# EQUATIONS DETERMINE COILED TUBING COLLAPSE PRESSURE

July 24, 1995
A set of equations has been developed for calculating pipe collapse pressure for oval tubing such as coiled tubing. Although work has been performed and reported showing methods of calculating pipe collapse pressure for round tubing, there is very little literature discussing collapse pressure calculations for oval tubing.
Halliburton Energy Services
Duncan, Okla.
A set of equations has been developed for calculating pipe collapse pressure for oval tubing (22612 bytes) such as coiled tubing. Although work has been performed and reported1 showing methods of calculating pipe collapse pressure for round tubing, there is very little literature discussing collapse pressure calculations for oval tubing. When coiled tubing is placed onto a reel, the tubing is forced into an oval shape and never again returns to perfect roundness because the coiling process exceeds the plasticity limits of the tubing. Straightening the tubing for the trip into the well does not restore roundness. The consequence of this physical property is that all coiled tubing collapse pressure calculations should be made considering oval tubing, not round tubing. Tubing collapse can occur when formation pressure against the coiled tubing exceeds the collapse resistance inherent in the coiled tubing. As coiled tubing becomes more oval in shape, it becomes more susceptible to collapse from outside pressure.

### Collapse pressure

Collapse pressure for oval thin wall and long tubing was derived by S. Timoshenko.2 3 Here are basic concepts of that solution using some new notations. The deviation of the mean tubing circle from a perfectly round shape is characterized by initial radial displacement, do, along the minor (or major) axis (Fig. 1)(60759 bytes). It can be expressed in terms of mean circle radius, R, or outside diameter, D, as in Equation 1 (see equation box). At current angular position, Upsilon (Variant), initial deviation (at no external pressure), d, is assumed to be defined by Equation 2. If radial displacement from pressure P is u, the current bending moment is shown in Equation 3. This bending moment is positive when it decreases the initial curvature of the tubing. Using M, tubing radial displacement, u, will be obtained from J. Boussinesq's differential equation, Equation 4,4 where Const is the flexural rigidity of the one-unit long tubing. Applying boundary conditions of continuity at Upsilon (Variant) = np/4, n = 1,3,5,7, radial displacement becomes as defined by Equation 5, where Pcr is a critical collapse pressure for perfectly round tubing. Maximum bending moment occurs at Upsilon (Variant) = 0 and Upsilon (Variant) = p (Fig. 1)(60759 bytes), Equation 6, and maximum compressive stress takes place on the outer fibers at Upsilon (Variant) = 0 (or Upsilon (Variant) = p) due to the bending moment, Mo, and the compressive force PR as shown in Equation 7. This equation can be resolved for outside pressure, P = Po, which produces compressive stress equal to the yield strength, s = sy in Equation 8, where the ovality index is introduced as Equation 9. In Equation 8, S. Timoshenko suggests using M. Levy's elastic collapse pressure equation for perfectly round tubing as a critical pressure, Pcr. This results in Equation 10. S. Timoshenko's Equation 8 is discussed and analyzed in Reference 6.

### Ovality

A large number of results have been assembled from experiments on tubes under external hydrostatic pressure, such as an all-round pressure having both a radial and an axial component. The experiments were performed with steel tubes at initial ovality of about (Dmax-Dmin)/D = 0.01 (1%). Test results are published in a data volume.7 For the analysis provided in Reference 6, Equation 8 was corrected to match the hydrostatic loading and the initial ovality. Comparison between theoretical and experimental results are plotted in graphs. The authors concluded that "the agreement appears to be remarkably good, with the equation providing, in general, safe and not overconservative predictions." For real tubing dimensions and material properties, both roots of Equation 8 are positive, and the smallest root is the practical collapse pressure for oval tubing as shown in Equation 11. The collapse pressure, Po, approaches Pcr as ovality, Ov, vanishes. That is, collapse pressure for perfectly round tubing becomes M. Levy's elastic collapse pressure (Equation 10) when following Timoshenko's recommendation.

### Collapse equations

Table (54757 bytes) For the oil field industry, it is normal to request collapse pressure for perfectly round tubing equal to the API collapse pressure.8 As discussed in Reference 1, throughout the 20th century the API has been a reliable source for collapse pressure evaluation. Equations were periodically updated whenever new data were available. The API collapse pressure is less than M. Levy's theoretical collapse pressure, and the upper limit of collapse pressure for thick-wall tubing is "truncated" by so-called yield strength collapse pressure, protecting tubing against yielding. Therefore, it is safe and more practical to use the API collapse pressure as the critical value, Pcr, Such replacement can be used until new data are obtained and new solutions can be proved. For analysis, and for computer and calculator applications, the collapse pressure tied to API data can be found by Equation 14.1 This collapse pressure equation is an approximation of the API collapse pressure data. It can be used with any measurement units, English or metric. The API collapse pressure can be evaluated as discussed in Reference 8. The API defines four collapse modes (yield, plastic, transition, and elastic) and recommends four equations for their evaluations. Each collapse pressure equation has its own a = D/t domain and equations for their boundary D/t ratios. The boundary ratios are functions of five intermittent factors (A, B, C, F, and G) and the tubing yield strength in psi. The equation structure (of the five factors, three boundary ratios, and plastic and transition collapse pressure) does not allow any pressure units other than psi. Even such multiples as kpsi are not usable, since they provide distortion of the results unless relative corrections are applied in the equations. Because of this complexity, Equation 14 for the critical pressure, Pcr, can be a practical and acceptable approximation for API collapse pressure in many instances. Using Equations 11-14, Figs. 2(63318 bytes) and 3(57845 bytes) are graphs of collapse pressure for oval tubing for 70 and 80 kpsi (483 and 552 MPa) material yield strengths. Collapse pressure can be defined as a function of DA ratio and ovality index Ov = (Dmax - Dmin)/D.

### Example

QT-700 coiled tubing has nominal outside diameter D = 1.50 in., wall thickness t = 0.109 in., and yield strength sy = 70 kpsi (483 MPa). Measurements indicate that tubing is oval and its section major diameter is Dmax = 1.55 in. and minor diameter is Dmin = 1.46 in. Steps to find the outside collapsing pressure are as follows:
1. Tubing ovality Ov = (Dmax - Dmin)/D = (1.55 - 1.46)/1.50 = 0.06
2. Ratio D/t = 1.50/0.109 = 13.76
3. The graphical solution is shown in Fig. 2,(63318 bytes) and collapse pressure is read as Po = 3,600 psi (24.5 MPa).
4. To obtain an analytical solution, the following steps are performed (see example box).
Collapse pressure for perfectly round tubing is defined by Equation 14 as Pcr = 8.603 kpsi = 59.32 MPa. From Equation 13, factor f = 94.38; and from Equation 12, factor e = 15.11. Collapse pressure for oval tubing from Equation 11 is then Po = 3.536 kpsi (24.38 MPa).

### Tension and compression

Collapsing pressure of oval tubing will be changed when external pressure is acting simultaneously with axial tensile or compression load. Reduction of collapse pressure for perfectly round tubing combined with axial tensile load is covered by API literature.8 The effect of combining external pressure with compression load is discussed in Reference 6. Combination of external pressure with axial loading is a special subject and is not addressed in this article.

### References

1. Avakov, V.A., "Equations shorten pipe collapse calculations," OGJ, Apr. 10, 1995, pp. 53-55.
2. Timoshenko, S., Strength of Materials, Part II-Advanced Theory and Problems, 2nd Edition, Van Nostrand, 1954.
3. Timoshenko, S., "Working Stresses for Columns and Thin-Walled Structures," Trans. ASME, Applied Mechanics, Vol. 1, 1933, pp. 173-83.
4. Boussinesq, J., "Resistance d'un anneau a la flexion, quand sa surface exterieure supporte one pression normale constants par units de longueur de sa fibre moyenne," Comptes rendus des seances de I'Academie des Sciences, 2 Semestre, Tome 97, No. 16, 1883, pp. 843-48.
5. Levy, M., journal d. Math. pure et Appl., Iiouville, Ser. 3, Vol. 10, 1884, p, 5.
6. Ellians, C.P., Supple, W.J., and Walker, A.C., Buckling of Offshore Structures, State-of-the-Art Review, Gulf Publishing Co., 1984.
7. Department of Energy (U.K.), Buckling of Offshore Structures, Data Volume, OT/0/8321, London, 1983.
8. API Bulletin on Formulas and Calculations for Casing, Tubing, Drill Pipe and Line Pipe Properties, API Bulletin 5C3, Sixth Edition, Oct. 1, 1994