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RESEARCH PAPERS

# Planning 3D Well Trajectories Using Spline-in-Tension Functions

[+] Author and Article Information
J. H. Sampaio

Department of Petroleum Engineering, Curtin University of Technology, Kensington, WA 6151, Australiaj.sampaio@curtin.edu.au, jrgsampaio@gmail.com

Although the table has been calculated with increments of 0.01 for the variable $u$, only rows where $u$ is multiple of 0.04 are presented.

Procedures to calculate the inclination, azimuth, curvature, and tool face at any point of the trajectory are shown in Appendix .

In order for this representation to satisfy the right hand convention, the positive vertical axis must point downwards.

The length along the trajectory.

Dots above the variable indicate differentiation with respect to the parameter $u$.

$A∙B$ represents the scalar product between $A$ and $B$.

J. Energy Resour. Technol 129(4), 289-299 (May 17, 2007) (11 pages) doi:10.1115/1.2790980 History: Received May 30, 2005; Revised May 17, 2007

## Abstract

This work presents the mathematical method to design complex trajectories for three-dimensional (3D) wells using spline in tension as coordinate functions. 3D spline-in-tension trajectories are obtained for various end conditions: free end, set end, free inclination/set azimuth, and set inclination/free azimuth. The resulting trajectories are smooth continuous functions which better suit the expected performance of modern rotary steerable deviation tools, in particular, point-the-bit and push-the-bit systems. A continuous and gradual change in path curvature and tool face results in the smoothest trajectory for 3D wells, which, in turn, results in lower torque, drag, and equipment wear. The degree of freedom and the associated parameters of the 3D curves express the commitment between the average curvature to the final length of the path that can be adjusted to fit the design requirements and to optimize the trajectory. Several numerical examples illustrate the various end conditions. This paper also presents the full mathematical expressions for the 3D path for four end conditions. The method is directly applicable to the well planning cycle as well as to automatic and manual hole steering’s. Spline-in-tension functions differ from the cubic functions in the extent that an additional parameter, which represent the “tension” of the curve, can be controlled. A totally “relaxed” curve is identical to a cubic curve, and as the tension increases a shorter curve length is obtained with a consequent effect in the curvature profile along the curve. In the limit, as the tension increases to infinite, the spline-in-tension approaches to a straight line. The tension offers an additional degree of freedom, which can be used to further optimize the final trajectory. The 3D spline-in-tension model provides the most versatile model to plan a 3D well trajectory to date. Suitable manipulation of the curve parameters, namely, $L0$, $L1$, and the three tensions, allows to give to the planned trajectory any desired behavior.

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## Figures

Figure 1

Free slope SiT function

Figure 2

Set slope SiT function

Figure 3

Free end 3D SiT trajectory

Figure 4

Set end 3D SiT trajectory

Figure 5

Free inclination and set azimuth 3D SiT trajectory

Figure 6

Set inclination and free azimuth 3D SiT trajectory

Figure 7

Effect of tension (λ=0.01)

Figure 8

Effect of tension (λ=5)

Figure 9

Effect of tension (λ=10)

Figure 10

Optimized trajectory for set end 3D SiT trajectory

Figure 11

Unit vectors at a point P of the trajectory

## Errata

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