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Cross-axis control system design for borehole drilling based on damping optimum criterion and utilization of proportional-integral controllers

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Abstract

This paper presents a control system design methodology for the drill-string rotary drive and draw-works hoist system aimed at their coordinated control for the purpose of establishing a fully-automated mechatronic system suitable for borehole drilling applications. Both the drill-string rotary drive and the draw-works hoist drive are equipped with proportional-integral (PI) speed controllers, which are readily available within modern controlled electrical drives. Moreover, the rotary speed control system is equipped with torsional active damping system and drill-string back-spinning prevention scheme for the case of stuck drill-bit scenario, whereas the draw-works-based drill-bit normal force control system is extended with the auxiliary control system aimed at timely prevention of the drill-string torsional overload. The design of proposed control systems has been based on suitable reduced-order control-oriented process models and a practical tuning methodology based on the damping optimum criterion aimed at achieving the desired level of closed-loop system damping. The functionality of the proposed cross-axis control system has been systematically verified, first by experimental tests of individual rotary/vertical axis control systems on a downscaled laboratory experimental setup, followed by thorough simulation study of the overall control system for realistic scenarios encountered in the field.

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Availability of data and material

The source code and simulation models can be made available upon request made via direct correspondence with the first author (danijel.pavkovic@fsb.hr), for non-commercial use only.

Abbreviations

BHA:

Bottom-hole assembly

HWDP:

Heavy-weight drill-pipes

PI:

Proportional-integral (controller)

VSD:

Variable speed drive

sat:

Saturation function

\(F_b\), \(F_t\) :

Buoyancy and drill-bit penetration force

\(F_r\), \(F_{ds}\) :

Steel wire rope force and drill-string dilatation force

\(m_1\), m, \({\hat{m}}_r\) :

Rotary drive motor torque, drill-string torque and estimated drill-string torque

\(m_{f1}\), \(m_{f2}\) :

Rotary drive motor-side and drill-bit friction torque

\(M_{m,op}\), \(M_{max}\) :

Operator’s torque limit and modified torque limit for rotary drive

\(v_B\), \(v_r\) :

BHA longitudinal speed and steel wire rope longitudinal speed

WoB, \(WoB_R\) :

Weight-on-Bit measurement and reference (target) value

\(\Delta {\hat{m}}_r\), \(\Delta {\hat{m}}_{Rr}\) :

Estimated drill-string torque variation and related reference value

\(\Delta v_r\), \(\Delta x_r\) :

Steel wire rope longitudinal dilatation speed and dilatation

\(\Delta v_d\), \(\Delta x_d\) :

Drill-string longitudinal dilatation speed and dilatation

\(\Delta \varOmega \), \(\Delta \alpha \) :

Drill-string motor vs. drill-bit (BHA) speed difference and torsional angle

\(\Delta WoB\), \(\Delta WoB_R\) :

Weight-on-Bit measurement and WoB reference from auxiliary controller

\(\omega _{Rw}\), \(\omega _w\) :

Draw-works winch speed reference and actual value

\(\omega _{R,op}\), \(\omega _{Rr}\) :

Operator’s speed target and modified speed target for rotary drive

\(\omega _1\), \(\omega _2\) :

Rotary drive motor speed and drill-bit (BHA) rotational speed

\(c_t\) :

Drill-bit penetration resistance coefficient

\(D_2, \ldots, D_n\) :

Damping optimum characteristic ratios

\(D_{4d,max}\) :

Non-dominant characteristic ratio maximum value (active damping system design)

g :

Gravity (free-fall) acceleration

i, \(i_g\), z :

Rotary drive, winch drive and pulley transmission ratio

\(J_1\), \(J_2\) :

Rotary drive motor-side and BHA-side moments of inertia

\(J_{bit}\), \(J_c\), \(J_{hw}\), \(J_{dp}\) :

Drill-bit, collar, HWDP and regular drill-pipe moments of inertia

k, c :

Drill-string torsional stiffness and damping coefficients

\(k_r\), \(c_r\) :

Steel wire rope longitudinal stiffness and damping coefficients

\(k_{ds}\), \(c_{ds}\) :

Drill-string longitudinal stiffness and damping coefficients

\(K_{c\omega }\), \(T_{c\omega }\) :

Rotary drive PI controller gain and integral time constant

\(K_{wob}\), \(T_{wob}\) :

WoB PI controller gain and integral time constant

\(K_{md}\), \(T_{IR}\) :

Torsional torque feedback proportional gain and reference correction term time constant

\(K_{mr}\), \(T_{mr}\) :

Auxiliary cross-axis PI controller gain and integral time constant

\(K_{pw}\), \(T_{pw}\) :

Equivalent gain and time constant of simplified drill-string model

\(m_h\), \(m_B\) :

Hook mass and BHA mass

\(m_{lim}\) :

Torsional torque limit (cross-axis control)

\(M_C\), \(M_S\) :

Rotary drive drill-bit side Coulomb and static friction torque

\(M_{cm}\) :

Rotary drive motor-side Coulomb friction torque

\(r_d\), \(r_t\) :

Draw-works drum radius and effective drill-bit radius

\(T_e\) :

Damping optimum closed-loop equivalent time constant

\(T_{ed}\) :

Active damping system closed-loop equivalent time constant

\(T_{ewob}\) :

WoB control system equivalent closed-loop time constant

\(T_f\), \(T_{eo}\) :

WoB measurement filter and drill-string torque estimator time constant

\(T_\Sigma \), \(T_{e\omega }\) :

Rotary drive torque lag and draw-works drive speed control loop lag

\(\delta \), \(\omega _s\) :

Stribeck coefficient and Striebeck speed

\(\Delta _\omega \) :

Zero-speed region (Karnopp friction model)

\(\kappa _{mr}\) :

Scaling factor (auxiliary cross-axis PI controller)

\(\mu (\omega _2)\) :

Drill-bit speed-dependent friction coefficient

\(\mu _C\), \(\mu _S\) :

Drill-bit side Coulomb and static friction coefficients

\(\varOmega _0\) :

Freely-oscillating drill-string resonance frequency

\(\varOmega _{01}\) :

Motor-side resonance frequency for the case of stuck drill-bit (BHA)

\(\varOmega _{02}\) :

BHA-side resonance frequency for the stiff motor speed control

\(\varOmega _{bw}\) :

Auxiliary cross-axis control system closed-loop bandwidth

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Acknowledgements

Authors gratefully acknowledge that the presented research has been carried out within the Centre of Research Excellence for Data Science and Advanced Cooperative Systems (ACROSS-DataScience). This work has also been conducted within the activities of the Scientific Council for Oil and Gas Economy and Energy of the Croatian Academy of Sciences and Arts (HAZU). Authors would also like to express their appreciation of the efforts of the journal Editor and anonymous reviewers whose comments and suggestions have helped to improve the quality of the presented subject matter.

Funding

Authors gratefully acknowledge that the presented research has been funded by the European Regional Development Fund under the Grant KK.01.1.1.01.0009 (DATACROSS).

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DP: Conceptualization, Investigation, Formal analysis, Writing - original draft, Funding acquisition. PŠ: Methodology, Experimental work. MC: Writing - review & editing, Supervision. MK: Validation.

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Correspondence to Pavle Šprljan.

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Pavković, D., Šprljan, P., Cipek, M. et al. Cross-axis control system design for borehole drilling based on damping optimum criterion and utilization of proportional-integral controllers. Optim Eng 22, 51–81 (2021). https://doi.org/10.1007/s11081-020-09566-z

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