## New Analytical Model Enhances Understanding of Connection Strengths

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The extra torsional capacity and clearance available when proprietary, double-shouldered connections are used instead of public-domain, single-shouldered connections enables well teams to drill farther, faster, and with less damage to the drillstring. No analytical model as yet can accurately calculate the tensile and torsional capacities of double-shouldered connections. This paper presents a set of equations that extends the approach of the original single-shouldered equation to account for a second shoulder, and helps to understand connection strengths better.

## Introduction

Single-shouldered connections such as API drillstem connections use a straightforward analytical equation to determine the capacities of any connection. This equation may not be perfect—it relies on linear assumptions that are probably not descriptive of the connection loading—but several decades of use have made the industry confident in the equation’s strengths and aware of its shortcomings.

Because no analytical model is available to calculate these capacities in proprietary double-shouldered connections, the connection designer or manufacturer typically creates an empirical formula that is calibrated through laboratory and field testing. Although this approach is acceptable, it hinders tool designers who need something different from what is offered on the public market. It is rarely cost-efficient to perform laboratory and field testing for a one-off connection design.

The complete paper states that what is needed is to extend the approach of the original single-shouldered equation to account for a second shoulder. According to the author, although the mathematical complexity increases, the assumptions are the same, lending confidence to the extended equation.

The complete paper presents the derivation of a set of equations that use the same basic assumptions as the original single-shouldered connection torque equations, in the same way as do the API equations, to advance understanding of the torsion and tension capacities for double-shouldered equations. The implicit assumptions present in the original equations are discussed; then the same ideas are applied to a double-shouldered connection. The full set of new equations is developed and described, including tips on their practical use gained from tool-design experience.

## Current Single-Shouldered Connection Model

Connection strength formulas contained in *API RP 7G Appendix A* are the basis for the connection torsion and tensile capacities in that standard and in many other oilfield references. These equations and resulting numbers have been in place for decades, creating industry confidence in their usefulness and awareness of their weaknesses.

These equations originated with the modified screw-jack equation outlined by A.P. Farr in 1957. That work starts with an energy balance: The work applied to the connection during an incremental amount of makeup is equal to the strain energy absorbed by the connection plus the friction energy dissipated. The author notes that the Farr paper used different nomenclature, but the ideas remain the same. The important point is Farr’s definition of “useful work,” or strain energy. This assumes that the makeup increment is small enough that the force remains constant, which is reasonable when considering the problem in differential terms. More importantly, the strain energy is assumed to be the axial cross-sectional force, either in the pin neck or the box shoulder, multiplied by the thread advancement. This implicitly assumes that only the unengaged areas of the connection absorb any strain energy; the actual threaded region does not strain. Farr, without explicitly stating such, assumed that the threaded region of the connection is rigid.

At first glance, this assumption may seem incorrect. However, it is generally accepted that most of the stresses occur in these areas, and a rigid-thread assumption will make the calculations conservative—i.e., the calculated stresses will be higher than actual stresses. The author states that this simplification is usually accepted and that history supports it.

Splitting the *F*_{Axial} term in Farr’s equation into stress×area, either for the pin or box side, makes it possible to find the torsional yield of the box cross section and pin cross section, named T1 and T2, respectively, by *API RP 7G.*

The next equation in the paper enables calculation of makeup torque by setting a target stress—that is, the stress achieved in whichever cross section is smaller, box or pin, after the makeup is complete.

In addition to torque, tensile capacity needs to be considered. At low makeup torques, the connection fails not by yielding the material, but by leaking because of loss of seal pressure as tension is applied to the connection. Increasing the makeup torque increases the tension required to separate the shoulders. This effect is calculated by assuming that the unengaged areas of the pin neck and box counterbore deform together with applied tension until the shoulders separate, so that these areas are essentially two springs in parallel. Importantly, this is a continuation of the presumption of a rigid threaded area. This leads to the shoulder-separation line expressed mathematically in *API RP 7G* as follows.

. . . . . . . (1)

The point where this shoulder-separation line intersects with the pin yield limit is T4, a special makeup torque where the shoulder will separate and the pin will yield at the same amount of tension applied. At makeup torques greater than T4, the pin stress increases because of the makeup torque applied, which decreases the pin’s ability to carry string weight.

Critical points from the industry-standard methods of calculating single-shouldered connection capacities include:

- Farr’s modified screw-jack equation is the result of an incremental energy balance.
- To determine the amount of energy absorbed by the connection, the engaged threads of the connection are assumed to be rigid. All the energy is absorbed by the pin neck and box counterbore.
- In defining the connection’s performance in tension and torque, the same rigid-thread assumption is used to require the pin neck and box counterbore to deform together when separating the shoulders or yielding the pin.

## Extension to Double-Shouldered Connections

Double-shouldered connections can be analyzed the same way that single-shouldered connections have been for decades. The assumptions presented previously are clearly not perfectly true, and may even be significantly inaccurate at some points. However, the historical use of those equations provides confidence that the results are reasonable and likely conservative.

The paper illustrates areas of interest for a generic double-shouldered connection. The outer shoulder is the one that is typical for API single-shouldered connections. The inner shoulder is the secondary pin-tip shoulder and the cross sections associated with it.

A full derivation of the equations of interest is given in the Appendix of the complete paper. Using the same rigid-thread assumption as the single-shouldered equation, the total torque contributes to force/stress in both the outer and inner connections (which is the reason double-shouldered connections provide higher torque capacities—the applied torque energy is shared at two stressed locations).

Additional equations and schematics are presented to explain and compute relationships between outer-side and inner-side forces, including the forces and torques required to close gaps between shoulder surfaces. Final torque equations are also provided. Schematics show the necessity of overlaying the curves for the outer and inner sides and identifying the acceptable operating region.

## Using the Equations

With the ability to analyze the combined connection capacities of a double-shouldered connection comes the ability to design a connection to meet whatever performance requirements are in place. The author uses his experience in designing these connections for tool companies to make the following suggestions.

- Smaller gaps are better. A larger gap consumes more of the available stress in the critical cross sections before engaging the second shoulder; this lowers the overall torque capacity.
- Longer unengaged lengths are better. The engagement forces are directly proportional to the gap size, but indirectly proportional to the unengaged length. If tolerances of two or three one-thousandths of an inch are unreasonable, lengthening the counterbore (presuming the gap is at the pin nose) can make a dramatic difference. With a longer counterbore, the strain required to close the gap is reduced, in turn reducing the stress in the connection.
- Both ends of the tolerances should be considered. In many cases designers will consider only the “worst case” when the pin-nose gap is as large as the tolerances allow. Typically, however, the smallest gaps can be a problem for torsional failure at the pin nose. A good design will consider all the alternatives and optimize within the given constraints.
- The gap should be placed only at one side. Designing for multiple gap sizes at both the outer and inner side of the connection almost always leads to decreased performance. Tolerancing the connection lengths such that the gap can only occur at one side (which is typical for most commercial designs) not only simplifies the design work, but usually increases the capacity limits.

## Conclusions

- A set of equations for understanding torsion and tension capacities for double-shouldered connections has been developed using the same basic assumptions as for single-shouldered torque connections—rigid threaded areas and an energy balance—and in the same way. These equations are presented and developed in detail in the complete paper.
- There is every reason to have the same confidence for the new equations as for the old.
- The new equations should help the industry understand connections better.

*JPT*Technology Editor Judy Feder, contains highlights of paper SPE 194180, “An Analytical Model for Double-Shouldered Connection Strengths,” by

**Grant Pettit,**SPE, Bureau Veritas, prepared for the 2019 SPE/IADC International Drilling Conference and Exhibition, The Hague, 5–7 March. The paper has not been peer reviewed.

New Analytical Model Enhances Understanding of Connection Strengths

01 July 2019

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