Deep Borehole Trajectory optimization on the Basis of the Lagrange Problem Statement

In the paper, the problem associated with optimization of deep oil and gas borehole outlines is posed. Vertically drilled boreholes usually aim at piercing several different producing levels to improve recovery from multiple horizons by combination of their production. An inclined deep borehole has a better luck to meet petroliferous layers than a vertical well, but there are limits to what trajectory of a curvilinear borehole can be drilled and what cost will it absorb. Solutions to these questions can be achieved through the use of methods of differential geometry, nonlinear programming and optimal control. On their basis, the mathematic model of the well trajectory tracking is elaborated. It includes differential equations, objective functional, and constraints, limiting phase variables and controlling function. The objective functionals are selected in the integral (Lagrange) form minimizing the trajectory curvature, length, or cost; the algorithm of their minimization is proposed; typical examples are considered.