# American Institute of Mathematical Sciences

June  2022, 15(6): 1547-1560. doi: 10.3934/dcdss.2022007

## Stability analysis of pressure and penetration rate in rotary drilling system

 LIM laboratory, Polytechnic School of Tunisia, BP 743, 2078 La Marsa, Tunisia

*Corresponding author: Rhouma Mlayeh

Received  August 2021 Revised  October 2021 Published  June 2022 Early access  January 2022

The purpose of this paper is to stabilize the annular pressure profile throughout the well bore continuously while drilling. A new nonlinear dynamical system is developed and a controller is designed to stabilize the annular pressure and achieve asymptotic tracking by applying feedback control of the main pumps. Hence, the paper studies the control design for the well known Managed Pressure Drilling system (MPD). MPD provides a closed loop drilling process in which pore pressure, formation fracture pressure, and bottomhole pressure are balanced and managed at surface. Although, responses must provide a solution for critical downhole pressures to preserve drilling efficiency and safety. Our MPD scheme is elaborated in reference to a nontrivial backstepping control procedure, and the effectiveness of the proposed control laws is shown by simulations.

Citation: Rhouma Mlayeh. Stability analysis of pressure and penetration rate in rotary drilling system. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1547-1560. doi: 10.3934/dcdss.2022007
##### References:
 [1] M. T. Alrifai, J. H. Chow and D. A. Torrey, A backstepping nonlinear control approach to switched reluctance motors, Proceedings of the 37th IEEE Conference on Decision and Control, (1998). doi: 10.1109/CDC.1998.762067. [2] J. J. Bailey and I. Finnie, An Analytical study of drill-string vibration, J. Eng. Ind., 82 (1960), 122-127.  doi: 10.1115/1.3663017. [3] G. C. Downton, Challenges of modeling drilling systems for the purposes of automation and control, IFAC Proceedings Volumes, 45 (2012), 201-210.  doi: 10.3182/20120531-2-NO-4020.00054. [4] S. Dwars, Recent dvances in Soft Torque Rotary Systems, Paper presented at the SPE/IADC Drilling Conference and Exhibition, London, England, UK, March 2015. doi: 10.2118/173037-MS. [5] G. O. Kaasa, Ø. N. Stamnes, O. M. Aamo and L. S. Imsland, Simplified hydraulics model used for intelligent estimation of downhole pressure for a managed-pressure-drilling control system, SPE Drilling and Completion, 27 (2012), 127-138.  doi: 10.2118/143097-PA. [6] E. Kreuzer, L. Krumm, M. A. Pick, E. Solowjow and M. Steidl, Active Vibration Isolation Via Decomposition of Traveling Waves, 11th International Conference on Vibration Problems, Lisbon, Portugal, 2013. [7] M. Krstic, I. Kanellakopoulos and P. Kokotovic, Nonlinear and Adaptive Control Design, Wiley-Interscience, 1995. doi: 10.0471127329. [8] X. Liu, N. Vlajic, X. Long, G. Meng and B. Balachandran, Coupled axial-torsional dynamics in rotary drilling with state-dependent delay: Stability and control, Nonlinear Dynamics, 78 (2014), 1891-1906.  doi: 10.1007/s11071-014-1567-y. [9] R. Lozano and B. Brogliat, Adaptive control of robot manipulators with flexible joints, IEEE Trans. Automat. Control, 37 (1992), 174-181.  doi: 10.1109/9.121619. [10] H. J. Marquez, Nonlinear Control Systems: Analysis and Design, Wiley, 2003. [11] E. Navarro-López and R. Suáre, Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings, Proceedings of the 2004 IEEE International Conference on Control Applications, 2 (2004), 1454-1460.  doi: 10.1109/CCA.2004.1387580. [12] B. Saldivar, S. Mondie, J. J. Loiseau and V. Rasvan, Stick-Slip oscillations in oillwell drillstrings: Distributed parameter and neutral type retarded model approaches, IFAC Proceedings Volumes, 44 (2011), 284-289.  doi: 10.3182/20110828-6-IT-1002.00084. [13] B. Saldivar, S. Mondié, S.-I. Niculescu, H. Mounier and I. Boussaada, A control oriented guided tour in oilwell drilling vibration modeling, Annual Reviews in Control, 42 (2016), 100-113.  doi: 10.1016/j.arcontrol.2016.09.002. [14] R. Skjetne and T. I. Fossen, On Integral Control in Backstepping Analysis of Different Techniques, Control Applications. Proceedings of the 2004 IEEE International Conference, (2004), 1899–1904. doi: 10.23919/ACC.2004.1386858. [15] Y. N. Stamnes, Thesis for the degree of philosophiae doctor, NTNU Norwegian Universty Science of Technology, Trondheim, Mathematics and Electrical Engineering, Department of Engineering Cybernetics, (2011). [16] S. Toumi, L. Beji and R. Mlayeh, Torsional vibration suppression with boundary impulsive conditions in rotary drilling system, IIEEE 58th Conference on Decision and Control (CDC 2019), (2019). doi: 10.1109/CDC40024.2019.9029411. [17] S. Toumi, L. Beji, R. Mlayeh and A. Abichou, Stabilization of stick-slip oscillations Integrating fluid injection in oilwell drillstring system, European Control Conference, (2016), 352–357. doi: 10.1109/ECC.2016.7810310. [18] S. Toumi, L. Beji, R. Mlayeh and A. Abichou, Boundary observer design for hyperbolic PDE in rotary drilling systems, IEEE 55th Conference on Decision and Control (CDC), Las Vegas, USA, (2016), 2128–2133. doi: 10.1109/CDC.2016.7798578. [19] S. Toumi, L. Beji, R. Mlayeh and A. Abichou, Stabilization of torsional vibration in oilwell drillstring system, Eur. J. Control, 35 (2017), 19-27.  doi: 10.1016/j.ejcon.2017.03.002. [20] S. Toumi, L. Beji, R. Mlayeh and A. Abichou, Stability analysis of coupled torsional vibration and pressure in oilwell drillstring system, International Journal of Control, 91 (2017), 241-252.  doi: 10.1080/00207179.2016.1278269. [21] S. Toumi, R. Mlayeh, L. Beji and A. Abichou, Stability analysis of oilwell drilling torsional vibrations, Control and Automation (MED), 24th Mediterranean Conference, (2016), 677–682. doi: 10.1109/MED.2016.7535945. [22] F. White, Fluid Mechanics, New York: McGraw-Hill (2008), 2008.

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##### References:
 [1] M. T. Alrifai, J. H. Chow and D. A. Torrey, A backstepping nonlinear control approach to switched reluctance motors, Proceedings of the 37th IEEE Conference on Decision and Control, (1998). doi: 10.1109/CDC.1998.762067. [2] J. J. Bailey and I. Finnie, An Analytical study of drill-string vibration, J. Eng. Ind., 82 (1960), 122-127.  doi: 10.1115/1.3663017. [3] G. C. Downton, Challenges of modeling drilling systems for the purposes of automation and control, IFAC Proceedings Volumes, 45 (2012), 201-210.  doi: 10.3182/20120531-2-NO-4020.00054. [4] S. Dwars, Recent dvances in Soft Torque Rotary Systems, Paper presented at the SPE/IADC Drilling Conference and Exhibition, London, England, UK, March 2015. doi: 10.2118/173037-MS. [5] G. O. Kaasa, Ø. N. Stamnes, O. M. Aamo and L. S. Imsland, Simplified hydraulics model used for intelligent estimation of downhole pressure for a managed-pressure-drilling control system, SPE Drilling and Completion, 27 (2012), 127-138.  doi: 10.2118/143097-PA. [6] E. Kreuzer, L. Krumm, M. A. Pick, E. Solowjow and M. Steidl, Active Vibration Isolation Via Decomposition of Traveling Waves, 11th International Conference on Vibration Problems, Lisbon, Portugal, 2013. [7] M. Krstic, I. Kanellakopoulos and P. Kokotovic, Nonlinear and Adaptive Control Design, Wiley-Interscience, 1995. doi: 10.0471127329. [8] X. Liu, N. Vlajic, X. Long, G. Meng and B. Balachandran, Coupled axial-torsional dynamics in rotary drilling with state-dependent delay: Stability and control, Nonlinear Dynamics, 78 (2014), 1891-1906.  doi: 10.1007/s11071-014-1567-y. [9] R. Lozano and B. Brogliat, Adaptive control of robot manipulators with flexible joints, IEEE Trans. Automat. Control, 37 (1992), 174-181.  doi: 10.1109/9.121619. [10] H. J. Marquez, Nonlinear Control Systems: Analysis and Design, Wiley, 2003. [11] E. Navarro-López and R. Suáre, Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings, Proceedings of the 2004 IEEE International Conference on Control Applications, 2 (2004), 1454-1460.  doi: 10.1109/CCA.2004.1387580. [12] B. Saldivar, S. Mondie, J. J. Loiseau and V. Rasvan, Stick-Slip oscillations in oillwell drillstrings: Distributed parameter and neutral type retarded model approaches, IFAC Proceedings Volumes, 44 (2011), 284-289.  doi: 10.3182/20110828-6-IT-1002.00084. [13] B. Saldivar, S. Mondié, S.-I. Niculescu, H. Mounier and I. Boussaada, A control oriented guided tour in oilwell drilling vibration modeling, Annual Reviews in Control, 42 (2016), 100-113.  doi: 10.1016/j.arcontrol.2016.09.002. [14] R. Skjetne and T. I. Fossen, On Integral Control in Backstepping Analysis of Different Techniques, Control Applications. Proceedings of the 2004 IEEE International Conference, (2004), 1899–1904. doi: 10.23919/ACC.2004.1386858. [15] Y. N. Stamnes, Thesis for the degree of philosophiae doctor, NTNU Norwegian Universty Science of Technology, Trondheim, Mathematics and Electrical Engineering, Department of Engineering Cybernetics, (2011). [16] S. Toumi, L. Beji and R. Mlayeh, Torsional vibration suppression with boundary impulsive conditions in rotary drilling system, IIEEE 58th Conference on Decision and Control (CDC 2019), (2019). doi: 10.1109/CDC40024.2019.9029411. [17] S. Toumi, L. Beji, R. Mlayeh and A. Abichou, Stabilization of stick-slip oscillations Integrating fluid injection in oilwell drillstring system, European Control Conference, (2016), 352–357. doi: 10.1109/ECC.2016.7810310. [18] S. Toumi, L. Beji, R. Mlayeh and A. Abichou, Boundary observer design for hyperbolic PDE in rotary drilling systems, IEEE 55th Conference on Decision and Control (CDC), Las Vegas, USA, (2016), 2128–2133. doi: 10.1109/CDC.2016.7798578. [19] S. Toumi, L. Beji, R. Mlayeh and A. Abichou, Stabilization of torsional vibration in oilwell drillstring system, Eur. J. Control, 35 (2017), 19-27.  doi: 10.1016/j.ejcon.2017.03.002. [20] S. Toumi, L. Beji, R. Mlayeh and A. Abichou, Stability analysis of coupled torsional vibration and pressure in oilwell drillstring system, International Journal of Control, 91 (2017), 241-252.  doi: 10.1080/00207179.2016.1278269. [21] S. Toumi, R. Mlayeh, L. Beji and A. Abichou, Stability analysis of oilwell drilling torsional vibrations, Control and Automation (MED), 24th Mediterranean Conference, (2016), 677–682. doi: 10.1109/MED.2016.7535945. [22] F. White, Fluid Mechanics, New York: McGraw-Hill (2008), 2008.
MPD in Rotary Drilling System
Stabilization of the state $y$
Stabilization of the pressure $P_2$
Stabilization of the penetration rate of the bit $v$
Stabilization of the rotation velocity of the drill string $\Omega$
Stabilization of the the flow rate from the tool $q_{bit}$
Stabilization of the control law $u_1$
Stabilization of the control law $u_4$
Different physical parameters
 Variable Value $L$ $2000~m$ $I$ $0.095~kg.m$ $\rho_1 = \rho_3$ $1250~kg. m^{-3}$ $M$ $8300~kg.m^{-4}$ $\beta_1 = \beta_3$ $24750~bar$ $V_0$ $110~ m^3$ $g$ $9.81~ m s^{-2}$ $S$ $\pi\times(0.25)^2~ m^2$ $c_d$ $0.61$ $T_a$ $0.003. 10^6~ \frac{bar. s^2}{m^6}$
 Variable Value $L$ $2000~m$ $I$ $0.095~kg.m$ $\rho_1 = \rho_3$ $1250~kg. m^{-3}$ $M$ $8300~kg.m^{-4}$ $\beta_1 = \beta_3$ $24750~bar$ $V_0$ $110~ m^3$ $g$ $9.81~ m s^{-2}$ $S$ $\pi\times(0.25)^2~ m^2$ $c_d$ $0.61$ $T_a$ $0.003. 10^6~ \frac{bar. s^2}{m^6}$
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