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RESEARCH PAPERS

# Planning 3D Well Trajectories Using Cubic Functions

[+] Author and Article Information
Jorge H. B. Sampaio

Curtin University of Technology, 24 Dick Perry Avenue, Kensignton, WA 6151 Australiajsampaio@peteng.curtin.edu.au

Directional drilling programs like Technical Toolboxes $3D3$, Landmark Compass, and Maurer Engineering Suite of Drilling Programs.

In order 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 $B$.

In fact, an expression exists in terms of generalized hypergeometric functions.

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 inclination, azimuth, curvature, and tool face at any point of the trajectory are shown in a following section.

The horizontal coordinate functions cannot be free slope cubic functions because for small inclinations they will reach the target coordinate much “faster” than the vertical coordinate and convergence will not be attained.

J. Energy Resour. Technol 128(4), 257-267 (Apr 05, 2006) (11 pages) doi:10.1115/1.2358140 History: Received August 19, 2004; Revised April 05, 2006

## Abstract

This work presents a mathematical method to design complex trajectories for three-dimensional (3D) wells. Three-dimensional cubic 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, that 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, which can be adjusted to fit the design requirements and to optimize the trajectory. Several numerical examples illustrate the various end conditions. The paper also presents the full mathematical results (expressions for the 3D path, actual curvature, and actual tool face). The method is directly applicable to the well planning cycle as well as to automatic and manual hole steering.

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## Figures

Figure 6

Set inclination and free azimuth

Figure 7

Unit vectors at a point P of the trajectory

Figure 8

Optimized trajectory for a set inclination and azimuth case

Figure 1

Free slope condition cubic function

Figure 2

Set slope condition cubic function

Figure 3

Free inclination and free azimuth

Figure 4

Set inclination and set azimuth

Figure 5

Free inclination and set azimuth

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