Tubing Stress and Movement Calculations

Basic Calculations for Packer 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, δL (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 δL* 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 = δL* - δL.

L

δL

δL* ∆L

Figure 1a

Figure 1b

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) Poisson’s 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 ∆L = ∆L1 + ∆L2 + ∆L3 + ∆L4 where ∆L1 = ∆L2 = ∆L3 = ∆L4 =

change in stretch due to elastic strain change in stretch due to helical buckling change in stretch due to Poisson's effect change in stretch due to temperature change

(1)

Movement from Elastic Strain ∆L1 According to Hooke’s Law for an elastic material, the strain ε or relative length change is proportional to the axial stress σa imposed. This proportionality is valid as long as the stress remains within the elastic range of the tubing material. The constant of proportionality is Young’s modulus of elasticity E so that:

σ a = Eε = E

δL L

(2)

Tubing axial stress σa is total axial force FR applied to the cross-sectional area AS:

σa =

FR AS

(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: ∆L1 = δ L1* − δ L1 =

∆FR L E AS

(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 FR = wS 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

FR = wS L − ( Ao − Ai ) Po

(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): FR = wS L − Po Ao + Pi Ai

(7)

Likewise, the axial force at any distance X above the bottom is given by:

FR , X = ws X − Po Ao + Pi Ai

(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: ∆FR = ∆Pi Ai − ∆Po 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 ρo and ρi and pressures Po,X and Pi,X at some distance X above the tubing bottom:

Po = Po , X + ρo X

(10)

Pi = Pi , X + ρ i X

(11)

Combining Eqns (8), (10)and (11) results in the following for axial force at position X:

FR , X = ( wS − ρ o Ao + ρ i Ai ) X − Po , X Ao + Pi , X Ai

(12)

Change in total axial force can be calculated in terms of density and surface pressures:

∆FR = ( ∆ρi Ai − ∆ρo Ao ) L + ∆Pi ,S Ai − ∆Po ,S Ao

(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

L Pi

Po Figure 2a

X

Po Figure 2b

Movement from Helical Buckling ∆L2 In 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, δL2 (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.1,2 Fluid 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 magnitude of -PoAo without buckling. (see Appendix for rigorous discussion)

Po,X

x δL2

F = PoAo

F = PoAo

F Figure 3a

Figure 3b

Figure 3c

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:

FR ,0 = − Ao Po + Ai Pi

(14)

So, effective force at the bottom by the tubing is converted from FR,0 thus:

FE ,0 = FR ,0 + Ao Po − Ai Pi

(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 :

FE , X = FR , X + Po , X Ao − Pi , X Ai

(16)

Combining Eqns. (12) and (14) results in the following complete expression for effective force:

FE , X = ( wS − ρ o Ao + ρ i Ai ) X

(17)

The terms ρoAo and ρiAi 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 wB = wS − wo + wi ,

(18)

then a simplified expression results for FE,X :

FE , X = wB X

(19)

Eqn. (19) implies that FE,0 = 0 at bottom. The total effective force at the top of the tubing string is then: FE = wB 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:

FVHB = −

8EIwB rc

(21)

where rc is the radial clearance between the tubing and inner casing diameter. The period or pitch of the helix, λ, varies with FE according to the following equation:

λ = 2π

2 EI , FE < FVHB . FE

(22)

For a short section of length l above the packer in which axial force remains relatively constant, change in length is given by: ⎡ ⎛ 2π r ⎞ 2 ⎤ c ⎥ δA = l ⎢ ⎜ 1 1 + − ⎟ ⎢⎣ ⎝ λ ⎠ ⎥⎦

(23)

Since FE and λ 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 ∆L2 = δ L*2 − δ L2 = −

F 2 rc2 , F < FVHB 8EIwB

(24)

In this case, the distance to the neutral point (i.e. the length of buckled tubing) is: n=

F wB

(25)

Movement from Poisson’s Effect ∆L3 The presence of fluid pressure in the tubing and annulus results in another type of strain known as Poisson’s Effect, or “ballooning.” Fluid pressure acting laterally on the inner and outer tubing surface induces radial and hoop stresses, σr and σh. 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 Lame’s equations for thick-walled cylinders. Integrating Lame’s equations across the tubing wall results in the total Poisson strain, ε3:

ε3 =

2ν ( Po Ao − Pi Ai ) ( Ao − Ai ) E

(26)

where ν = 0.3 is the accepted Poisson’s ratio for steel and where Po and Pi are the average external and inner fluid pressures along the tubing length L. The tubing stretch from the Poisson effect, δL3, is calculated from the preceding strain ratio thus:

δ L3 = ε 3 L =

2ν L ( Po Ao − Pi Ai ) ( Ao − Ai ) E

(27)

Since Eqn (27) is based on average fluid pressures, then the calculation of δL3 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 Poisson’s effect is then: ∆L3 = δ L*3 − δ L3 =

2ν L ( ∆Po Ao − ∆Pi Ai ) ( Ao − Ai ) E

(28)

Movement from Temperature Change ∆L4 Often 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:

δ L4 = C (T − TS ) 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:

∆L4 = δ L*4 − δ L4 = 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).

Ao

Ao

Ao

Ai

Ai

Ai

Po Po

Po Pi

Pi

Pi

Ap

Ap

Ap

Figure 4a

Figure 4b

Figure 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:

Fp = Po ( Ap − Ao ) − Pi ( Ap − Ai )

(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:

∆FR = ∆Po ( Ap − Ao ) − ∆Pi ( Ap − Ai ) Referring to Eqn.(13), this same result may be expressed in terms of change in surface pressures and change in fluid density: ∆FR = ( ∆ρo L + ∆Po ,S )( Ap − Ao ) − ( ∆ρ i L + ∆Pi ,S )( Ap − Ai )

(32)

(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):

FE ,P = FP + Po Ao − Pi Ai

(34)

Eqns.(31) and (34) combined yield a simple equation for effective force at the packer, FE,P:

FE ,P = ( Po − Pi ) Ap

(35)

Change in effective force given at the packer is given by: or equivalently by:

∆FE ,P = ( ∆Po − ∆Pi ) Ap

(36)

∆FE ,P = ⎡⎣ ( ∆ρ o − ∆ρ i ) L + ∆Po ,S − ∆Pi ,S ⎤⎦ Ap

(37)

For a vertical well, tube movement from change in fluid pressures or density is given by:

∆L2 = −

( ∆FE ,P ) 2 rc2 , for ∆Pi > ∆Po 8EIwB

(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 which shortens 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 < 0 |∆L| > LPSB

L PSB ∆L > 0

Figure 5a

Figure 5b

Figure 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:

FE ,0 = FE ,P + FTP

(39)

∆L

∆LTP FTP

Figure 6a

Figure 6b

Figure 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: ∆LTP = −∆L + ∆Lno− go (40) Upper tubing shoulder ∆L no-go > 0 (limit up)

∆L no-go < 0 (limit down)

Bottom tubing shoulder Figure 7 Surface Slack-off and Over-pull In some situations additional tubing weight is slacked-off from surface at installation or later. For a packer limiting downward movement, this means that some of the tubing surface weight is transferred down-hole to increase FTP in compression. Likewise, over-pull may be exerted so that more tubing weight is supported at surface, reducing FTP in compression. If the packer limits tubing movement upward, then eventually enough weight may be picked up to exert upward force on the packer so that FTP becomes tensile. Slack-off and over-pull can be specified as length change at surface ∆L5 or a change in weight. The length of tubing slacked off or pulled up is equivalent to a corresponding change in elastic stretch and buckling down-hole. If slack-off/over-pull are given as a force F (i.e. change in weight), the associated length change ∆L5 may be calculated using Eqns(42) -(44) in the next section. The calculation of total tube movement in Eqn.(1) must be modified to account for this additional length change: ∆L = ∆L1 + ∆L2 + ∆L3 + ∆L4 + ∆L5

(41)

In general, once ∆L5 has been included in ∆L, the preceding analysis proceeds as described all ready above.

Analytic Solution for FTP (Vertical Well) As suggested, solving for FTP involves an iterative approach in which the value of FTP is varied until the required tube movement requirements are satisfied. Typically this involves a numerical simulation, especially in the case of deviated wells as will be discussed subsequently. However, in the interest of verifying such numerical simulations, an analytical method developed by Lubinski may be used to calculate FTP in the simple case of a vertical well. Assuming a vertical well, Eqns.(4) and (24) describe the relationship between tube movement and mechanical force applied at the tubing bottom. Eqn.(4) provides a linear relationship between the applied force and movement due to elastic strain. Eqn.(24) provides a non-linear relationship between the applied force and movement due to buckling. Taken together, the resultant tube movement ∆LF is a non-linear function of applied force F:

∆LF = aF 2 + bF

(42)

where

⎧ rc2 , F 0 0, ⎩

(43)

and b=

L E AS

(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. + ∆LF

-F

+F

- ∆LF Figure 8

Lubinski’s 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 for fixed/limited motion packers, proceed as follows: 1) Determine the effective packer pressure force FE,P from Eqns.(37) and (38). 2) Calculate theoretical tube movement ∆LE,P by substituting FE,P for F in Eqn.(42): ∆LE , P = aFE2,P + bFE ,P

(45)

3) Calculate ∆LTP = -∆L from Eqn.(41) for total movement assuming a free packer. 4) Let ξL be the net resultant of ∆LE,P and ∆LTP :

ξ L = ∆LE ,P + ∆LTP

(46)

5) Calculate the FE,0 by solving the quadratic equation in Eqn.(42) with ∆LF = ξL:

−b + b2 + 4aξ L 2a 6) Solve for the restoring tubing-packer force FTP in Eqn.(39): FTP = FE ,P − FE ,0 FE ,0 =

(47) (48)

Note: although this method is straight-forward procedurally, the reasoning behind it may not be immediately obvious. ∆LE,P is a hypothetical length change associated with FE,P, but included in it is the actual tubing movement due to hydraulic buckling. Substitution of FE,P into Eqn. (42) is physically justified in the sense that a mechanical force of magnitude –FE,P would have to be applied to the tubing to move the neutral point to the packer and remove any hydraulic buckling. Likewise, ξL is a hypothetical length change associated with FE,0, but included it is the final actual tubing movement due to buckling. Even though both ∆LE,P and ξL include hypothetical elastic strain, the difference between them is the restoring tube movement ∆L effected by the packer. The actual restoring force FTP is what accomplishes that movement. The results of this method may sometimes be surprising. For example, suppose high internal pressure exists in tubing fixed at the packer. The high internal pressure results in an extremely compressive FE,P which hydraulically buckles the tubing. The force required to move the neutral point to the packer position can be greater in magnitude than the tensile force FTP required to virtually restore the tubing to packer depth. In this case ξL < 0 and FTP < FE ,P with the result that FE,0 < 0. Thus, the tubing will still be buckled above the packer even though both tubing and packer are subject to a tensile contact force.

3. Deviated Well Paths and Friction The calculations discussed thus far have assumed that the tubing and packer installation reside in a vertical well. The assumption of a vertical well allows for simplified analysis and in some cases the derivation of analytic solutions. However, many wells are characterized by some amount of deviation from the vertical. Horizontal wells are also an increasingly common. Inclination and curvature of the well-path change the way in which tubing axial force is transferred between the down-hole end and surface. For a tubing forces calculation, the tubing string is typical divided into calculation segments. The segmentation is done in such a way that the well-bore and tubing geometry are constant for each given segment. Given an initial condition at the down-hole end and segmentation of the string into n discrete sections, incremental effective axial force dFE is calculated for the each tubing segment and then summed over all segments to attain cumulative axial force (see Figure 9). For a segmentation of the string into n subsections, then the total effective axial force at surface is given by: n

FE ,L = FE ,0 + ∑ dFE ,k

(49)

k =1

F E,L k=n k=n-1

k=2

k=1

F E,0

Figure 9 In vertical sections of the well, the only force acting on the tubing segment is that of buoyant weight due to gravity. In this case, the incremental axial force for section k is based on Eqn.(19) applied to the incremental segment length dsi:

dFE ,k = wB dsk

(50)

In a deviated or horizontal section of the well at an inclination angle θ from the vertical, the gravitational weight of the tubing is divided into an axial component and a normal component which generates frictional resistance with the well-bore wall:

dFE ,k = ( wB cos θ ± µ wB sin θ ) dsk

(51)

where the sign is determined by assumed directionality (+ uphole, - downhole) and µ is an empirical coefficient of sliding friction (typically around .25 or .30 for wet casing).

In a curved well section characterized by changing inclination and azimuth, then incremental change in axial force is given by the following non-linear differential equation: dFE ,k = wB cos θ dsk ± µ dFn ,k

(52)

The term dFn,k is the non-linear change in wall contact force normal to the well-bore wall:

dFn ,k =

(F

E ,k −1

sin θ d γ ) + ( FE ,k −1dθ + wB sin θ dsk ) 2

2

(53)

where dθ is change in inclination, dγ is change in azimuth, and θ average inclination. For curved sections axial, this change in wall contact force is non-linear and is referred to as the belt-friction (or “rope” friction) effect. In this case increasing axial force results in increased wall contact friction. Due to the non-linear nature of Eqn.(53), calculation of axial force for curved well sections must be accomplished by numerical techniques. Once again compressive axial force in excess of the helical buckling load causes helical buckling of the tubing. This buckling increases wall contact forces and hence frictional resistance with the well bore wall. At highly compressive force levels, increasing wall contact friction due to helical buckling may eventually exceed any additional force slackedoff from surface, an effect known as helical lockup. This conditions results in a maximum limit on the compressive force that can be applied at the packer. In the case of deviated or curved well bore sections, the helical buckling load is different than for vertical well sections in Eqn.(21). The general helical buckling load is given by the following non-linear equation: 2 EI 4 ⎛ dγ ⎞ ⎛ dθ ⎞ + wB sin θ ⎟ ⎜ FHB sin θ ⎟ + ⎜ FHB rc ⎝ ds ⎠ ⎝ ds ⎠ 2

FHB = − 2

2

(54)

For helically buckled tubing, normal wall contact force increases by the following term: dFnhb,k

rc FE2,k = 4 EI

(55)

Including Friction Effects: An important question in regard to the above discussion is whether frictional “drag” effects should be included at all. It can be argued that over long periods of time various well bore conditions such as vibration due to fluid flow act to mitigate the frictional contact. Thus for long-term analysis friction effects may be ignored, i.e., µ = 0. However, in some situations it is possible that frictional effects cannot be ignored. Clearly this is the case for tripping of the completion string into the well. In the case of an initial minimum requirement for compression applied to the packer, helical lockup due to friction effects as mentioned above must be considered. If friction effects are to be included in calculation of axial force, then a directionality of movement or attempted movement must be specified. A simplified approach is to assume directionality based on the final slack-off or overpull direction at installation. In a completely realistic simulation of forces and movement, it is possible that various tubing segments may attempt to move in opposite directions (downhole vs. uphole). This is perhaps the case in certain situations with localized thermal effects in which axial loads are not uniformly distributed.

4. Fluid Flow Effects The calculations discussed thus far have assumed only hydrostatic conditions for the fluids in the tubing and annulus. It has been shown that fluid flow in the annulus results in fluid shear drag on the tubing which changes effective axial force and can affect buckling of the tubing. Internal fluid drag on the other hand does not change effective force or buckling of the tubing. For tubing installed a packer there is typically no significant annular fluid flow. However, well conditions and operations over the service life of the production tubing often involve internal fluid flows. The most important cases are production of reservoir fluids and stimulation and hydraulic fracturing operations. High flow rates inside the tubing can result in a significant change in real axial force and elastic stretch of the tubing. Diffential real force due the fluid shear drag is given by: dF f = ∆Pf Ai ds

(56)

where ∆Pf is the internal frictional pressure drop for the current tubing section. For a vertical well, the analytical solution should use the average of this force over the tubing length: ∆FR , f = ∆Pf Ai L 2

(57)

5. Safe Operating Limits After tubing movements have been calculated, resultant tubing forces and stresses and any forces induced on the packer need to be evaluated relative to safe operating limits. Tubing lengths, material grades, and initial slack-off and over-pull at installation must be chosen to accommodate a realstic range of expected operating conditions. Tubing Yield Limits Calculation of tubing movements includes induced forces due to the presence of any pressure differential and movement restrictions at the packer. Once a complete profile of axial force has been determined, the distribution of axial stress σa in the tubing string may be calculated based on Eqn.(3):

σa =

FR AS

(58)

If axial stress is being calculated at a point where the tubing is helically buckled, then the axial stress is increased to account for the resulting bending stress. This bending stress is greatest at the outer tubing surface, so that axial stress becomes:

σa =

FR FE rc ro + , for FE < FHB AS 2I

(59)

As mentioned in the discussion of Poisson’s effect, the tubing is subject to two additional stresses: radial stress σr and hoop stress σh (also known as the tangential stress). These stresses result from fluid pressure in the tubing and annulus. These stresses vary across the wall of the tubing, and so it is usual to calculate stress values at both the inner and outer tubing surface and select the worst case. Their relative orientation for a thick-walled cylinder such as production tubing is shown in Figure 10. σa

σr

σh

σh σa

σr Figure 10

The general Lame equation for radial stress at a radius r from the tubing center is given by:

σr =

ri 2 Pi − ro2 Po ( Pi − Po ) ri 2 ro2 − 2 2 2 ro2 − ri 2 ( ro − ri ) r

(60)

Solving Eqn.(60) at the internal tubing radius ri results in the following radial stress:

σ r ,i = − Pi

(61)

Solving Eqn.(60) at the outer tubing radius ro results in the following radial stress:

σ r ,o = − Po

(62)

The general Lame equation for hoop stress at a radius r from the tubing center is given by:

σh =

ri 2 Pi − ro2 Po ( Pi − Po ) ri 2 ro2 + 2 2 2 ro2 − ri 2 ( ro − ri ) r

(63)

Solving Eqn.(63) at the internal tubing radius ri results in the following radial stress:

σ h ,i =

Pi ( ri 2 + ro2 ) − 2 Po ro2 ro2 − ri 2

(64)

Solving Eqn.(63) at the outer tubing radius ro results in the following radial stress:

σ h ,o =

2 Pi ri 2 − Po ( ri 2 + ro2 ) ro2 − ri 2

(65)

These three primary stresses may be combined into a single composite stress to furnish a yield criterion for onset of tubing yield. The standard accepted formulation for the composite stress is furnished von Mises theory combined equivalent stress:

σ VM =

1 ⎡(σ − σ ) 2 + (σ − σ ) 2 + (σ − σ ) 2 ⎤ r a r a r ⎦ 2⎣ a

(66)

In practice onset of tubing yield is assumed to occur when the von Mises combined equivalent stress exceeds the tensile yield stress of the tubing material. Typically a yield safety factor YSF is included on the order of 80%. Thus the effective yield criterion is:

σ VM ≥ YSF ⋅ σ y

(67)

A standard procedure is used to depict the current stress state of the tubing according to the von Mises yield theory. The von Mises combined stress equation is set equal to the tubing yield stress and solved algebraically. This yields a surface in 3-dimensions. In order to render a 2-dimensional graph, either the internal pressure Pi or external pressure Po is set to a constant minimum or maximum value. The resulting quadratic equation may be solved in terms of the two free variables of real axial force FR and differential fluid pressure Pi - Po. The resulting graph is familiarly known as the “von Mises Ellispe” where typically an inner working curve is given based on YSF (see Figure 11).

YSF Working Curve

Pi – Po > 0, Burst Zone

FR < 0

FR > 0

Pi – Po < 0, Collapse Zone Figure 11

Packer & Casing Safety Limits Given the tubing-to-packer contact force FTP determined iteratively or via Eqns.(45) -(47) and the packer pressure force FP, the real axial force at packer FR,P is given by:

FR , P = FTP + FP

(68)

The real force on the packer should be compared against any maximum rated compression or pull on the packer configuration. In addition, a force is induced on the casing by the packer. This packer-to-casing force FPC is the resultant of the real tubing force on packer and the pressure differential acting on the packer itself. This is given by the following: FPC = FR ,P + ( Po − Pi )( AID − Ap ) where AID is the cross-sectional area of the well bore at the packer depth. The force FPC should be compared against any rated limits of packer-casing contact force.

(69)

6. Summary of Calculations for Vertical Wells In the case of vertical wells, an appropriate sequence of calculations for the various components of tube movement and induced forces and stresses is summarized below. 1. Based on the changes in internal and external pressure and fluid density, calculate change in real axial force, ∆FR, from Eqn.(32) or (33): ∆FR = ∆Po ( Ap − Ao ) − ∆Pi ( Ap − Ai )

(70)

∆FR = ( ∆ρ o L + ∆Po ,S )( Ap − Ao ) − ( ∆ρ i L + ∆Pi ,S )( Ap − Ai )

(71)

2. The real pressure force at the packer after pressure changes is given by Eqn.(31): Fp = Po ( Ap − Ao ) − Pi ( Ap − Ai ) 3. If there is a fluid flow and frictional pressure drop associated with the pressure changes, then calculate additional change in real force from flow by Eqn.(57): ∆FR , f = ∆Pf Ai L 2

(72)

(73)

4. Calculate tube movement from elastic stretch including any effects of fluid flow based on Eqn.(4): ( ∆FR + ∆FR , f ) L ∆L1 = (74) E AS 5. Calculate the change in effective axial force at the packer using Eqn.(36) or (37): ∆FE ,P = ( ∆Po − ∆Pi ) Ap

(75)

∆FE ,P = ⎡⎣ ( ∆ρ o − ∆ρ i ) L + ∆Po ,S − ∆Pi ,S ⎤⎦ Ap

(76)

6. Calculate tube movement from helical buckling by Eqn.(38) using ∆FE,P from Step 4 and the new tubing buoyant weight per foot, wB: ⎧ ( ∆FE ,P ) 2 rc2 , ⎪− ∆L2 = ⎨ 8 EIwB ⎪ 0 ⎩

∆Pi > ∆Po

(77)

∆Pi < ∆Po

7. Calculate tube movement from the Poisson effect using Eqn.(28): ∆L3 =

2ν L ( ∆Po Ao − ∆Pi Ai ) AS E

(78)

8. Calculate tube movement from thermal stretch using Eqn.(30) where temperature is averaged between surface and packer depth, T = (TS + TP ) 2 : ∆L4 = C ⋅ ∆T ⋅ L (79) 9. Sum the tube movement components from Steps 1-7, assuming that the packer allows free movement of the tubing: ∆L = ∆L1 + ∆L2 + ∆L3 + ∆L4

(80)

10. Update total tube movement to include any slack-off or over-pull length from surface: ∆L ← ∆L + ∆L5

(81)

11. Based on the updated internal and external pressures at the packer depth, calculate the effective packer pressure force, FE,P: FE ,P = ( Po − Pi ) Ap

(82)

12. In the case of free tube movement at packer, set FE ,0 = FE ,P and proceed to Step 15. 13. In the case of fixed tubing or limited tubing movement at the packer, determine the net virtual movement to restore the tubing to the packer limit depth as in Eqn.(40): (Note: that for fixed tubing at packer ∆L no-go is 0) ∆LTP = −∆L + ∆Lno− go

(83)

14. For fixed and limited movement packers, determine the tubing-to-packer force according to the procedure involving Eqns.(45) -(47): a. Calculate the tube movement ∆LE,P associated with the effective force at the packer depth: ∆LE , P = aFE2,P + bFE ,P

(84)

⎧ rc2 − , FE ,P < 0 ⎪ a = ⎨ 8 EIwB ⎪ FE ,P > 0 0, ⎩

(85)

b=

L E ( Ao − Ai )

(86)

b. Calculate ξL, the resultant of ∆LE,P and ∆LTP:

ξ L = ∆LE ,P + ∆LTP

(87)

c. Calculate the net effective force at packer:

FE ,0

⎧ −b + b2 + 4aξ L , ξL < 0 ⎪⎪ 2a =⎨ ξL ⎪ ξL ≥ 0 , ⎪⎩ b

(88)

15. Calculate the restoring tubing-to-packer force from Eqn.(39) FTP = FE ,0 − FE ,P

(89)

16. If FE,0 is compressive and exceeds the helical buckling load, calculate the helix period just above the packer based on Eqn.(22):

λ = 2π

2 EI FE ,0

(90)

17. Calculate the distance from the packer to the neutral point, i.e. the actual length of buckled tubing with Eqn.(25): FE ,0 n= , FE ,0 < 0 (91) wB 18. Total effective force is given by the sum of Eqn(20) and FE,0 from Step 12 or 14: FE = wB L + FE ,0

(92)

19. Surface weight is given by FE in Eqn.(92) and any snubbing pressure at surface: Surface Weight = FE − Ao Po ,S

(93)

20. Real surface tension is calculated from FE in Eqn.(92) based on Eqn(16): FR ,S = FE − Ao Po ,S + A i Pi ,S

(94)

21. Real force at the packer is calculated using Eqn.(68) FR ,P = FTP + FP

(95)

22. Axial stress just above the packer is given by Eqns.(58) or (59): ⎧ FR , P FE ,0 rc ro ⎪ A + 2I , ⎪ σa = ⎨ S FR , P ⎪ , ⎪⎩ AS

for FE ,0 < 0

(96) for FE ,0 > 0

23. Radial stress just above the packer on the inner and outer tubing surfaces is given by Eqns.(61) and (62): σ r ,i = − Pi (97) σ r ,o = − Po 24. Hoop stress just above the packer on the inner and outer tubing surfaces is given Eqns.(64) and (65): P (r2 + r2 ) − 2P r 2 σ h ,i = i i 2o 2 o o ro − ri 2 P r 2 − P ( r 2 + ro2 ) σ h ,o = i i 2 o 2i ro − ri 25. The von Mises combined stress just above the packer is given by Eqn.(66) and computed by taking the maximum stress at the outer and inner tubing surfaces:

σ VM =

1 ⎡(σ − σ ) 2 + (σ − σ ) 2 + (σ − σ ) 2 ⎤ r a r a r ⎦ 2⎣ a

(98)

(99)

26. The packer-to-casing force is given by Eqn.(69): FPC = FTP + ( Pi − Po )( AID − Ap )

(100)

7. Example Calculations Some example scenarios are presented below to illustrate and verify the analytical solutions provided above. In order to assist in verification of the calculations, the scenarios are based on examples formulated in the frequently cited paper by Lubinski. Consider a vertical well with completed with 7” OD, 32 lb/ft casing. Suppose that 2.875” 6.5 lb/ft workover tubing is to be installed with a packer. The packer has a seal bore diameter of 3.25” and is installed at a depth of 10,000. The packer is assumed to be fixed, so as to prevent motion of the tubing in either upward or downward directions. The following exact numerical values are assumed: casing ID - 6.094”, tubing OD - 2.875”, tubing ID - 2.441”. The tubing is assumed to possess the following material characteristics: thermal expansion coefficient, C: 6.9 × 10-6 1/F° Modulus of elasticity, E: 3.0 × 107 psi Possion’s ration, ν: 0.30 The following intermediate calculation values result from the inputs provided above: ro: 1.438” rc: 1.610” Ai: 4.680 in2 I: 1.611 in4 r i: 1.221” As: 1.812 in2 Ao: 6.492 in2 Ap: 8.296 in2 ws: 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 constant fluid density of 7.315 lbs/gal. Initial surface pressures in tubing and annulus are Po,S = 0 psi and Pi,S = 0 psi. Based on simple hydrostatics, the pressures at the packer depth are thus Po = Pi = 3800 psi. Temperature at surface is taken to be an ambient temperature of TS = 70 F° and temperature at packer depth is TL = 150 F°. Scenario A A cement squeeze operation is performed though the tubing. In this case the tubing inner fluid 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 operation. The pumping operation results in a cooling effect so the final temperatures are Ts = 65 F° and TL = 115 F°. Note that effect of fluid flow and the resulting frictional pressure drop is disregarded which in this case gives a more conservative result.

The following intermediate calculation values result: ∆Po,S: 1000 psi ∆ρo: 0 psi/ft ∆ρi: 0.0333 psi/ft ∆Pi,S: 5000 psi wb: 7.680 lb/ft ∆Tavg: -20 F°

Po: Pi:

4800 psi 12792 psi

The resulting real force at the packer is Fp = -37,598 lbs. The change in real force at the packer is then ∆FR = -30,712 lbs. Effective force at the packer and change in effective force at packer are FE,P = ∆FE,P = -66,302 lbs.

Thus the basic components of tubing movement may be calculated as follows: ∆L1 = -5.650 ft ∆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 < 0, this indicates a shortening of the tubing and seal bore length of at least |∆L| would be required. However in this scenario the packer was specified as allowing no movement. Thus, a virtual lengthening of the tubing of ∆LTP = 13.762 ft is effected by the packer which means a tensile force will be applied by the packer on the tubing, or alternatively, an upward force will be applied by the tubing on the packer. Applying FE,P to Eqns.(84)-(89) where the coefficients of the force/length curve are a = -8.725 × 10-10 and b = 1.840 × 10-4, we calculate a theoretical movement ∆LEP = -16.032 ft and induced tubing-to-packer force FTP = 54,610 lbs. The final real and effective forces at the packer are: FE,0 = -11,690 lbs FR,P = 17,013 lbs This indicates that the tubing is completely in tension due to the tensile effect of the tubing to packer contact forcd. However, the effect of hydraulic buckling is still significant as indicated by a negative effect force. Thus, a significant portion of the tubing is still buckled. Namely, the distance from the packer to the neutral point, i.e. the length of buckled tubing, is n = 1522 ft and the helical buckling period just above the packer is λ = 47.6 ft. The total surface weight of the tubing is 58,614 lbs. The components of stress just above the packer are as follows: σa = 994 psi σr,i = -12,792 psi σr,o = -4800 psi σh,i = 44,474 psi σh,o = 36,482 psi σVM,i = 51,769 psi σVM,o = 38,711 psi σVM = 51,769 psi The resultant packer to casing force is FPC = 221,420 lbs.

Scenario B Suppose that exactly the same operation is performed as in scenario A with the exception that at the beginning of the operation a slack-off of -20,000 lbs (or equivalently 4.028 ft) is applied at surface. In this case, many of the initial calculations remain unchanged.

The slack-off length means that ∆L5 = 4.028 ft. So now net tube movement is ∆L = -9.734 ft. This results in the following force values: FE,0 = -29,975 lbs FTP = 36,327 lbs FR,P = -1270 lbs As would be expected, the extent of helical buckling has increased: n = 3900 ft, λ = 29.7 ft. The new weight at surface is 40,330 lbs. Since the pressures have not changed, then radial and hoops stresses are the same as before. However, the axial stress and hence von Mises stress have changed as follows: σa = -22,227 psi σVM,i = 62,520 psi σVM,o = 52,223 psi σVM = 62,520 psi The resultant packer to casing force is FPC = 203,136 lbs. Scenario C: A common operation scenario that provides a good example of the phenomenon of hydraulic buckling is a simple tubing pressure test. Suppose that prior to the squeeze operation described above, a simple pressure test is conducted to make verify the integrity of the tubing. In this case, ALL input values are left constant at the initial conditions during installation, with the exception that internal tubing pressure at surface is set to 5000 psi.

Now all differential values are nil, except that ∆Pi,S = 5000 psi and wb = 5.811 lb/ft. The resulting real force at the packer is Fp = -24,965 lbs. The change in real force at the packer is then ∆FR = 18,080 lbs. Effective force at the packer and change in effective force at packer are FE,P = ∆FE,P = -41,479 lbs. The basic components of tubing movement may be calculated as follows: ∆L1 = -3.326 ft ∆L2 = -1.984 ft ∆L3 = -2.583 ft ∆L4 = 0.0 ft This indicates that if the packer allowed free movement of the tubing, a shortening of 7.982 ft would take place. This would seem to make sense since high internal pressure would cause a significant Poisson effect as well as increased axial compression on the tubing.

Since the packer is actually fixed, then a positive virtual movement is implied so that the tubing remains fixed in net length at the packer position: ∆LTP = 7.982 ft. This of course implies a positive, i.e. tensile, packer to tubing force. Taking note that the effective packer force FE,P is still negative and applying Eqns.(84)-(89), we get the following: a = -1.153 × 10-9 b = 1.840 × 10-4 ∆LEP = -9.614 ft FE,0 = -8867 lbs FTP = 32,612 lbs FR,P = 7646 lbs N = 1525 ft λ = 54.7 ft And so in this case, the packer and tubing are tension from the bottom up. Indeed a significant tensile reaction is occurring between tubing and packer. However, effective force at and above the packer is still negative. In fact, 1525 ft of the tubing above the packer is still buckled. This may be a surprising result, given that the tubing is fixed at the packer, the tubing is pulling up on the packer, and that the only change is increased internal pressure. The new weight at surface is 49,247 lbs and the real net tension at surface is 72,646 lbs. The von Mises stress at the packer is σVM = 33,007 psi. The resultant packer to casing force is FPC = 136,969 lbs.

References 1. Lubinski, A., “Influence of Tension and Compression on Straightness and Buckling of Tubular Goods in Oil Wells,” Proc. 31rst Annual Meeting API, Production Sec. IV (1951). 2. Lubinski, A., Althouse, W.S., and Logan, J.L., “Helical Buckling of Tubing Sealed in Packers,” JPT (June 1962), 655-670.

Appendix – Influence of Hydrostatic Pressure on Buckling One of the key points asserted in the body of this technical note which needs to be more thoroughly justified is that lateral fluid pressure forces acting against the outer and inner walls of the tubing have an effect on buckling. This assertion can be justified based on some simple geometrical observations. Also, the magnitude of this influence must be determined and compared with any other forces acting on the tubing. (On both points this appendix summarizes arguments in the well known references of Lubinski.) First, consider the influence of lateral external annular pressure on the buckling of tubing. Suppose a string of tubing is deployed in some vertical portion of a well. The conditions on the end of the tubing are ignored for now. Suppose that the annulus is full of fluid and that the tubing is somehow perturbed so as to be buckled, initially in a 2-dimensional plane. The buckled or curved portion of the tubing may be subdivided into sections by cutting it through the points of inflection A, B, C, D along the deflected centerline, as in Figure 12.

A

B FAB FBC

C

FCD

D

Figure 12 The deflection means that walls on opposite sides of each curved section (AB, BC, CD) are no longer equal in length, since once side has been stretched and the other compressed. As a result, a non-zero net pressure force is developed against each component: FAB, FBC, and FCD. These forces are directed towards the center of curvature of each section, and hence the action of the resultant moments is to negate the deflection, i.e. to straighten the tubing.

Similarly, suppose that the annulus is empty but that the same tubing is now filled with fluid. In the same deflected state as before, the same tubing sections again result. Each section is subject to a net reaction due to the hydrostatic pressure acting against the internal walls of the tubing – see Figure 13. In this case, the resulting forces FAB’, FBC’, and FCD’ act in a direction opposite from the center of curvature for each section. This means that internal pressure forces act to exacerbate any initial deflection.

A FAB’ B FBC’

C FCD’ D

Figure 13 It may be argued that if there is no initial deflection of the tubing, then the internal net forces FAB’, FBC’, and FCD’do not exist. Any force due to hydrostatic pressure acting against a point on the internal wall of the tubing is offset by an equal and oppositely directed force caused by the same pressure acting on the point symmetrically opposed to the first. If the tubing is truly vertical and straight, then clearly all hydrostatic pressure forces acting against the inner walls of the tubing are in equilibrium with each other. However, this must be characterized as an example of unstable equilibrium. In reality, the tubing will always be subject to disturbances or imperfections such that some initial deflection will be present. An analogous physical example to tubing subject to external or internal pressure is a ball set to rest on a circular bowl. If the bowl is sitting normally, then its cross-section is concave up. This means if the ball is somehow perturbed so as to move away from the center point O of the bowl, it enters a state of disequilibrium (Figure 14a). The forces of gravity and the curved inner surface of the bowl act to bring the ball back into a state of stable equilibrium positioned once again at the center point (Figure 14b).

In a similar manner, when tubing surrounded by external fluid is perturbed so as to initially deflect and buckle, the external hydrostatic pressure results in forces which act to move the tubing back into an equilibrium state by straightening it.

O

O

Figure 14a

Figure 14b

If the bowl in Figure 14 is now inverted so as to be concave down, the center position now becomes a position of unstable static equilibrium. This means that it is theoretically possible that the ball would remain at the point O if it were positioned perfectly above that point, if it were perfectly isolated from even the smallest perturbations, and if both it and the bowl did not possess the slightest asymmetric imperfection (Figure 15a). However, in any real situation, such imperfections or disturbances would exist and the ball would immediately be perturbed from its equilibrium position (Figure 15b). Now the forces of gravity and the curvature of the bowl exacerbate the disequilibrium of the ball, and it will roll away until it finds a different position of stable or perhaps neutral equilibrium.

O

Figure 15a

O

Figure 15b

Likewise, tubing filled with fluid may be positioned in a relatively straight position. However, it resides in a state of unstable equilibrium when straight. With any disturbance or symmetrical imperfection, force is developed from internal pressures which acts to accentuate the slightest deflection of the tubing centerline. This can cause the tubing to buckle and assume a new equilibrium in a buckled position. In order to determine whether the tubing is ultimately buckled or not, the effect of these developed lateral pressure forces as described above must be quantified and related to all axial forces acting on the tubing. However, in any real situation the magnitude and orientation of the developed lateral pressure forces is practically impossible to calculate. This presents a great dilemma. These are real forces which exist and which often critically impact buckling. Yet, they are essentially impossible to calculate directly. The key to this dilemma is to make some hypothetical assumptions which do not change the final result, and which allow for a simple method to back-calculate the unknown lateral pressure forces and determine the net effect on buckling. The analysis for freely suspend tubing and tubing set in a packer differs in details but not in general approach.

Consider first a string of tubing suspended freely in a well where both the tubing and the well are filled with fluid. Assume that the tubing is deflected temporarily from the straight position. Can we determine whether the tubing is buckled at some arbitrary position O distance X from the tubing bottom? From Figure 16a, we see that the tubing is subject to the following force vectors: WS - the weight of steel of the tubing section below O, acting on the center of gravity G – net developed force on the outer wall from external hydostatic pressure H – net developed force on the inner wall from internal hydratic pressure J – external fluid pressure on the bottom surface (open or closed tubing) K – internal fluid pressure on the bottom surface (open or closed tubing) FR – the reaction on the remainder of the tubing string above O M

FR O

FE O L

H

H G

G K

K

J

J WS

WS

Figure 16a

Figure 16b

The following equilibrium of forces acting above and below the cross-section at O results: K K K K K K FR = WS + G + H + J + K (101) As already discussed, G and H are real forces acting on the tubing that impact buckling. However, the specific magnitudes of G and H and indeed their orientation and ultimate effect on buckling are unknown and impractical to calculate directly. However, it is not necessary to know these values individually. To determine whether the tubing straightens or buckles at O, we simply need to look at the net moment around O of all forces acting on the tubing: K K K K K K M O ( FR ) = M O (WS ) + M O ( G ) + M O ( H ) + M O ( J ) + M O ( K ) (102) Assume now that the tubing section below O is physically cut-away and that the tubing section is completely immersed in the external and internal fluids(Figure 16b). This exposes the cross-section at O to two hypothetical forces L and M. The magnitudes of these are the fluid pressures at O applied to the tubing cross-section: ||L|| = Pi,X Ai and ||M|| = Po,X Ao.

The new vector sum at O, FE, including the new hypothetical pressure forces is thus: K K K K K K K K FE − M = WS + G + H + J + K + L or K K K K K K K K FE = WS + G + H + J + K + L + M

(103) (104)

At first, the addition of these hypothetical forces may seem to confuse rather than clarify the analysis. However, a very important observation is that the moment of these hypothetical forces about O is nil, so that moment balance is unchanged from Eqn.(102): K K K K M O ( L ) = M O ( M ) = 0 ⇒ M O ( FR ) = M O ( FE ) (105)

So, straightening or buckling at O is determined by the moment of FE about O, where K K K K K K K K M O ( FE ) = M O (WS ) + M O ( G ) + M O ( H ) + M O ( J ) + M O ( K ) + M O ( L ) + M O ( M ) (106) Furthermore, formulating the problem in this way allows us to apply the fundamental physical principle of Archimedes: the sum of all pressure forces acting on the surface of a totally submerged body is equal to the weight of the fluid displaced by the body volume. So, K K K K K K K K (107) M O ( FE ) = M O (WS ) + M O ( G + J + M ) + M O ( H + K + L ) or

K K K K M O ( FE ) = M O (WS ) + M O ( B ) + M O (Wi )

(108)

where B = G + J + M is the buoyant force of the annular fluid acting upward on the center of gravity of the tubing section opposite to WS and Wi = H + K + L is the weight of the internal fluid volume in the tubing below O. The vector sum WB = WS + B + Wi is net buoyed tubing weight below the point O, so that K K (109) M O ( FE ) = M O (WB ) Thus, the freely suspended section of tubing will straighten again at the position O as long as moment of buoyed weight of the section with respect to O is positive (assuming the righthand rule for moments in the plane in Figure 16), that is, if K M O (WB ) > 0 (110) This will be the case regardless of changes in the hydrostatic pressure profile if the combined density of the tubing and internal fluid is greater than the density of the annular fluid. Any change in pressure affecting G or H 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 magnitude of FR is real force FR,X. The relationship between FE and FR is given by K K K K FE = FR + 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 for freely suspended string. As discussed in the main text, the tubing is subject at the packer to a real force Fp given by Eqn(31). Once again, it is required to determine in general whether the net effect of all forces acting on the tubing result in straightening or buckling of the tubing at some point O, a distance X above the packer (Figure 17a). FE

FR

O O

G

G

H

H WS

M

L

WS

K Fp Figure 17a

J Figure 17b

The vector sum of all forces acting on the section of tubing below the position O is given by: K K K K K FR = WS + G + H + Fp (112) As with the freely suspended tubing, buckling or straightening at O is determined by the resultant moment at O and the magnitude and orientation of forces G and H is unknown: K K K K K M O ( FR ) = M O (WS ) + M O ( G ) + M O ( H ) + M O ( Fp ) (113) Once again to avoid a direct calculation of the developed pressure forces G and H, we hypothetically cut-away the tubing at the cross-section at 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 hypothetical pressure forces acting at the cross-section at O due to the assumed cut-away. The new vector sum is again denoted by FE: K K K K K K K K FE = WS + G + H + J + K + L + M (114) Reasoning as before, the resulting moment of FE about O associated is thus given by: K K K K K K K K M O ( FE ) = M O (WS ) + M O ( G ) + M O ( H ) + M O ( J ) + M O ( K ) + M O ( L ) + M O ( M ) (115)

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: K K K K B=G+J +L (116) K K K K (117) Wi = H + K + M Eqn(115) may be restated in the equivalent form K K K K M O ( FE ) = M O (WS ) + M O ( B ) + M O (Wi ) or

K K M O ( FE ) = M O (WB )

We can now re-express G and H in terms of these theoretical force vectors: K K K K G=B−J −L K K K K H = Wi − K − M

(118) (119)

(120) (121)

If we substitute Eqns.(120) and (121) into Eqn.(113) and observe again that the moments of L and M about O are nil, then we get that K K K K K K K M O ( FR ) = M O (WS ) + M O ( B ) + M O (Wi ) + M O ( Fp − J − K ) (122) or where

K K K M O ( FR ) = M O ( FE ) + M O ( FE ,P )

(123)

K K K K FE ,P = FP − J − K

(124)

defines a new hypothetical force vector acting at the packer. What is the significance of Eqns.(123) and (124)? Initially, to calculate the moment of real forces at O in Eqn.(113) and determine buckling, we needed to know the magnitude and direction of G and H – which is impossible directly. However, Eqn.(123) says that we can reach the same conclusion by using the effective force FE which is easily calculated and a modified 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 tubing? Because by doing so we are able to apply the simple yet powerful Archimedes principle and thereby avoid calculation of important real forces which cannot be determined directly, namely the lateral pressure forces acting on the buckled tubing. Furthermore, the tubing will indeed actually buckle or straighten as if it were subject to these hypothetical forces.