In both deviated and vertical wells, coiled tubing takes a helical shape. The conventional approach to describing coiled tubing constrained inside a wellbore relies upon the relationship between the axial force and the helical pitch. However, in the field and in experiments, it is hard to measure accurately the helical pitch for very long tubing. Instead, the tubing operators collect data on the force and shortening of coiled tubing in the axial direction of wells. This paper transforms the classical expression for coiled-tubing deformation into an alternative form that relates the axial force to the axial shortening of the tubing. The new model is consistent with experimental data from several previous studies. Moreover, through this new model, a simple technique is developed to predict the lockup length of the coiled tubing resulting from friction. This leads to an improved prediction of lockup length for coiled tubing, allowing more accurate modeling of downhole depth.


Since 1950, many analyses have been performed on the mechanics of tubing in oil and gas wells. The model of a helically post-buckled tubing within a circular wellbore was first developed by Lubinski et al. in 1962.1 They described the tubing configuration by "pitch." Pitch is the tubing length in the wellbore axial direction as the tubing circumferential displacement advances by 2p around the wellbore. Since the groundbreaking work of Lubinski et al., the majority of work on helically post-buckled tubing has been focused on modified expressions of the force-pitch relationship.

Several experiments have been conducted to categorize the behaviors of post-buckled tubings. Earlier work measured the pitch of helically buckled tubing under continuously varying axial force.2 Recent experiments measured the axial shortening of tubing due to helical buckling.3,4 The results of these later experiments are more attractive from the field operator's perspective. This is because a wellbore is usually very long, and it is hard to measure the pitch along the wellbore. On the other hand, the axial shortening of the tubing can be easily measured as the operator loads the tubing into the wellbore.

The intent of this study is to transform the axial force-helical pitch relationship into an axial force-axial shortening expression for helically buckled tubings. Our current analysis verifies the experimental results published earlier.3,4 Moreover, we demonstrate the use of the simple formula developed here in predicting the lockup length of helically buckled tubings.

Section 2 summarizes the results of some existing experimental work. The classical formulation for helically buckled tubing is described in Section 3, followed by a derivation in Section 4 that transforms this classical formulation into a new form. A key concept in Section 4, the spring constant of the tubing, is then correlated to previous experimental results in Section 5. Finally, we show in Section 6 that the concept of a spring constant enables us to reach a new method of predicting lockup length for coiled tubings. Section 7 concludes this analysis.

Previous Experimental Study

In their experimental work, Wu et al.3 plotted the axial force vs. axial shortening for sinusoidal and helical buckling of brass rods. They used clear plastic pipes as the wellbore so that they could observe the transition of the rod's configuration from sinusoidal to helical shape. The two tests conducted in their experiment used a 3.2-mm-OD brass rod in an 8.3-mm-ID plastic pipe, and a 2.4-mm-OD brass rod in a 25.7-mm-ID plastic pipe, respectively. They stated that, "the axial load increases almost linearly with respect to rod shortening in the simulated wellbore axis direction (the axial movement of the loading end of the rod) during the helical buckling process" [pp 193].3 Analytical justification for this experimental result was not provided, however. Similar experimental results for helically buckled rods are described in a related work.4

Both of these works included the same expression describing the axial shortening of a rod in terms of the helical pitch. This expression was also reported in an earlier paper by Cheatham and Pattillo2 to describe the strain energy and external work in the system. In this paper, we will further extend this expression to justify the previous experimental results.

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