Deep Well Trajectory Optimization Using the Lagrange problem
DOI:
https://doi.org/10.32347/2410-2547.2026.116.336-343Keywords:
curvilinear drilling, deep boreholes, trajectory optimization, optimal controlAbstract
In the conditions of the modern oil and gas industry, the construction of wells of various types is carried out, including vertical, two-dimensional and three-dimensional directional, as well as branched wells. The selection and formation of the trajectory of the well is a complex engineering and technical task and is carried out taking into account the depth and geological structure of the oil and gas-bearing deposit, the physical and mechanical properties of rocks, the degree of their hardness, structural heterogeneity, permeability and other geological and technological factors.
The set of these parameters has a decisive influence on the technical and economic indicators of drilling, in particular the final cost of well construction and its operational productivity.
In order to increase the overall efficiency of the functioning of wells, reduce the costs of drilling them, and increase the mechanical speed of drilling, methods of optimal nonlinear control are widely used in the practice of drilling operations. The use of these methods makes it possible to more accurately control the process of formation of the well trajectory, to adapt the drilling parameters to the changing geological conditions and, as a result, to achieve higher indicators of economic and technological effectiveness.
Using the application of differential geometric correlations, a nonlinear mathematical model of the well contour was developed in the form of a system of nonlinear ordinary differential equations. Different objective functions are selected, reflecting the full integral curvature of the well axis, its length and the cost of its penetration; additional restrictions separating permitted and prohibited areas are selected. Trajectory curvature and torsion functions are used as control variables. Discrete continuous correlations are considered, as well as non-linear programming methods are applied.
Based on the method of projection of the gradient (antigradient) of the objective function onto linearized constraints on the plane, a step-by-step algorithm for approaching the optimal state has been developed. Newton's method is used for correcting corrupted constraints at each step of calculations. The results of the numerical analysis are discussed.
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Andrusenko O.M., Haidaichuk V.V., Kotenko K.E., Lazareva M.V. Neliniine optymalne keruvannia pobudovoiu traiektorii hlybokoi sverdlovyny (Nonlinear optimal control of the construction trajectory of a deep well) // Opir materialiv i teoriia sporud: nauk.-tekh. zbirn. – K.: KNUBA, 2025. – No.114 (2025). – P. 145 - 154.
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