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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