Critical buckling of drill strings in curvilinear channels of directed bore-holes

The problem about theoretic modeling incipient stage of a drill string (DS) buckling in a curvilinear channel of a deep directed bore-hole is stated. On the basis of the theory of curvilinear flexible rods, non-linear differential equations describing elastic bending of the DS inside the bore-hole cavity are deduced. The effects of the DS curvature, its axial non-uniform prestressing, action of torque, and interaction between the DS and the bore-hole surface are taken into consideration. Owing to the use of curvilinear coordinates in the constraining channel surface and a specially chosen concomitant reference frame, it became possible to separate the desired variables and to reduce the total order of the equations system. With the aim to analyze critical states of the DS equilibrium and to construct modes of its stability loss, these equations are linearized in the vicinity of the considered state and eigen-value problem is formulated. Techniques for its numerical solution are proposed. Via its use the critical loads and shapes of buckling DSs are found for their different locations inside the bore-hole with circular trajectory. It is shown that the buckling proceeds with generation of irregular harmonic wavelets localized inside the well or in the neighborhood of its boundaries. Influence of boundary conditions, torque, clearance value, and channel axis curvature on the buckling process is analyzed. The presented results illustrate the applicability of the method proposed for the regimes of drilling and the drill string lowering and raising.