A nonsmooth approach for the modelling of a mechanical rotary drilling system with friction

Samir Adly; Daniel Goeleven

doi:10.3934/eect.2020042

Article Contents

2020, Volume 9, Issue 4: 915-934. Doi: 10.3934/eect.2020042

This issue Previous Article Preface Next Article Measurable solutions to general evolution inclusions

A nonsmooth approach for the modelling of a mechanical rotary drilling system with friction

Samir Adly^1, , and
Daniel Goeleven^2,

1.
Laboratoire XLIM, Université de Limoges, 87060 Limoges, France
2.
Laboratoire PIMENT, Université de La Réunion, 97400 Saint-Denis, France

^* Corresponding author: Samir Adly
^* Corresponding author: Samir Adly

Dedicated to 70th birthday of Professor Meir Shillor.

Received: August 2019

Revised: November 2019

Early access: March 2020

Published: December 2020

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Abstract

In this paper, we show how the approach of nonsmooth dynamical systems can be used to develop a suitable method for the modelling of a rotary oil drilling system with friction. We study different kinds of frictions and analyse the mathematical properties of the involved dynamical systems. We show that using a general Stribeck model for the frictional contact, we can formulate the rotary drilling system as a well-posed evolution variational inequality. Several numerical simulations are also given to illustrate both the model and the theoretical results.

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Mathematics Subject Classification: 34A60, 46N10, 49J52, 70F40.

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Figure 1. General friction model as in (6) and the function $ \varphi $ as in (7)

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Figure 2. Coulomb friction model

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Figure 3. Stribeck friction model

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Figure 4. Stiction model

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Figure 5. Consistent stiction model as in (6) with $ \varphi_+(x) = F_C +(F_S-F_C)e^{-\frac{x}{v_s}} $

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Figure 6. Oil drilling rig illustration-1. Mud tank, 2. Shale shakers, 3. Suction line (mud pump), 4. Mud pump, 5. Motor or power source, 6. Vibrating hose, 7. Draw-works (winch), 8. Standpipe 9. Kelly hose, 10. Goose-neck, 11. Traveling block, 12. Drill line, 13. Crown block 14. Derrick-Author: Tosaka-Attribution 3.0 Unported (CC BY 3.0) - https://creativecommons.org/licenses/by/3.0/deed.en (https://commons.wikimedia.org/wiki/File:Oil_Rig_NT.PNG)

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Figure 7. Rotary drilling system

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Figure 8. Graph of the function $ V(t) $ in Table 3

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Figure 9. Numerical solution of the evolution variational inequality (19) with the initial conditions given in Table 5

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Figure 10. Numerical solution of the evolution variational inequality (23) with the initial conditions given in Table 5

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Table 1. Parameters

$ J_1 $	$ 999.35 \; (kg.m^2) $
$ J_2 $	$ 127.27\; (kg.m^2) $
$ d_1 $	$ 51.38\; (N.m.s/rad) $
$ d_2 $	$ 39.79 \;(N.m.s) $
$ k $	$ 481.29 \; (N.m/rad) $
$ R $	$ 0.01 \;(\Omega) $
$ L $	$ 0.005 \;(H) $
$ K_M $	$ 6 \;(N.m/A) $
$ N $	$ 7.20 $
$ K = NK_M $	$ 43.20 \;(N.m/A) $
$ E $	$ 130 \;(MJ/m^3) $
$ \delta $	$ 0.64 \times 10^{-3}\;(m/rad) $
$ R_B $	$ 0.10 \; m $
$ \mu_C $	$ 0.4 $
$ \mu_S $	$ 0.6 $

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Table 2. Empirical coefficients

$ \sigma $	$ 1 $
$ \omega_s $	$ 10^{-3} $ $ (rad/s) $

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Table 3. Motor voltage. Augmentation of DC motor voltage from 125 (V) to 150 (V) at $ t = 30 \; (s) $ (see Figure 8)

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Table 4. Weight-On-Bit and corresponding friction torques

$ W $	$ 15 000 \; (kg) $
$ T_C = \frac{1}{2}\mu_CR_BW $	$ 300 \; (kg.m) $
$ T_S = \frac{1}{2}\mu_SR_BW $	$ 450 \; (kg.m) $
$ T_{{\rm CUT}} = \frac{1}{2} \delta R_B^2E $	$ 4.16\, 10^{-4}\; (MJ/rad) $

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Table 5. Initial conditions for problems (19) and (23)

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