Constrained optimization assists deviated wellbore trajectory selection from families of quadratic and cubic Bezier curves

Abstract

The ease of calculation of families of cubic and quadratic Bezier curves using just one or two control variables makes them well-suited for identifying smooth, deviated wellbore trajectories using easy-to-implement, constrained optimization techniques. Most wellbore trajectory optimization methods currently applied during the wellbore planning stages aim to minimize measured depth (MD) while constraining dog-leg severity (DLS) and/or torque and drag to select drillable, smooth wellbores. This study identifies the benefits of conducting MD optimization of Bezier curves using multiple constraints related to different characteristics of DLS and absolute changes in tool-face angle (ΔTFA). Whereas DLS is a major influence on wellbore tortuosity on the multi-meter scale, ΔTFA influences wellbore roughness on a finer scale and in a different way than DLS. As rotary-steerable bottom-hole assemblages are now the standard for drilling the multiple horizontal wells required for tight gas and oil reservoirs, minimizing the magnitude of TFA changes can speed up drilling and reduce borehole roughness. This study identifies the benefits of applying a sequence of up to four constraints (maximum DLS combined with average DLS and/or average or maximum ΔTFA) to minimize MD while achieving the smoothest possible wellbore trajectory from a family of cubic Bezier curves, configured to drill from a vertical starting location to a horizontal finishing location offset in 3D space. On the other hand, families of quadratic Bezier curves drilling some relatively simple trajectories between two points located on a two-dimensional plane can be effectively optimized for MD and smoothness using solely a maximum DLS constraint.