Whirl Interaction of a Drill Bit with the Bore-Hole Bottom

Abstract

This paper deals with the theoretic simulation of a drill bit whirling under conditions of its contact interaction with the bore-hole bottom rock plane. The bit is considered to be an absolutely rigid ellipsoidal body with uneven surface. It is attached to the lower end of a rotating elastic drill string. In the perturbed state, the bit can roll without sliding on the bore-hole bottom, performing whirling vibrations (the model of dynamic equilibrium with pure rolling when maximum cohesive force does not exceed the ultimate Coulombic friction). To describe these motions, a nonholonomic dynamic model is proposed, constitutive partial differential equations are deduced. With their use, the whirling vibrations of oblong and oblate ellipsoidal bits are analyzed, the functions of cohesive (frictional) forces are calculated. It is shown that the system of elastic drill string and ellipsoidal bit can acquire stable or unstable whirl modes with approaching critical Eulerian values by the parameters of axial force, torque and angular velocity. The analogy of the found modes of motions with ones of the Celtic stones is established. It is shown that the ellipsoidal bits can stop their whirling vibrations and change directions of their circumferential motions in the same manner as the ellipsoidal Celtic stones do. As this takes place, the trajectories of the oblate ellipsoidal bits are characterized by more complicated paths and irregularities.

Critical Buckling of Drill Strings in Cylindrical Cavities of Inclined Bore-Holes

“…Later, a lot of refinements were introduced in theoretical models of the buckling effects [7][8][9][10][11][12]. Analysis of whirl interaction of a drill bit with the bore-hole bottom was performed by Musa et al [13].…”

Elastic Bending Deformation of the Drill Strings in Channels of Curve Wells

“…The paper discussed the analogy between the phenomena of nonholonomic motions of the Celtic stones and the ellipsoidal bits. Also techniques of solution of the equations of the bit whirling were proposed [7].…”

“…Such elaborated software permits one to construct its trajectory securing the smallest values of resistance forces and to choose the least energy-consuming and safe regimes of drilling [6]. Nabil W. Musa, et al (2015) discussed the problem of simulation of a drill bit whirling under conditions of its contact interaction with the bore-hole bottom rock plane. It was showed that the problem on rolling of a rigid ellipsoidal drill bit on a bore-hole bottom plane, the bit is supposed to be attached to the lower end of an elastic rotating drill string.…”