The control of drilling vibrations: A coupled PDE-ODE modeling approach

Saldivar, Belem; Mondié, Sabine; Ávila Vilchis, Juan Carlos

doi:10.1515/amcs-2016-0024

Abstract

The main purpose of this contribution is the control of both torsional and axial vibrations occurring along a rotary oilwell drilling system. The model considered consists of a wave equation coupled to an ordinary differential equation (ODE) through a nonlinear function describing the rock-bit interaction. We propose a systematic method to design feedback controllers guaranteeing ultimate boundedness of the system trajectories and leading consequently to the suppression of harmful dynamics. The proposal of a Lyapunov-Krasovskii functional provides stability conditions stated in terms of the solution of a set of linear and bilinear matrix inequalities (LMIs, BMIs). Numerical simulations illustrate the efficiency of the obtained control laws.

References

Anabtawiii, M. (2011). Practical stability of nonlinear stochastic hybrid parabolic systems of Itô-type: Vector Lyapunov functions approach, Nonlinear Analysis: Real World Applications 12(1): 1386-1400.10.1016/j.nonrwa.2010.09.029
Search in Google Scholar
Bailey, J. and Finnie, I. (1960). An analytical study of drillstring vibration, Journal of Engineering for Industry, Transactions of the ASME 82(2): 122-128.10.1115/1.3663017
Search in Google Scholar
Ben-Tal, A. and Zibulevsky, M. (1997). Penalty/barrier multiplier methods for convex programming problems, SIAM Journal on Optimization 7(2): 347-366.10.1137/S1052623493259215
Search in Google Scholar
Boussaada, I., Mounier, H., Niculescu, S. and Cela, A. (2012). Analysis of drilling vibrations: A time delay system approach, 20th Mediterranean Conference on Control and Automation MED, Barcelona, Spain, pp. 610-614.
Search in Google Scholar
Canudas-de Wit, C., Rubio, F. and Corchero, M. (2008). D-OSKIL: A new mechanism for controlling stick-slip oscillations in oil well drillstrings, IEEE Transactions on Control Systems Technology 16(6): 1177-1191.10.1109/TCST.2008.917873
Search in Google Scholar
Challamel, N. (2000). Rock destruction effect on the stability of a drilling structure, Journal of Sound and Vibration 233(2): 235-254.10.1006/jsvi.1999.2811
Search in Google Scholar
Detournay, E. and Defourny, P. (1992). A phenomenological model for the drilling action of drag bits, International Journal of Rock Mechanics, Mining Science and Geomechanical Abstracts 29(1): 13-23.10.1016/0148-9062(92)91041-3
Search in Google Scholar
Fliess, M., Lévine, J., Martin, P. and Rouchon, P. (1995). Flatness and defect of non-linear systems: Introductory theory and examples, International Journal of Control 61(6): 1327-1361.10.1080/00207179508921959
Search in Google Scholar
Fridman, E. and Dambrine, M. (2010). Control under quantization, saturation and delay: A LMI approach, Automatica 45(10): 2258-2264.10.1016/j.automatica.2009.05.020
Search in Google Scholar
Fridman, E., Dambrine, M. and Yeganefar, N. (2008). Input to state stability of systems with time-delay: A matrix inequalities approach, Automatica 44(9): 2364-2369.10.1016/j.automatica.2008.01.012
Search in Google Scholar
Fridman, E., Mondié, S. and Saldivar, B. (2010). Bounds on the response of a drilling pipe model, IMA Journal of Mathematical Control and Information 27(4): 513-526.10.1093/imamci/dnq024
Search in Google Scholar
Grujić, L.T. (1973). On practical stability, International Journal of Control 17(4): 881-887.10.1080/00207177308932431
Search in Google Scholar
Halsey, G., Kyllingstad, A. and Kylling, A. (1988). Torque feedback used to cure slip-stick motion, Proceedings of the 63rd Society of Petroleum Engineers Drilling Engineering Annual Technical Conference and Exhibition, Houston, TX, USA, pp. 277-282.
Search in Google Scholar
Jansen, J. (1993). Nonlinear Dynamics of Oilwell Drillstrings, Ph.D. thesis, Delft University of Technology, Delft.
Search in Google Scholar
Jansen, J. and van den Steen, L. (1995). Active damping of self-excited torsional vibrations in oil well drillstrings, Journal of Sound and Vibration 179(4): 647-668.10.1006/jsvi.1995.0042
Search in Google Scholar
Javanmardi, K. and Gaspard, D. (1992). Application of soft torque rotary table in mobile bay, Technical Report IADC/SPE 23913, International Association of Drilling Contractors/Society of Petroleum Engineers, Dallas, TX.10.2118/23913-MS
Search in Google Scholar
Khalil, H. (2002). Nonlinear Systems, Third Edition, Prentice-Hall, Upper Saddle River, NJ.
Search in Google Scholar
Knuppel, T., Woittennek, F., Boussaada, I., Mounier, H. and Niculescu, S. (2014). Flatness-based control for a non-linear spatially distributed model of a drilling system, in A. Seuret et al. (Eds.), Low Complexity Controllers for Time Delay Systems: Advances in Delays and Dynamics, Volume 2, Springer, Cham, pp. 205-218.10.1007/978-3-319-05576-3_14
Search in Google Scholar
Kŏcvara, M. and Stingl, M. (2003). PENNON-a code for nonlinear and convex semidefinite programming, Optimization Methods and Software 8(3): 317-333.10.1080/1055678031000098773
Search in Google Scholar
La Salle, J. and Lefschetz, S. (1961). Stability by Lyapunov’s Direct Method: With Applications, Academic Press, New York, NY.
Search in Google Scholar
Lakshmikantham, V., Leela, S. and Martynyuk, A. (1990). Practical Stability of Nonlinear Systems, World Scientific Publishing Company, Singapore.10.1142/1192
Search in Google Scholar
Levinson, N. (1944). Transformation theory of non-linear differential equations of the second order, Annals of Mathematics 45(4): 723-737.10.2307/1969299
Search in Google Scholar
Lu, H., Dumon, J. and de Wit, C.C. (2009). Experimental study of the D-OSKIL mechanism for controlling the stick-slip oscillations in a drilling laboratory testbed, 2009 IEEE Control Applications (CCA) & Intelligent Control (ISIC), St. Petersburg, Russia, pp. 1551-1556.
Search in Google Scholar
Ma, R., Dimirovski, G. and Zhao, J. (2013). Backstepping robust H∞ control for a class of uncertain switched nonlinear systems under arbitrary switchings, Asian Journal of Control 15(1): 41-50.10.1002/asjc.512
Search in Google Scholar
Navarro-López, E. and Cortés, D. (2007a). Avoiding harmful oscillations in a drillstring through dynamical analysis, Journal of Sound and Vibration 307(1): 152-171.10.1016/j.jsv.2007.06.037
Search in Google Scholar
Navarro-López, E. and Cortés, D. (2007b). Sliding-mode control of a multi-DOF oilwell drillstring with stick-slip oscillations, Proceedings of the 2007 American Control Conference, New York, NY, USA, pp. 3837-3842.10.1109/ACC.2007.4282198
Search in Google Scholar
Navarro-López, E. and Licéaga-Castro, E. (2009). Non-desired transitions and sliding-mode control of a multi-DOF mechanical system with stick-slip oscillations, Chaos, Solitons and Fractals 41(4): 2035-2044.10.1016/j.chaos.2008.08.008
Search in Google Scholar
Navarro-López, E. and Suárez, R. (2004). Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings, Proceedings of the 2004 IEEE International Conference on Control Applications Taipei, Taiwan, pp. 1454-1460.
Search in Google Scholar
Pavone, D. and Desplans, J. (1994). Application of high sampling rate downhole measurements for analysis and cure of stick-slip in drilling, Technical Report SPE 28324, Society of Petroleum Engineers, Dallas, TX.10.2118/28324-MS
Search in Google Scholar
Rasvan, V. (2006). Three lectures on dissipativeness, IEEE International Conference on Automation, Quality and Testing, Robotics, Cluj-Napoca, Romania, pp. 167-177.
Search in Google Scholar
Saldivar, B., Knuppel, T., Woittennek, F., Boussaada, I., Mounier, H. and Niculescu, S. (2014). Flatness-based control of torsional-axial coupled drilling vibrations, 19th World Congress of the International Federation of Automatic Control, Cape Town, South Africa, pp. 7324-7329.
Search in Google Scholar
Saldivar, B. and Mondié, S. (2013). Drilling vibration reduction via attractive ellipsoid method, Journal of the Franklin Institute 350(3): 485-502.10.1016/j.jfranklin.2012.12.010
Search in Google Scholar
Saldivar, B., Mondié, S., Loiseau, J. and Rasvan, V. (2013). Suppressing axial torsional coupled vibrations in oilwell drillstrings, Journal of Control Engineering and Applied Informatics 15(1): 3-10.
Search in Google Scholar
Serrarens, A., van de Molengraft, M., Kok, J. and van den Steeen, L. (1998). H∞ control for suppressing stick-slip in oil well drillstrings, IEEE Control Systems 18(2): 19-30.10.1109/37.664652
Search in Google Scholar
Skaugen, E. (1987). The effects of quasi-random drill bit vibrations upon drillstring dynamic behavior, Technical Report SPE 16660, Society of Petroleum Engineers, Dallas, TX.10.2118/16660-MS
Search in Google Scholar
Suh, Y., Kang, H. and Ro, Y. (2006). Stability condition of distributed delay systems based on an analytic solution to Lyapunov functional equations, Asian Journal of Control 8(1): 91-96.10.1111/j.1934-6093.2006.tb00258.x
Search in Google Scholar
Timoshenko, S. and Young, D. (1955). Vibrations Problems in Engineering, Third Edition, D. Van Nostrand Company, Princeton, NJ.
Search in Google Scholar
Tucker, R. and Wang, C. (1999). On the effective control of torsional vibrations in drilling systems, Journal of Sound and Vibration 224(1): 101-122.10.1006/jsvi.1999.2172
Search in Google Scholar
Weaver, W., Timoshenko, S. and Young, D. (1990). Vibrations Problems in Engineering, Fifth Edition, John Wiley & Sons, New York, NY.
Search in Google Scholar
Wu, J., Li, S. and Chai, S. (2010). Exact controllability of wave equations with variable coefficients coupled in parallel, Asian Journal of Control 12(5): 650-655.10.1002/asjc.179
Search in Google Scholar
Yang, L. and Wang, J. (2014). Stability of a damped hyperbolic Timoshenko system coupled with a heat equation, Asian Journal of Control 16(2): 546-555.10.1002/asjc.739
Search in Google Scholar
Yoshizawa, T. (1960). Stability and boundedness of systems, Archive for Rational Mechanics and Analysis 6(1): 409-421.10.1007/BF00276172
Search in Google Scholar
Yoshizawa, T. (1966). Stability Theory by Lyapunov’s Second Method, The Mathematical Society of Japan, Tokyo.
Search in Google Scholar
Zhang, X. and Zuazua, E. (2004). Polynomial decay and control of a 1-D hyperbolic-parabolic coupled system, Journal of Differential Equations 204(2): 380-438.10.1016/j.jde.2004.02.004
Search in Google Scholar
Zhou, Z. and Tang, S. (2012). Boundary stabilization of a coupled wave-ode system with internal anti-damping, International Journal of Control 85(11): 683-693. 10.1080/00207179.2012.696704
Search in Google Scholar

The control of drilling vibrations: A coupled PDE-ODE modeling approach

Abstract

Paradigm

My account