Borehole Trajectory Modeling Using Circular Arcs, Constant Curvature and Toolface, and Sections With Constant Build-Up and Turn Rates

Abstract Wellbore trajectories are often represented by numerous survey stations, either reflecting measurement data or generated by planning-stage algorithms using generative techniques. Finely discretized trajectories can strain computational resources required for torque, drag, or hydraulic calculations. A smoother representation using segmented sections provides a more efficient approach for hydro-mechanical analyses. This method identifies trajectory sections within acceptable tolerances using three types of curves: circular arcs (employed in minimum curvature calculations), constant curvature and toolface sections (representing the behavior of rotary steerable systems, RSS), and constant build-up and turn rate curves (modeling rotary assemblies or positive displacement motors, PDM, in rotating mode). By minimizing the number of sections and selecting optimal curve types, the method efficiently approximates the original trajectory while enabling reconstruction during the planning stage, throughout drilling, or when analyzing completed wells. The difference between the smoothed and original trajectories also provides an accurate estimate of wellbore micro-tortuosity. Additionally, the method supports long-step calculations for hydro-mechanical evaluations, accelerating engineering analyses, enhancing generative trajectory algorithms, and facilitating stochastic approaches to model accuracy estimation. This approach effectively balances speed and precision, enabling efficient design and operational decision-making.