Optimal control of deep petroleum borehole trajectory tracking

Abstract

• A new problem of optimal tracking of a deep borehole trajectory is posed. • Target functions related to the hole smoothness, cost, and length are considered. • Optimal nonlinear control methods and gradient projection method are jointly used. • Examples of the optimal smoothing of deep projected trajectories are discussed. This paper is concerned with the application of optimal control theory to the problem of tracking deep oil and gas borehole trajectories. Based on the methods of differential geometry, the mathematical model of the trajectory curve with its curvature representing controlling variable is elaborated in the form of ordinary differential equations: The objective functional chosen as integral curvature, length or cost of the borehole are considered. The techniques for the optimization problem solving are developed with the use of the continuous version of the step-by-step anti-gradient projection on the hyper-planes of linearized constraints. At every step of the minimization procedure, the constraints spoilt by the linearization operations are restored through the use of the Newton method. Some examples are considered for a borehole with fixed and shifting boundary positions under conditions of minimizing its total curvature and length. It is shown that it is possible to improve the smoothness of the borehole trajectory using the outlined approach, and in so doing, reduce the friction and resistance forces impeding the drill string motion.