Proportional-Derivative Control of Stick-Slip Oscillations in Drill-Strings

Stick-slip oscillation in drill-string is a universal phenomenon in oil and gas drilling. It could lead to the wear of drill bit, even cause catastrophic failure of drill-strings and measurement equipment. Therefore, it is crucial to study drilling parameters and develop appropriate control method to suppress such oscillation. This paper studies a discrete model of the drill-string system taking into account torsional degreeof-freedom, drill-string damping, and highly nonlinear friction of rock-bit interaction. In order to suppress the stick-slip oscillation, a new proportional-derivative controller, which can maintain drill bit’s rotation at a constant speed, is developed. Numerical results are given to demonstrate its efficacy and robustness.


Introduction
For conventional rotary drilling, drill-string is a principal component for the entire drilling system. As shown in Fig.1 (a), a typical drilling system is composed of a rotary table, a slender drill pipe, a drill collar, and a drill bit. The drill-string could run several kilometres deep, but the diameter of the drill pipe does not exceed 0.3 metres.
When drill pipes transmit the drive torque from rotary table to drill bit, deformations in drill pipes are very complex, especially considering the constraints of the borehole. The slenderness of the drill pipe makes it prone to exhibit undesired oscillations, including bit bouncing, lateral oscillation, torsional oscillation, and whirling motion. In oil and gas drilling, stick-slip behavior exists in 50% of drilling time [1], and the whipping and high rotation of the drill bit may cause bit bouncing and whirling motion [2]. Moreover, formation change is also very complex during drilling procedure, so the interactive model between rock and bit is highly nonlinear [3]. In addition, the interaction between drill-strings and borehole also makes drilling system uncertain [4].    [5][6][7]. Balakumar [5] developed a lumped-parameter model to study the coupled axial, torsion, and lateral dynamics of a drill-string dynamics, including stick-slip and delay effect. Canudas-de-Wit et al. [6] proposed a lumped parameter model to use the weight on bit as an additional variable for suppress stick-slip oscillation.
Liping Tang [7] proposed a lumped torsional pendulum model of the drilling system, and studied the stick-slip dynamics and negative damping effect. However, most of the dynamics model could not be guaranteed to suppress the stick-slip oscillation steadily and appropriate to the parameters changing. Since this paper proposed a proportional-derivative (PD) controller to suppress stickslip oscillations using the lumped-mass model.
The remaining sections of this paper are organized as follows. In Section 2, a discrete dynamic model is developed for the drill-string system, and a new PD controller is studied. In Section 3, simulation results are presented to validate the effectiveness of the controller.
Finally, conclusions are drawn in Section 4.

Drill-String Model and Proportional-Derivative Controller
The drilling system, which is illustrated in Fig.1 (a), is comprised of rotary table, along with drill pipe, drill collar and drill bit. In order to simplify the system, a discrete drill-string model is developed for analytical and numerical study. The mechanical schematic of drill-string system shown in Fig.1  Making use of the dynamic theory, the government equation of motion for the discrete system [8] can be written as Where Φ is the rotational position of lumped mass, J is the rotational inertia matrix, C is the torsional damping matrix, and K is the torsional stiffness matrix, T is the torque of friction between drill bit and rock, and U is the control torque applied on the rotary table. According to the finite element method, matrixed (J, C, and K) are comprised as follows.
The frictional torque for the rock-bit contact is modelled as a combination of Stribeck [9] and the Karnopp's [10] model described as In the sticking phase(|̇| < ζ and | | < ), the bit velocity is less than a small positive constant ξ, and the Wob is the weight on bit (WOB), Rb is the bit radius.
In the process of stick to slip(|̇| < ζ and | | > ), the drill bit velocity is still less than the small constant ξ, but the reaction torque Tr is greater than the static friction torque Ts and the drill bit start to move.
In the slip phase(|̇| > ζ), the drill bit starts to rotate, and the friction torque includes the effects of drill bit radius, the WOB, and the dynamic friction coefficient [11].
For the purpose of control stick-slip oscillation, a new proportion-differential controller is designed.
Where u is the control torque applied on rotary table, k1 is proportion coefficient, k2 and k3 is the differential coefficient, ̇ is the desired angular velocity, ̇ is the rotary table velocity, and ̇ is the drill bit velocity.

Simulation Results
Numerical studies were conducted with the earlier described discrete system model by applying the PD controller. The geometry and physical parameters for the drill-string are given in Table 1. In the follow subsections, the stick-slip oscillation, control of stick-slip, and the respond to WOB oscillation are brought forth.  Fig.2 (a), the peak speed of the drill bit is about 3.5 rad/s, which is much greater than the rotary table velocity. As is show in Fig.2 (b), the angular acceleration of drill bit also changes between positive and negative. In this rapidly changes of angular speed and angular acceleration, the tooth of the drill bit will be more likely to wear out than it is in constant speed.
Therefore, it is necessary to study the stick-slip oscillation and suppress it. control torque is about 8.5 kN·m, as seen in Fig.3 (b). In Fig.4, we can see that drill bit velocity converges at 5.26 rad/s, which is a little less than the desired velocity 6 rad/s.
The result for torsional oscillation in Fig.3 and Fig.4 indicate that the PD controller can suppress stick-slip oscillation and keep the velocity on a constant desired velocity.     Future work will focus on improving controller's accuracy and verify through experimentation. Borehole constraints and axial bit movement will also be considered in the discrete model.