## Abstract

In modern oil/gas producing industry, vertical, 2D and 3D directed, and multilateral (branched) boreholes are drilled. Their trajectories are designed depending on the petroleum deposit depth and structure, properties of mining rocks, their hardness, heterogeneity, fracturing anisotropy, permeability, and so on. Therefore, the borehole cost and its productivity are determined by the length, smoothness, and configuration of its trajectory. To enhance efficiency of a borehole and to reduce cost of its drivage, to enlarge rate and volume of the reservoir depletion, it is proposed to use methods of optimal control for the best tracking of its trajectory. Through application of the differential geometry correlations, the mathematic model of the borehole outline in the form of nonlinear ordinary differential equations system is elaborated. Different objective functions, representing total integral curvature of the borehole axis line, its length, and cost of its drivage, are selected; additional constraints, separating allowed and forbidden zones of passing, are chosen. The functions of the trajectory curvature and torsion are used as controlling variables.

The continuous correlations of the model are discretized and, further, the techniques of nonlinear programming and optimal control are employed.

On the basis of the method of objective function gradient (antigradient) projection on the linearized constraint planes, the step-by-step algorithm of approaching to the optimal trajectory is elaborated. To correct the spoilt constraints, at every step of calculations, the Newton method is used. The elaborated approach is applied to optimization of deep curvilinear borehole outlines. The results of numerical analysis are discussed. It is shown that smoothing the hole trajectory permits also to diminish the contact and frictional interaction between the drill string and bore-hole wall and, by this, to decrease the resistance forces acting on the string during tripping in/out operations performing and to diminish energy expenditures for these operations fulfillment; to decrease the rate of the drill string tube wear; and to reduce the drill string sticking occurrence probability.

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