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Nonlinear Dynamic Model and Characterization of Coiled Tubing Drilling System Based on Drilling Robot

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Abstract

Purpose

Due to the uncontrollable dragging force and velocity of the drilling robot, the weight on bit (WOB) and rate of penetration (ROP) can not be controlled. So, there is not an application of in the drilling engineering even through downhole robots have been developed for many years. Dynamic characteristics are the theoretical basis for controlling WOB and ROP.

Methods

In this paper, the power source of the driving force of the drilling robot is found, which is the pressure difference between inside and outside of the drilling robot. The calculation model of the dragging force of the drilling robot is obtained. On the basis, a fluid–solid coupling nonlinear dynamic model (CT-drilling robot-hydraulic motor-bit-rock-drilling fluid) is established. The nonlinear dynamic characterization is simulated by Runge–Kutta method.

Results

The nonlinear dynamic model of the coiled tubing drilling system introduces the characteristics of drilling fluid, rock mechanics and ROP for the first time. The effects of robot position, friction coefficient, displacement fluctuation of drilling fluid and WOB fluctuation on the dynamic characteristics of drilling system are studied. It is found that the displacement of drilling fluid, position of drilling robot and friction coefficient are the main factors affecting the dynamics of the CT drilling system based on drilling robot.

Conclusion

This paper finds out power source of the drilling robot, which is the pressure difference between the internal and external of the drilling robot. The acquisition of dynamic characteristics will provide a theoretical basis for the control method of downhole robot, rock breaking mechanism based on drilling robot, bit selection and design of bottom hole assembly, etc. It will also promote the development of downhole robots and solve the bottleneck problem of coiled tubing “buckling” and “locking”.

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Acknowledgements

This work was funded by the National Natural Science Foundation of China (Nos. 5200041991, U19A2097), the Research Foundation of Sichuan Province (2019YFG0458, 2019105) and the Open Foundation of Sichuan Province Key Laboratory of Process Equipment and Control Engineering (GK202006). The authors also sincerely thank the editors and the reviewers for their efforts in improving this paper.

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Appendix A

Appendix A

The pressure loss of other tools between robot and bit, hydraulic motor, bit and annulus are Eq. (16) [25, 26], Eq. (17) [27], Eq. (18) [28], Eq. (19) [28], respectly.

$$\Delta P_{\text{rm}} = \frac{{0.2399\rho^{0.75} \tau^{0.25} Q^{1.75} L_{\text{rm}} }}{{d_{{{\text{rm}}\_{\text{i}}}}^{4.75} }}$$
(16)

where ΔPrm is pressure loss of other tools, ρ is density of drilling fluid, τ is viscosity of drilling fluid, Lrm is length of other tools between robot and hydraulic motor and drm_i is inner diameter of other tools.

$$\Delta P_{\text{m}} = \Delta P_{{{\text{noload}}\_{ \text{max} }}} \left\{ {\frac{{Q\left[ {1 - \mu_{f} \left( {\delta_{1} - \delta } \right)} \right]}}{{Q_{ \text{max} } }}} \right\}^{1.8} + k\left( {aW + bW^{2} } \right)^{f}$$
(17)

where \(a{ = }\frac{{T_{1} - T_{\text{max} } \left( {\frac{{W_{1} }}{{W_{\text{max} } }}} \right)^{2} }}{{W_{1} - \frac{{W_{1}^{2} }}{{W_{\text{max} } }}}}\),\(b{ = }\frac{{T_{1} - aW_{1} }}{{W_{1}^{2} }}\),\(f{ = }\frac{{{ \ln }\frac{{\Delta P_{1} - \Delta P_{0} }}{{\Delta P_{\text{max} } - \Delta P_{0} }}}}{{{ \ln }\frac{{T_{1} }}{{T_{\text{max} } }}}}\),\(k{ = }\frac{{\Delta P_{1} - \Delta P_{0} }}{{T_{1}^{f} }}\), ΔPm is pressure loss of motor, ΔPnoload_max is maximum pressure loss of moto in no load condition, T1 is work torque, N m, Tmax is max torque, N m, W1 is work WOB, N, Wmax is max WOB, N, ΔP1 is work pressure loss, Pa and ΔPmax is max pressure loss, Pa.

$$\Delta P_{\text{bit}} = \frac{1}{2}\frac{{\rho Q^{2} }}{{C^{2} \left( {\frac{\pi }{4}\sum\limits_{i = 1}^{n} {d_{i}^{2} } } \right)^{2} }}$$
(18)

where ΔPbit is pressure loss of bit C is flow coefficient of drilling fluid, 0.81–0.98 and di is diameter of the ith nozzle of bit, m.

$$\Delta P_{\text{a}} = \frac{{0.8106\rho L_{\text{a}} Q^{2} \left\{ {{ \ln }\left[ {\frac{\Delta }{{3.715\left( {d_{\text{w}} - d_{{{\text{rm}}\_{\text{o}}}} } \right)}} + \frac{{4.6\tau^{0.9} \left( {d_{\text{w}} + d_{{{\text{rm}}\_{\text{o}}}} } \right)^{0.9} }}{{\rho^{0.9} Q^{0.9} }}} \right]} \right\}^{ - 2} }}{{\left( {d_{\text{w}} - d_{{{\text{rm}}\_{\text{o}}}} } \right)^{3} \left( {d_{\text{w}} + d_{{{\text{rm}}\_{\text{o}}}} } \right)^{2} }}$$
(19)

where ΔPa is pressure loss of annulus, L4 is length of annulus, m, Δ is absolute roughness of wall of wellbore, m, dw is diameter of wellbore, m and dtool_o is diameter of drilling tools, m.

According to Eq. (19), it can be get:

$$\begin{aligned} F_{\text{d}} &= \frac{{0.2399\rho^{0.75} \tau^{0.25} Q^{1.75} L_{\text{rm}} A_{\text{r}} }}{{d_{{{\text{rm}}_{\text{i}} }}^{4.75} }} + \frac{{\rho Q^{2} A_{\text{r}} }}{{2C^{2} \left( {\frac{\pi }{4}\mathop \sum \nolimits_{i = 1}^{n} d_{i}^{2} } \right)^{2} }} + \frac{{0.8106\rho L_{\text{a}} Q^{2} \left\{ {{ \ln }\left[ {\frac{\Delta }{{3.715\left( {d_{\text{w}} - d_{{{\text{rm}}_{\text{o}} }} } \right)}} + \frac{{4.6\tau^{0.9} \left( {d_{\text{w}} + d_{{{\text{rm}}_{\text{o}} }} } \right)^{0.9} }}{{\rho^{0.9} Q^{0.9} }}} \right]} \right\}^{ - 2} A_{\text{r}} }}{{\left( {d_{\text{w}} - d_{{{\text{rm}}_{\text{o}} }} } \right)^{3} \left( {d_{\text{w}} + d_{{{\text{rm}}_{\text{o}} }} } \right)^{2} }} \hfill \\&\quad + \Delta P_{{{\text{noload}}\_{ \text{max} }}} \left\{ {\frac{{Q\left[ {1 - \mu_{\text{f}} \left( {\delta_{1} - \delta } \right)} \right]}}{{Q_{ \text{max} } }}} \right\}^{1.8} A_{\text{r}} + k\left( {aW + bW^{2} } \right)^{f} A_{\text{r}} \hfill \\ \end{aligned}$$
(20)

As can be seen from Eq. (20), only Q and W are variables. In order to simplify the equation and facilitate the derivation of the dynamic model, Eq. (20) can be divided into two parts. One part contains W, the other part do not has W. If the part without W is equal to f1 (Q), then:

$$F_{\text{d}} = f_{1} \left( Q \right) + A_{\text{r}} k(aW + bW^{2} )^{f}$$
(21)

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Zhao, J., Liu, Q., Zhu, H. et al. Nonlinear Dynamic Model and Characterization of Coiled Tubing Drilling System Based on Drilling Robot. J. Vib. Eng. Technol. 9, 541–561 (2021). https://doi.org/10.1007/s42417-020-00246-x

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