Casing and tubing buckling

As installed, casing usually hangs straight down in vertical wells or lays on the low side of the hole in deviated wells. Thermal or pressure loads might produce compressive loads, and if these loads are sufficiently high, the initial configuration will become unstable. However, because the tubing is confined within open hole or casing, the tubing can deform into another stable configuration, usually a helical or coil shape in a vertical wellbore or a lateral S-shaped configuration in a deviated hole. These new equilibrium configurations are what we mean when we talk about buckling in casing design. In contrast, conventional mechanical engineering design considers buckling in terms of stability (i.e., the prediction of the critical load at which the original configuration becomes unstable).

Analysis

Accurate analysis of buckling is important for several reasons:

• Buckling generates bending stresses not present in the original configuration. If the stresses in the original configuration were near yield, this additional stress could produce failure, including permanent plastic deformation called “corkscrewing.”
• Buckling causes tubing movement. Coiled tubing is shorter than straight tubing, and this is an important consideration if the tubing is not fixed.
• Tubing buckling causes the relief of compressive axial loads when the both ends of tubing are fixed. This effect is not as recognized as the first two buckling effects, but is equally important. The axial compliance of buckled tubing is much less than the compliance of straight tubing. Casing movement, because of thermal expansion or ballooning, can be accommodated with a lower increase in axial load for a buckled casing.

The accuracy and comprehensiveness of the buckling model is important for designing tubing. The most commonly used buckling solution is the model developed by Lubinski in the 1950s. This model is accurate for vertical wells, but needs modification for deviated wells. Tubing bending stress, because of buckling, will be overestimated for deviated wells using Lubinski’s formula. However, Lubinski’s solution, applied to deviated wells, will also overpredict tubing movement. This solution overestimates tubing compliance, which might greatly underestimate the axial loads, resulting in a nonconservative design.

Casing buckling in oilfield operations

Buckling should be avoided in drilling operations to minimize casing wear. Buckling can be reduced or eliminated with the following methods:

• Applying a pickup force when landing the casing in surface wellhead after cement set
• Holding pressure while wait on cement (WOC) to pretension the string
• Raising the top of cement
• Using centralizers to increasing casing bending stiffness

In production operations, casing buckling is not normally a critical design issue. However, a large amount of buckling can occur, because of increased production temperatures in some wells. A check should be made to ensure that plastic deformation or corkscrewing will not occur. This check is possible using triaxial analysis and including the bending stress because of buckling. Corkscrewing occurs only if the triaxial stress exceeds the yield strength of the material.

Tubing buckling in oilfield operations

Buckling is typically a more critical design issue for production tubing than for casing. Tubing is typically exposed to the hottest temperatures during production. Pressure/area effects in floating seal assemblies can significantly increase buckling. Tubing is less stiff than casing, and annular clearances can be quite large. Buckling can prevent wireline tools from passing through the tubing. Buckling can be controlled by:

• Tubing-to-packer configuration (latched or free, seal bore diameter, allowable movement in seals, etc.)
• Slackoff or pickup force at surface after packer set
• Cross-sectional area changes in tubing
• Packer fluid density
• Tubing stiffness which is influenced by tubing diameter and wall thickness
• Centralizers
• Hydraulic set pressure for packers

As in casing design, a triaxial check should be made to ensure that plastic deformation or corkscrewing will not occur.

Buckling models and correlations

Buckling occurs if the buckling force, F_b, is greater than a threshold force, F_p , known as the Paslay buckling force. The buckling force, F_b, is defined as

....................(1)

where

F_b = buckling force, lbf,

F_a = axial force (tension positive), lbf,

p_i = internal pressure, psi,

A_i = r_i² , where r i is the inside radius of the tubing, in.²,

p_o = external pressure, psi,

and

A_o = r_o² , where r_o is the outside radius of the tubing, in.²

The Paslay buckling force, F_p , is defined as

....................(2)

where

F_p = Paslay buckling force, lbf,

w_c = casing contact load, lbf/in.,

w_e = distributed buoyed weight of casing, lbf/in.,

Φ = wellbore angle of inclination, radians,

Θ = wellbore azimuth angle, radians,

EI = pipe bending stiffness, lbf-in.² ,

and

r = radial annular clearance, in.

Table 1 gives the relationship between the buckling force F_b, the Paslay buckling force F_p, and the type of buckling expected for the tubing.

• 

  Table 1- Buckling Criteria

An increase in internal pressure acts on the buckling force in two ways. It increases F_a because of ballooning, which tends to decrease buckling, and increases the p_iA_i term, which tends to increase buckling. The second effect is much greater, as an increase in internal pressure will result in an increase in buckling.

A temperature increase results in a reduction in the axial tension (or increase in the compression). This reduction in tension may transition the tubing into compression and result in buckling. The onset and type of buckling is a function of hole angle. Because of the stabilizing effect of the lateral distributed force of a casing lying on the low side of the hole in an inclined wellbore, a greater force is required to induce buckling. In a vertical well, F_p = 0, and helical buckling occurs at any F_b > 0. For production tubing that is free to move in a seal assembly, the upward force, because of pressure/area effects in the seal assembly, will decrease F_a, which, in turn, increases buckling. In order to give the correlations for tubing stresses and movement, definitions are made. The lateral displacements of the tubing, shown in Fig. 1, are given by

• 

  Fig. 1—Coordinate system for buckling analysis.

....................(3)

and

....................(4)

where ϴ is the helix angle.

The quantity ϴ´, where ´ denotes d/dz , is important and appears often in the next analysis. It can be related to the more familiar quantity, pitch through Eq. 5.

....................(5)

where P_hel = pitch of helically buckled pipe, in.

Other important quantities, such as pipe curvature, bending moment, bending stress, and tubing length change are proportional to the square of ϴ´. Nonzero ϴ´ indicates that the pipe is curving, while zero ϴ´ indicates that the pipe is straight.

Correlations for maximum buckling dogleg

The correlation for the maximum value of ϴ´ for lateral buckling, with 2.8 F_p > F_b> F_p, can be expressed by ....................(6)

For F_b > 2.8 F_p, the corresponding helical buckling correlation is ....................(7)

The region 2.8 F_p > F_b> 1.4 F_p may be either helical or lateral; however, 2.8 F_p is believed to be the lateral buckling limit on loading, while 1.4 F_p is believed to be the helical buckling limit on unloading from a helical buckled state. An important distinction between Eq. 6 and Eq. 7 is that Eq. 6 is the maximum value of ϴ´, while Eq. 7 is the actual value of ϴ´. The equation for a dogleg curvature for a helix is

....................(8)

assuming ϴʺ is negligible. The dogleg unit for Eq. 8 is radians per inch. To convert to the conventional unit of degrees per 100 ft, multiply the result by 68,755.

Correlations for bending moment and bending stress

Given the tubing curvature, the bending moment is determined by

....................(9)

The corresponding maximum bending stress is

....................(10)

where d_o is the outside diameter of the pipe.

The following correlations can be derived with Eqs. 6 and 7. M = 0, for F_b < F_p; ....................(11)

....................(12)

σ_b = 0, for F_b < F_p; ....................(13)

....................(14)

Correlations for buckling strain and length change

The buckling "strain," in the sense of Lubinski, is the buckling length change per unit length. The buckling strain is given by

....................(15)

For the case of lateral buckling, the actual shape of the ϴ´ curve was integrated numerically to determine the relationship,

....................(16)

for 2.8 F_p > F_b> F_p, which compares to the helical buckling strain given by ....................(17)

for F_b > 2.8 F_p. The lateral buckling strain is roughly half the conventional helical buckling strain. To determine the buckling length change, ΔL_b, we must integrate Eqs. 16 and 17 over the appropriate length interval, which is written as ....................(18)

where z₁ and z₂ are defined by the distribution of the buckling force, F. For the general case of arbitrary variation of F_b over the interval ΔL = Z₂ – Z₁ , Eq. 18 must be numerically integrated. However, there are two special cases that are commonly used. For the case of constant force, F_b, such as in a horizontal well, Eq. 18 is easily integrated.

....................(19)

where e_b is defined by either Eq. 6 or Eq. 7 . The second special case is for a linear variation of F_b over the interval.

....................(20)

The length change is given for this case by Eqs. 21 and 22.

....................(21)

for 2.8 F_p > F₂> F_p. ....................(22)

for F > 2.8 F_p.

Correlations for contact force

From equilibrium considerations only, the average contact force for lateral buckling is

....................(23)

The average contact force for the helically buckled section is

....................(24)

When the buckling mode changes from lateral to helical, the contact force increases substantially.

Sample buckling calculations

The basis of the sample calculations is the buckling of tubing (2 ⁷/₈ in., 6.5 lbm/ft) inside of casing (7 in., 32 lbm/ft). The tubing is submerged in 10-lbm/gal packer fluid with no other pressures applied. The effect of the packer fluid is to reduce the tubing weight per unit length through buoyancy. w_e = w + A_iγ_i – A_oγ_o , where w_e is the effective weight per unit length of the tubing, A_i is the inside area of the tubing, γ_i is the density of the fluid inside the tubing, A_o is the outside area of the tubing, and γ_o is the density of the fluid outside the tubing. The calculation gives

Other information useful for the buckling calculations are radial clearance = r = 1.61 in.; moment of inertia = I = 1.611 in.⁴, and Young ’ s modulus = 30 × 10⁶ psi.

Sample buckling length calculations

From Eq. 2, we can calculate the Paslay force for a variety of inclinations. First, we calculate the value for a horizontal well, which is written as

This means that the axial buckling force must exceed 7,500 lbf before the tubing will buckle. We can evaluate other angles by multiplying the horizontal F_p by the square root of the sine of the inclination angle. Table 2 was developed with this procedure. Of particular notice in Table 2 is how large these buckling forces are for relatively small deviations from vertical. For a well 10° from the vertical, the buckling forces are nearly half of the horizontal well buckling forces.

• 

  Table 2- Buckling Forces

With Table 2, the total buckled length of the tubing can be calculated, as well as maximum and minimum lateral buckling or helical buckling. Assume an applied buckling force of 30,000 lbf is applied at the end of the tubing in a well with a 60° deviation from vertical. The tubing will buckle for any force between 6,939 lbf and 30,000 lbf. The axial force will vary as w_e cosΦ (i.e., w_a = w_e cos(60) = 5.56 lbf/ft (0.50) = 2.78 lbf/ft). Therefore, the total buckled length, L_bkl, is L_bkl = (30,000 – 6939)lbf/(2.78 lbf/ft) = 8,295 ft. The maximum helically buckled length, L_helmax, is L_helmax = (30,000 – 9,813)lbf/(2.78 lbf/ft) = 7,262 ft. The minimum helically buckled length, L_helmin, is L_helmin = (30,000 – 19,626)lbf/(2.78 lbf/ft) = 3,732 ft.

Sample buckling bending stress calculations

The maximum bending stress, because of buckling, can be evaluated with Eq. 16. σ_b = .25(2.875 in.)(1.61 in.)(30,000 lbf)/(1.611 in.⁴) = 21,550 psi. This stress is fairly large compared to tubing yield strengths of about 80,000 psi, so buckling bending stresses can be important for casing and tubing design. At the buckling load of 19,626 lbf, both helical and lateral buckling can occur. The lateral bending stress is given by Eq. 7.35. σ_b = .3151 (2.875 in.)(1.61 in.)/(1.611 in.⁴) (6,939 lbf)^.08(19,626 – 6,939 lbf)^0.92 = 10,945 psi. The equivalent calculation for helical buckling gives σ_b = .25(2.875 in.)(1.61 in.)(19,626 lbf)/(1.611 in.⁴) = 14,097 psi, so helical buckling produces approximately 29% higher stresses than lateral buckling. This indicates that determination of buckling type can be important in casing design where casing strength is marginal.

Sample buckling length change calculations – tubing movement

Tubing length change calculations involve two calculations for this case, tubing movement because of lateral buckling and tubing movement because of helical buckling. Eqs. 21 and 22 are used to calculate tubing movement, and these equations assume the minimum amount of helical buckling. A third calculation is made to show the movement because of pure helical buckling. The lateral buckling tubing movement is given by

The helical buckling tubing movement is given by

The total tubing movement is 0.297 ft plus 1.242 ft, which equals 1.539 ft. Pure helical buckling produces the length change,

Tubing movement is a design consideration for packer selection. Seal length is an important criterion for tubing well completion design. The use of pure helical buckling produces a 41% error in the calculation of tubing movement. When designing seal length in a deviated well, use of pure helical buckling can produce significant error.

Nomenclature

Ai	= the inside area of the tubing, πri2, in.2
Ao	= the outside area of the tubing, πro2, in.2
eb	= buckling strain, in./in.
F	= constant in transition collapse equation, dimensionless
Fa	= axial force (tension positive), lbf
Fb	= buckling force (compression positive), lbf
Fp	= Paslay buckling force, lbf
gc	= gravity constant, 32.2 ft/sec2
L	= engaged thread length, in.
M	= bending moment, lbf-ft
pi	= internal pressure, psi
po	= external pressure, psi
Phel	= pitch of helically buckled pipe, ft
r	= radial annular clearance, in.
ro	= outside radius of the pipe, in.
we	= the effective (buoyant) weight per unit length of the tubing, lbm/ft
Wn	= lateral contact force, lbf/in.
ϴ´	= rate of change of helix angle with respect to pipe length, radians/ft
ϴ	= helix angle, radians
κ	= curvature, radians/ft
σb	= stress at the pipe’ s outer surface, psi
φ	= wellbore angle with the vertical, radians

See also

Internal pressure loads on casing and tubing strings

Thermal loads on casing and tubing strings

Mechanical loads on casing and tubing strings

External pressure loads on casing and tubing strings

PEH:Casing_Design

References

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