Actual 3D Shape of Wellbore Trajectory: An Objective Description for Complex Steered Wells

Abstract All conventional methods for calculating the trajectory of a wellbore involve assumptions. Most calculations assume that segments of the trajectory can be approximated as straight lines, polygonal lines, cylindrical helixes, circular arcs, or combination of these approximations. Conventional methods calculate the course coordinates of a well survey interval according to the predetermined shape for the trajectory. But, what does the real shape of a trajectory in each surveyed interval really look like? Why? What parameters play decisive roles in determining the true shape of a surveyed interval? Is it reasonable to assume that every survey interval has the same kind of approximate shape? Until now, the drilling industry has not found satisfactory solutions to these questions. This paper presents the results of research about the true shape of a wellbore trajectory relative to its survey stations. It also provides a universal equation for a wellbore trajectory in a surveyed interval and presents an objective approach for describing and calculating a wellbore trajectory in space. The new method does not assume that the shape of wellbore trajectory is a set of typical approximations. The study shows that the wellbore curvature and torsion at survey stations determine the shape of the wellbore trajectory at subsequent intervals, and that 3D coordinates of wellbore trajectory in a local coordinate system are linear, quadratic, and cubic functions vs. curve length, respectively. The new method yields a continuous wellbore trajectory. Extensive simulations have been carried out and are reviewed in this paper. The paper compares the results predicted by mathematical simulation using the new method with actual, observed trajectories and describes the results from other methods to show the accuracy of the new, improved method. The new method shows excellent precision in calculation and reliability. This study has also proved that the minimum curvature method and the natural curve method, the most commonly used methods, are related to the new method.