Modification of the Infinite-Dimensional Neutral-Type Time-Delay Dynamic Model for the Coupled Axial–Torsional Vibrations in Drill Strings With a Drag Bit

Abstract

Drill strings are subjected to complex coupled dynamics. Therefore, accurate dynamic modeling, which can represent the physical behavior of real drill strings, is of great importance for system analysis and control. The most widely used dynamic models for such systems are the lumped element models, which neglect the system distributed feature. In this paper, a dynamic model called neutral-type time delay model is modified to investigate the coupled axial–torsional vibrations in drill strings. This model is derived directly from the distributed parameter model by employing the d'Alembert method. Coupling of axial and torsional vibration modes occurs in the bit–rock interface. For the first time, the neutral-type time delay model is combined with a bit–rock interaction model that regards cutting process in addition to frictional contact. Moreover, mistakes made in some of the related previous studies are corrected. The resulting equations of motion are in terms of neutral-type delay differential equations with two constant delays, related to the oscillatory behavior of the system, and a state-dependent delay, induced by the bit–rock interaction. Illustrative simulation results are presented for a representative drill string, which demonstrates intense axial and torsional vibrations that may lead to system failure without a controller.

Axial–Torsional Dynamics of a Drilling Bit with Non-uniform Blade Arrangement

“…Then, the equations of motion are obtained by solving d'Alembert's solution and employing Riemann variables [28,29]. Subsequently, the following NTD model is given to represent the coupled axial-torsional dynamics of a drill-string [11,24,[28][29][30]:…”

“…However, the significance of the state-dependent delay model was not realized until much later [4,5]. The state-dependent delay model have been employed by many researchers to investigate the root cause of these unwanted vibrations [6][7][8][9][10][11][12]. However, this model assumes that the cutting operation is in progress whenever the bit rotates and hence, cannot capture the multiple-regenerative effects.…”

Prediction-Based Control for Mitigation of Axial–Torsional Vibrations in a Distributed Drill-String System

“…Along the line of [9], the procedure of obtaining the axial and torsional NTD models is explained in the following.…”

“…Different approaches have been employed to model these vibrational phenomena: lumped-parameter models [1]- [3], distributed parameter models [4]- [6], and neutral-type time-delay (NTD) models [7]- [9]. NTD models strike a favorable balance between modeling accuracy and complexity of simulation, analysis, and control.…”

“…Such an internal NTD model is obtained directly from the distributed parameter model by neglecting the damping along the drill string. The NTD model was established in [7], [8], where the bit-rock interaction is modeled as a mere frictional contact, and modified in [9] with consideration of the cutting process as well as the frictional contact in the bit-rock interaction. A review of mathematical modeling of axial and torsional self-excited drilling vibrations can be found in [11].…”