Abstract: In this paper,a four order nonlinear differential equation with a parameter β, which describes the buckling deformation of pipes inhorizontal wells,is derived by using the principle of minimum potential energy and the variational method. Two bifurcation points (β₁=1,β₂=1.47) as well as the two critical buckling loads Fcrsand Fcrhare obtained and the three different types of bifurcation solutions are also analysized. When β<β₁, the nonlinear equation has a zero solution,which is related to the stable state of the pipes. When β₁≤β<β₂, the equation has a peoriodic solution,which is related to the sinusoidal buckling state. And when β≥β₂, it has a helix solution,which is related to the helical buckling state. The results obtained here are very important for correctly designing pipes and reasonably selecting working parameters in horizontal well drilling,completing and stimulating operations.

Key words: horizontal well, pipe, stability, deformation, non-linear differential equation

摘要： 运用最小势能原理和变分方法,导出了描述水平井眼中管柱屈曲变形的微分方程——一个含有参数β的四阶非线性微分方程。求得了该非线性系统的两个分叉点(β₁=1,β₂=1.47)及相应的两个临界屈曲载荷F_crs和F_crh;并对其三种不同性态的分枝解进行分析。当β＜β₁时,微分方程有零解,对应于管柱的直线稳定状态;当β₁≤β≤β₂时,微分方程有周期解,对应于管柱的正弦屈曲状态;而当β≥β₂时,微分方程有螺线解,对应于管柱的螺旋屈曲状态。本文的结果对水平井钻井、完井和增产改造等井下作业管柱的正确设计和作业参数的合理选择等具有重要的参考价值。

关键词: 水平井, 管柱, 稳定性, 变形, 非线性微分方程