双靶点三维井眼轨道设计的解析法（I）

Analytical method of design of three-dimensional double-target-point wellbore trajectory

鲁港 刘乃震 余雷 夏泊洢 佟长海

LU Gang LIU Naizhen YU Lei XIA Boyi TONG Changhai

辽河油田公司勘探开发研究院，辽宁盘锦　124010 中国石油长城钻探工程有限公司，北京　100010 中国石油长城钻探工程有限公司工程技术研究院，辽宁盘锦　124010

Exploration and Development Research Institute of Liaohe Oilfield Company, Panjin 124010, China CNPC Greatwall Drilling Engineering Company Limited, Beijing 100010, China Liaohe Petroleum Exploration Bureau of PetroChina, Panjin 124010, China

五段制双靶点三维井眼轨道设计问题需要求解多元非线性方程组，通常使用的数值迭代算法具有初值依赖性强、收敛速度慢、迭代过程可能发散等缺陷，所得到的是近似解。通过复杂的消元技巧，将多元非线性方程组化简为两个4次特征多项式，进而得到了轨道设计问题的解析解。提出了基于特征多项式实数根的解析求解算法。轨道设计问题是否有解取决于特征多项式是否有小于1的正实数根，在有解的情况下，用特征多项式的正实数根和已知设计参数的解析计算公式可以直接求出解。计算速度快于数值迭代算法，计算结果精确。消元技巧对推导其他类型轨道设计问题的解析计算公式具有指导意义。

Five sections design of three-dimensional double-target-point wellbore trajectory is complex and need a numerical iterative method for solving nonlinear simultaneous equations, and the method has the disadvantages of dependence on initial conditions,a lower convergence rate and diverging easily, and has a approximate solution. A new method can transform nonlinear simultaneous equations into two characteristic polynomial of degree 4 by elimination, which solution is a analytical solution of the nonlinear simultaneous equations. The solution algorithm is given in this paper, and the characteristic polynomial is solvable depends on whether there are less than 1 positive real root. In the case of solvability, the solution can be directly obtained by positive real root of characteristic polynomial and analytical formula of the known design parameters. The method is faster than the numerical iterative calculation method, the computational results accurate. Elimination skill has important significance for other wellbore trajectory design.