates the area seen moving in the negative direction from the connection, remembering that tension is positive. We can calculate the change in pressure load at the end of the pipe, which will be the change in axial load Fa:

iipboopba p)AA(p)AA(F = . . . . . .(A-8) for tubing sealed in a packer, with packer bore area Apb, and otherwise:

ooia p)AA(F = . . . . . .(A-9) where internal pressure must equal annular pressure. If there are N area changes, there are N+1 sections of constant area. We can calculate the force change in each section backward from the end condition using the force increments given in equation (A-7):

1Nj,F

Nj,FFF

a

N

jm

pmaj

+===

= . . . . . .(A-10)

The Hookes Law length change due to pressure changes is given by:

+=

=

1N

1mmp

mmm

p1 AELFL . . . . . .(A-11)

where Em is the Youngs modulus of section m, Apm is the cross-sectional area of section m, and Lm is the length of section m. Given an initial value of Fa, equation (A-11) need be evaluated only once. If the tubing string is fixed at both ends, we must find an additional force Fa that will satisfy displacement compatibility, that is, L equal zero. The remainder of the Hookes Law length change needed for fixed tubing strings is:

+=

=

1N

1mmp

mm

af1 AELFL . . . . . .(A-12)

The total Hookes Law length change term is then:

f1p11 LLL += . . . . . .(A-13) where only the last term might be unknown. The final length change term is due to buckling. A significant difference is that we cannot use incremental forces to determine the buckling strain (length change per unit length), rather, we need the complete solution to equation (A-2) for both the initial conditions and the final conditions. Given these solutions, we can calculate the buckling force Fb:

ooiiab ApApFF += . . . . . .(A-14) Then the calculation of L2 becomes:

= initialBfinalB2 |dse|dseL . . . . . .(A-15) The initial part of equation (A-15) need be determined only once, but notice that the final part may depend on Fa. Appendix B gives the basic relations of eB as a function of Fb. If we

6 SPE/IADC 105067

know how Fb varies with depth, we can use some of the special cases described in Appendix B, but in general, equation (A-14) must also be evaluated by numerical means. Appendix B: Tubing Buckling Correlations The full equations for buckling in deviated wells are non-linear and sufficiently difficult to be unsuitable for design calculations (Mitchell, 1997). Examination of the results of numerical analysis of these equations (Figure B-1), however, suggest that simple formulas match the numerical results well enough to develop useful correlations for buckling analysis (Mitchell, 1999). Lateral Buckling Criteria Two types of buckling are assumed to take place in wellbores, lateral or snake-like buckling (Figure B-2) and helical buckling (Figure B-3). Initial buckling is assumed to be lateral buckling, with transition to helical buckling at a higher buckling force. The minimum force necessary, i.e. the critical force FL, needed to initiate lateral buckling for a pipe lying in the bottom of a deviated well is given by (He, 1995):

c

cL r

EIw4F = . . . . . .(B-1) where:

222)( zbpbzbpc bwFnww += . . . . . .(B-2) and

gAAww ooiipbp )( += . . . . . .(B-3) Result (B-2) is determined using equation (C-6) from wellbore trajectory parameters described in Appendix C. Helical Buckling Criteria In practice, pipe will transition from lateral to helical buckling at some intermediate value between 2 FL and 2.8FL because of irregularities in the wellbore geometry, and will fall out of the helix for values of Fb less than 2 FL. Only helical buckling is expected for buckling forces greater than 2.8FL. Experimental studies have verified these results, at least qualitatively, and further experimental studies are continuing to investigate this behavior.

Buckling Strain and Length Change The buckling strain, in the sense of Lubinski (1962) is the buckling length change per unit length. The buckling strain is given by the following relationship:

2c2

1B )r(e = . . . . . .(B-4)

For the case of lateral buckling, we must determine the average strain in terms of the maximum strain. The shape of the curve for lateral buckling (Figure B-1) was integrated numerically to determine the following relationship:

( ) 92.Lb08.b2

cB FFFEI4

r7285.e = . . . . . .(B-5) which compares to the helical buckling strain:

b

2c

B FEI4re = . . . . . .(B-6)

To determine the buckling length change Lb, we need to integrate equations B-5 and B-6 over the appropriate length interval:

= 2s1s

B2 dseL . . . . . .(B-7)

where measured depths s1 and s2 are defined by the distribution of the buckling force Fb. For that case of constant force Fb, such as in a horizontal well, equation B-7 is easily integrated:

=2s1s

BB Ledse . . . . . .(B-8)

The second special case is for a linear variation of Fb over the interval:

0cos)( bbpb FswzF += . . . . . .(B-9) For constant inclination angle , we can now evaluate equation (B-8) by change of variables:

=2s

1s

2Fb

1FbB

bpB dFecosw

1dze . . . . . .(B-10)

For lateral buckling, the following is an accurate approximation for the integration of equation (B-5), with Fb1 equal to FL:

( )[ ]L2bL2bbp

2c

2 F3668.F3771.FFcosEIw4rL

= . . . . . .(B-11) The integration of equation (B-6) gives the familiar Lubinski result for helical buckling:

[ ]21b22bbp

2c

2 FFcosEIw8rL = . . . . . .(B-12)

Appendix C: Calculating the Wellbore Trajectory The normal method for determining the well path )(srv is to use some type of surveying instrument to measure the inclination and azimuth at various depths and then to calculate the trajectory. At each station, inclination angle and azimuth angle are measured, as well as the course length s between stations. These angles have been corrected to true north, if a magnetic survey, or for drift, if a gyroscopic survey. The survey angles define the tangent t

v to the trajectory at each station, where

the tangent vector is defined in terms of inclination and azimuth in the following formula: { })cos(,)sin()sin(,)sin()cos(t =v (C-1)

SPE/IADC 105067 7

The method most commonly used to define a well trajectory is called the Minimum Curvature method (Sawaryn, 2003). In this method, we connect two tangent vectors with a circular arc, as illustrated in Figure C-1. In this Figure we have a circular arc of radius R over angle , connecting the two tangent vectors 1t

v at measured depth s1,

and 2tv

at measured depth s2. The arc length is R = s2-s1 = s. Notice that the angle is also the angle between the tangents 1t

v and 2t

v. From this we can immediately determine

R:

=== /1)tt(cos/s/sR 211 vv . . . .(C-2) The following equations define a circular arc:

111

1111

1111

11111

bnt)s(b

)]ss(cos[n)]ss(sin[t)s(n)]ss(sin[n)]ss(cos[t)s(t

r)]}ss(cos[1{Rn)]ss(sin[Rt)s(r

vrrrvvv

rrvvrv

==+=

+=++=

. . . . . .(C-3) The vector 1r

v is just the initial position at s = s1, and = 1/R

is the curvature. The vector 1tv

is the initial tangent vector. The vector 1n

v is the initial normal vector. If we evaluate equation (C-3)b at s = s2, we find:

2112 tssinnscost)s(tvvvv =+= . . . . . .(C-4)

which we can solve for 1n

v:

)scot(t)scsc(t)ssin(

)scos(ttn 12121 == vv

rvv

. . . . . .(C-5) The vertical components of equation (C-3) are used in equation (B-2).

1bnt

ibi)nt()s(b

inn

itt

)]ss(cos[n)]ss(sin[t)s(n)]ss(sin[n)]ss(cos[t)s(t

2z

2z

2z

z1z11z

z1z1

z1z1

1z11z1z

1z11z1z

=++==

==

+=+=

vvvrrvv

vv

. . . . . .(C-6)

Ab

Fa

Fsp

Pi

Po

Ai

Ao

C/L

Figure 1: Shear Pin Force Balance

Figure 2: Tubing Pressures

0

1000

2000

3000

4000

5000

6000

7000

8000

90000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

pressure psi

dept

h ft

RunningPressure TestInjectionProduction

Figure 3: Annulus Pressures

0

1000

2000

3000

4000

5000

6000

7000

8000

90000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

pressure psi

dept

h ft

Running, Injection, ProductionPressure Test

8 SPE/IADC 105067

Figure 4: Tubing Temperatures

0

1000

2000

3000

4000

5000

6000

7000

8000

90000 50 100 150 200 250

Temperature F

Dep

th ft

Running, Pressure testsInjectionSeries3

Figure 5: Installation Loads

0

1000

2000

3000

4000

5000

6000

7000

8000

9000-50000 0 50000 100000 150000 200000 250000

Axial Load lbf

Dep

th ft

Running EJ SettingPBR Setting

Figure 6: Installation Triaxial Loads

6500

6700

6900

7100

7300

7500

7700

7900

8100

8300

85000.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

von Mises stress ksi

Dep

th ft

Series1EJ SettingPBR Setting

Figure 7: Pressure Test Loads

0

1000

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9000-150000 -100000 -50000 0 50000 100000 150000 200000 250000 300000

Axial Force lbf

Dep

th ft

EJ Tubing PressurePBR Tubing PressureEJ Annulus PressurePBR Annulus Pressure

Figure 8: Pressure Test Triaxial Stress

0

1000

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900030 35 40 45 50 55 60 65 70

von Mises Stress psi

Dep

th ft

EJ Tubing PressurePBR Tubing PressureEJ Annulus PressurePBR Annulus Pressure

Figure 9: Injection and Production Loads

0

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9000-100000 -50000 0 50000 100000 150000 200000

Axial Force lbf

Dep

th ft

EJ InjectionPBR InjectionEJ ProductionPBR Production

Figure 10: Injection and Production Triaxial Stress

0

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900010 15 20 25 30 35 40

von Mises Stress ksi

Dep

th ft

EJ InjectionPBR InjectionEJ ProductionPBR Production

SPE/IADC 105067 9

Aoj+

Fpj

Pi

Po

Aij-

Aoj-

Aij+

Figure A-1: Force Due to Area Change

C/L

sj

Figure A-2: Packer Force Boundary Condition

Ao

Po

Pi

Ai

Apb

Packer

Fa

C/L

Figure B-2: Lateral Buckling Deformation

Figure B-3: Helical Buckling Deformation

10 SPE/IADC 105067

Figure C-1: Inclination and Azimuth

iN

iE

iZ

t

t1

t2

s

R

R = 1/

t1

t2

Figure C-2: Circular Arc

Figure B-1: Buckling in Deviated Wells

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

0 0.5 1 1.5 2 2.5

Deviated WellCorrelation

2 5.

FFp

Lubinski

SPE/IADC 105067 11

Table I: Expansion Joint Tubing Length Change

Hookes Buckled Law Buckling Ballooning Thermal TOTAL Length LOAD CASE (ft) (ft) (ft) (ft) (ft) (ft) Setting -0.31 0 0 0 -0.31 362 Tubing P -2.69 -0.93 -2.73 0 -6.35 8298 Annulus P 0.65 0 3.5 0 4.15 0 Injection -1.79 -0.34 -1.87 -4.17 -8.18 6430 Production -1.06 -0.05 -0.91 4.81 2.79 3212

Table II: PBR Tubing Length Change

Hookes Buckled Law Buckling Ballooning Thermal TOTAL Length LOAD CASE (ft) (ft) (ft) (ft) (ft) (ft) Setting 0 0 0 0 0 0 Tubing P -2.38 -0.85 -2.73 0 -5.96 8298 Annulus P 0.96 0 3.5 0 4.46 0 Injection -1.49 -0.29 -1.87 -4.17 -7.82 6031 Production -0.75 -0.04 -0.91 4.81 3.11 2841