Buckling of a drill-string in two-sectional bore-holes

The nature of the phenomena of drill-strings buckling in channels of deep curvilinear bore-holes is essentially complicated by complex combination of acting distributed forces (gravity, contact and friction), large length of the string (leading to a virtual loss of its bending stiffness), and its intricate geometry. Therefore, theoretical modelling of these effects turns out to be difficult to implement. In this article, the touched problem is stated with the use of the curvilinear flexible rods theory, differential geometry concepts, basic foundations of channel surfaces theory, and methods of numerical mathematics. The elaborated software is applied to the global analysis of the incipient stage of the drill-string buckling inside a deep bore-hole consisting of two nearly rectilinear sections bridged with curvilinear interjacent part. To solve the problem, firstly, the critical stress-strain state of the drill-string (eigenfunction) is established, and then, the shape of the buckled drill-string (eigenmode) is built. It is shown that the stated problem refers to the singularly perturbed type and so, depending on the knee angle and friction coefficient, the mode of stability loss represents a short wavelet localized at the lower end of the lower section or it is shifted to the knee zone of the bore-hole. The features of critical behavior of the drill-strings associated with the values of the knee angle of the bore-hole, system clearance, and friction coefficient, as well as critical loads and buckling mode wavelets are discussed.