基于状态依赖时滞的钻柱动力学稳定性分析

Numerical stability analysis of the rotary drilling system on the basis of state-dependent delay

在石油深井钻探中,钻头与岩石相互作用诱发钻柱系统的自激振动,严重时导致钻头磨损和钻柱疲劳失效,影响钻井效率。钻柱动力系统不稳定的主要原因是钻头与岩石的相互作用引起钻柱动力系统中出现了状态依赖时滞变量,该研究考虑钻井液阻尼的影响,给出了控制钻头轴向和扭转运动的状态依赖时滞微分方程(state-dependent delay differential equation, SDDDE),引入钻头运动轨迹函数对状态依赖时滞变量进行描述,将状态依赖时滞微分方程转化为非线性耦合的偏微分方程(partial differential equation, PDE)和常微分方程(ordinary differential equations, ODE),并利用谱方法和伽辽金方法分析了上述耦合方程的线性稳定性,给出了钻柱动力学系统随钻井参数变化的稳定性图谱,对多自由度钻柱动力学系统的稳定性进行了分析。最后,与相关文献中的结果及数值模拟结果进行对比,验证了该研究稳定性数值分析方法的准确性,并研究了阻尼对钻柱动力学系统稳定性的影响。通过基于状态依赖时滞的钻柱动力学稳定性分析,有助于控制或避免钻柱的自激振动,进而延长钻头使用寿命,提高机械钻速。

The bit-rock interaction introduces state-dependent delays in a rotary drilling system, which cause the instability of the system and thus the self-excited vibration of the drill string. The vibration leads to the wear of the drill bit as well as fatigue failure of the drill string and hence decrease the drilling efficiency. Therefore, the stability analysis of the rotary drilling system helps to optimize the drilling parameters and bit design so as to reduce or eliminate undesirable self-excited vibrations. The state-dependent delay differential equations(SDDDEs) were derived considering the torsional damping of drilling mud. The delay-related variable was reformulated via the introduction of the bit trajectory function, which transformed the SDDDEs into a system of nonlinear coupled partial differential equation(PDE) and ordinary differential equations(ODEs). The combination of spectral method and Galerkin method was utilized to analyze the linear stability of the coupled PDE-ODEs, and the stability map of the system in the plane of drilling parameters was obtained. The fidelity of the proposed method was confirmed by comparing with the published results and the numerical simulation results. Finally, the effects of damping on the stability of the system were studied. The proposed method could be extended to analyze the stability of a more generic lumped-parameter model with multiple degrees of freedom.