VIBRATION CHARACTERISTICS ANALYSIS OF MULTI-SPAN DRILL STRING SYSTEM BY MEANS OF GREEN'S FUNCTIONS
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摘要: 随着油气勘探开发向着深层、深水及非常规等复杂领域的不断扩展，钻井面临的井况与约束条件更加苛刻，钻柱的动力学特性更加复杂，失效问题频发。该文应用格林函数理论对多跨旋转钻柱双向耦合动力学特性进行了定量分析和研究。考虑多稳定器及不同约束条件，以钻柱整体为研究对象，基于Euler-Bernoulli梁模型和Hamilton原理建立了具有广义边界约束条件及多稳定器的旋转钻柱双向耦合动力学方程。采用分离变量法、Laplace变换及Laplace逆变换求解所获得的振动微分方程，得到了旋转钻柱系统横向振动的格林函数解以及以格林函数为基础的多跨旋转钻柱系统的闭合形式的模态函数及隐式的频率方程。定量地分析了稳定器位置、弹簧刚度系数与稳定器个数对钻柱系统振动特性的影响。数值结果表明：稳定器位置与固有频率的关系曲线中有相应阶次数目的峰值；随着等效弹簧的刚度系数的增大，系统的固有频率随之增大，但当刚度增加到一定值时，系统的一阶和二阶频率将趋于稳定。研究结果有助于深化对多跨旋转钻柱的动力学特性规律的认识，为提高钻速、减少钻柱失效及钻柱钻井技术的应用提供了新的研究方法和理论依据。Abstract: With the continuous expansion of petroleum exploration and development toward deep, deep water, and unconventional areas, the well conditions and constraints faced by drilling have become more stringent, the dynamics of the drill string have become more complex, and well failures frequently occur. In this paper, the two-way coupling dynamic characteristics of multi-span spinning drill string are quantitatively analyzed and studied by using Green’s function theory. Considering the multi-stabilizers and different constraints, taking the whole drill string as the research object, the bidirectional coupling dynamic equation of spinning drill string with generalized boundary constraints and multi-stabilizers is established by the grounds of Bernoulli-Euler beam theory and Hamilton principle. The vibration differential equations are solved by using the separation variable method, Laplace transform and, inverse Laplace transform. The Green’s functions of the transverse vibration of the spinning drill string system and the closed-form mode shapes and the implicit frequency equation of the multi-span rotating drill string system based on the Green’s functions are obtained. The influence of stabilizer position, of spring stiffness coefficient and of the number of stabilizers on the vibration characteristics of the drill string system is quantitatively analyzed. The numerical results show that the positions of stabilizers and the equivalent spring stiffness have great impacts on the natural frequencies and mode shapes of the system. The relationship curve between the stabilizer position and the natural frequency has corresponding order peaks. With the increase of the stiffness coefficient of the equivalent spring, the natural frequency of the system increases, but when the stiffness increases to a certain value, the first and second order frequencies of the system will tend to be a constant. The research results are helpful to deepen the understanding of the dynamic characteristics of multi-span rotary drill string, and to provide a new research method and theoretical basis for improving drilling speed and, reducing drill string failure, and to extend the application of drill string drilling technology.
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图 1 具有稳定器和弹性边界的旋转钻柱系统的模型示意图
Figure 1. Schematic of the spinning drill string system with stabilizers and elastic boundary conditions
图 2 具有固定-自由边界条件的钻柱前三阶固有频率
Figure 2. The first three natural frequencies of the drill string with Clamped-Free boundary conditions
图 3 具有简支-简支边界条件的钻柱前三阶固有频率
Figure 3. The first three natural frequencies of the drill string with Pinned- Pinned boundary conditions
图 4 具有简支-简支边界条件的钻柱前四阶固有频率
Figure 4. The first four natural frequencies of the drill string with Pinned- Pinned boundary conditions
图 5 具有一个稳定器的钻柱系统的第一阶模态函数
Figure 5. The first-order mode shapes of the drill string system with a stabilizer
图 6 具有一个稳定器的钻柱系统的第二阶模态函数
Figure 6. The second order mode shapes of the drill string system with a stabilizer
图 7 具有不同弹簧刚度系数的钻柱系统的前两阶固有频率
Figure 7. The first two natural frequencies of the drill string system with different spring stiffness coefficients
图 8 具有不同弹簧刚度系数的钻柱系统的一阶模态函数
Figure 8. The first-order mode shapes of the drill string system with different spring stiffness coefficients
图 9 具有不同弹簧刚度系数的钻柱系统的二阶模态函数
Figure 9. The second-order mode shapes of the drill string system with different spring stiffness coefficients
图 10 前两阶固有频率与两个稳定器位置
$ {\xi _1} $ 及$ {\xi _2} $ 的关系图Figure 10. The relationships between the first two natural frequencies of the two stabilizers and the positions
$ {\xi _1} $ and$ {\xi _2} $ 表 1 钻柱系统的参数
Table 1. Parameters of the drill string system
参数 数值 钻柱长度$ L/{\text{m}} $ 200 井筒直径${D_{{\rm{ch}}} }/{\text{m} }$ 0.28 钻柱外径$ D/{\text{m}} $ 0.1270 钻柱内径$ d/{\text{m}} $ 0.1016 内流密度${\rho _{\rm{L}}}$/(kg·m−3) 24 外流密度${\rho _{\rm{M}}}$/(kg·m−3) 1200 钻柱密度${\rho _{\rm{p}}}$/(kg·m−3) 7380 弹簧刚度$ k_n^{} $/(kN·m−1) 17500 附加质量系数$ \chi $ 1.5 
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