基于多项式实数根的三维井眼轨道设计理论

Three Dimensional Trajectory Design Theory Based on the Real Roots of Polynomial

三维井眼轨道设计问题需要求解多元非线性方程组,由于未知数多、方程的非线性强,一般难以求出解析解,通常使用数值迭代方法求数值解.对三维s型轨道设计问题依据已知设计参数进行了分类,发现了一套有效的数学化简技巧,求出了第1类初值问题的解析解和第Ⅱ-Ⅳ类初值问题的拟解析解.提出了轨道设计问题的特征多项式的新概念,并证明了轨道设计问题是否有解取决于特征多项式是否有实数根,解的个数不多于实数根的个数或个数的二倍.所提出的基于特征多项式实数根的拟解析算法对于求解轨道设计问题具有计算速度快、计算可靠性高、易于计算机编程实现等优点,在三维水平井轨道设计、三维绕障井轨道设计、防碰设计等方面具有比数值迭代方法更好的计算性能.

For the problem of a three dimensional borehole trajectory design,it’s needed to solving a system of non-linear equations with multivariate.Because of so much variable and strong non-linear,it’s difficult to finding a analytic solution,and usually solved using the numerical iterative method.The design problem of S-type 3D trajectory is classfied as eight standard initial value problems,and the analytic solution for the initial value problem of class-Ⅰ and the quasi-analytic solutions for initial value problem of class-Ⅱ to class-Ⅳ by using a effective reduction we finded.A new terminology,characteristic polynomial of the trajectory design problem,is proposed,and it’s proved that the probles’ solvability depends on the count of real roots of the characteristic polynomial,and that the problems’ solutions is not more than the count or double of the real roots.The new algorithm based on the real roots of characteristic polynomial has the advantages of fast calculation,reliability,and easy to realize computer programming etc,and it’s more effective than the numerical algorithm on the problems such as 3D trajectory design for horizontal well,bypassing obstacles,anticollision,etc.