Analytic Solution the Design Problem of a Hyperboloidal Arch Type Trajectory of Equal Curvature for Step-Horizontal Hole Sections
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摘要: 为了快速、精确地求解阶梯形水平井段等曲率双圆弧形三维井眼轨道设计问题,对设计约束方程组的解析求解方法进行了研究.利用三维井眼轨道设计问题的拟解析解理论,推导了圆弧曲率相等且同为未知数时的特征多项式的数学表示式;利用C++面向对象编程语言,实现了多项式的数值计算.利用构造理论模型计算出该模型的各种关键数据,再以此为已知设计数据、以圆弧曲率为未知数进行求解验证;设计约束方程组的解析法不仅能够正确区分出真解与增解,而且真解与已知数据完全相同.求出了阶梯形水平井段等曲率双圆弧形三维井眼轨道设计问题的解析解,解决了该问题有解、无解的判定问题,以及快速求解的问题.研究结果表明,设计约束方程组的解析法是一种可以用计算机实现的新算法,计算速度快、结果精确、可靠性高,可应用于钻井设计软件开发.Abstract: This study investigated the design constraints on the analytical method of solving equations for quickly and accurately solving the system of restricted equations for the design problem of hyperboloidal arch type trajectory of equal curvature for step-horizontal sections.An explicit formulation of the characteristic polynomial was given by taking advantage of the known result of the theory of quasi-analytic solution for three dimensional trajectory design.An implementation strategy for polynomial numerical calculation was given by taking advantage of object-oriented C++ programming techniques.To test the correctness of the analytic solution formulation,a theoretical model was constructed,and the key data of trajectory was calculated using this model,then they solved the system of restricted equations in which the equal curvature was one of the unknowns and above key data were known.The analytic solution method was not only able to discriminate exactly the true solution from pseudo solution,but also giving the true solution the same to known data.An analytic solution formulation for the solution’s existence was given for the trajectory design problem,and could be used for quick calculation.The result showed that the analytic method was a new programmable algorithm for solving the system of restricted equations of trajectory design,which had the advantages of quickness,accuracy,reliability,and usefulness in drilling software development.
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Keywords:
- drilling design /
- well trajectory /
- step-horizontal section /
- mathematical model /
- analytic solution
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