Designing Three-Dimensional Directional Well Trajectories Using Bézier Curves

A relatively simple, general, and very flexible method to design complex, three-dimensional hole trajectories can be obtained by using a 3D extension for Bézier curves. This approach offers superior results in terms of coding, use, and flexibility compared to other methods using double-arc, cubic functions, spline-in-tension functions, or constant curvature. The mathematics is surprisingly simple, and the method can be used to obtain trajectories for any of the four typical end conditions in terms of inclination and azimuth, namely: free-end, set-end, set-inclination/free azimuth, and free-inclination/set-azimuth. The resulting trajectories are smooth, with continuous and smooth change of curvature and toolface, better exploiting the expected delivery of modern rotary steerable deviation tools, particularly the point-the-bit and the push-the-bit systems. With the relevant parameters at any point of the trajectory (curvature and toolface angle) an automated system can steer the hole toward the defined targets in a smooth fashion. The beauty of the method is that the description of the trajectory is obtained with one single expression that handles the three space coordinates, instead of working with three separate coordinate functions. It uses a generalization of the well-known 2D Bézier curve. The concept is easy to understand, and implementation even using spreadsheets is straightforward. Besides, the conditions at both ends (coordinates and inclination/azimuth for set ends) the trajectory curve has up to two independent parameters. By playing suitably with these parameters, one can obtain a curve that favors the reduction of drag and torque during drilling, tripping, and casing running. In addition to the formulation for trajectory calculation, the paper presents the expressions to calculate the inclination, azimuth, curvature, and toolface at any point along the trajectory. Proper numerical examples illustrate the various end-conditions. The method can be used during the hole planning cycle as well as during the hole drilling for automatic and manual steerage.