Critical Buckling Force
As discussed previously, the critical buckling force is the compressive force at which casing equilibrium changes from stable to unstable. In other words, pipe buckling occurs when the total effective axial force exceeds the average of radial and tangential (or stability) forces. Hence, the buckling force. Fi, uc, can be expressed as follows:
Fbuc = Fa, ~ A, (2/206)
where:
= average of radial and tangential stresses at any depth x
X
Considering the changes of fluid pressure gradient and surface pressures inside and outside the pipe, the average of radial and tangential forces, Fr /, at any depth x can be expressed as follows:
Fr, t — Asx — Ai(pi + A pi + A psl) — A„(p0 + A p0 + A pso) (2.207)
or
FT, t = Ai (xGp> + xAGPt + Apsl) — A0 (xGPo + xAGPo + Apso) (2.208)
Substituting Eqs. 2.204 and 2.207 in Eq. 2.206, the following expression is obtained:
Ftuc = Wn(D-x)-[A0{xGPc + (D-x)GPcm}-AiGPtD}
+v {A^AG’p, x + 2 Apsi) — A0 [AGPo x + 2 Aps0)}
FGPi 77дд5(/1ир1 A[ou,, ) + (AGp, D±д. + Apsl) {Aupi — AtOWl) — ASET AT + Fas — A,(xGp< + xAGPt + Apsl)
+A0(x GPo + x AG’Po + Apso) (2.209)
To prevent buckling the value of F(,uc, in the above equation, must be equal to or greater than zero. It is, however, important to note that Eq. 2.209 does not define the point at which the existing buckling force exceeds the casing critical buckling force.
Lubinski (1951) presented a formula for determining the critical buckling force based on the assumption that the pressure forces are vertically distributed and concentrated at the lower end of the casing. Hence, by applying Euler’s column theory, the critical buckling force, FbuCcr. is given by:
SHAPE * MERGEFORMAT
|
(2.210) |
|
• bucc |
= 3.5 [EI{WnBF)2lF
Although the above equation has been used extensively to predict the critical buckling force, there are several other relationships available in the literature to determine the critical buckling force. Dawson and Paslay’s (1984) equation describes the buckling force more precisely. They used the energy method to obtain the stability criteria and proposed the following equation to predict critical buckling force for casing in both vertical and deviated wells:
|
1/2 |
|
— 2 |
|
(2.211) |
|
EIW„ BF sir |
|
12 r. |
where:
WnBF — buoyant weight of casing, lb/ft.
— 7s As
rc = radial clearance between hole and casing, in.
в = angle of inclination, measured from the vertical.
For the derivation of Eqs. 2.210 and 2.211, the readers are referred to the original papers (Lubinski, 1951; Dawson and Paslay, 1984).
Determine the critical buckling force in Example 2-13 using: (i) Lubinski’s equation, (ii) Dawson and Paslay’s equation. Assume a borehole size of 12T in. and в = 3°.
Average weight, Wn:
(2,000 x 58.4) + (5,500 x 47) + (2,500 x 58.4)
|
10,000 |
= 52.13 lb/ft Average internal area, A;:
(4,500 x 55.88) + (5,500 x 59.19)
57.70 sq. in.
Thus, the buoyant weight of the string is:
0. 702 144
489.5 J = 41.36 lb/ft
Moment of inertia of the cross-section is:
/ = JL(9.6254 — 8.5714)
= 156.37 in.4
(i) From Lubinski, Eq. 2.210 yields:
FbuCcr = 3.5 ^30 x 106 x 156.37 ^52.13 1 — = 70,078 lbf
(ii) From Dawson and Paslay, Eq. 2.211 yields: ’30 x 10® x 156.37 x 41.36 sin3°
|
2 ’/a |
|
0.702 x 144 489.5 1/2 |
WnBF = 52.13 1
|
• buecr |
12 0.5(12.25 — 9.625)
= 50,783 lbf