三维圆弧型井眼轨道设计模型的拟解析解

为了研究三维圆弧型井眼轨道设计方程组是否存在解析解以及如何有效求解,使用消去法从多元非线性方程组中推导出其中的某个未知数满足一个特征多项式方程,并且证明了该方程组的拟解析解可以由特征多项式方程的实数根和已知设计参数构成的解析公式计算得到;如果极点多项式非恒等于0,则当特征多项式次数不超过5时,如果有解则为解析解,否则为数值解。通过使用C++类实现多项式加减法、乘法的计算方法,提出了多项式实数根的分隔算法、并用二分法快速求解出多项式全部实数根的计算方案。该方法在数学上经严格推导,计算方案简洁、稳健,在三维井眼轨道设计、智能钻井定向控制中具有应用价值。

Q uasi-analytical solution of the design model of three-dimensional arc type well trajectory

To study whether there is an analytical solution to the design equation set of three-dimensional arc-shaped well trajectory and how to solve it effectively, this paper uses the elimination method to deduce an unknown number from the multivariate nonlinear equation set which satisfies a characteristic polynomial equation, and proves that the approximate analytical solution of the equation set can be calculated from the analytic formula including the real roots of characteristic polynomial equation and the known design parameters. If the polar polynomial is not identically vanishing, when the degree of the characteristic polynomial does not exceed 5, if there is a solution, it is an analytical solution. Otherwise, it is a numerical solution. The authors explore the methods of applying polynomial addition, subtraction and multiplication in computer programming using C++ classes, propose the real root insulating algorithm for the real number roots of a polynomial, and quickly determines the calculation scheme for all real roots of polynomials by dichotomy. The proposed method is rigorously deduced in mathematics, and the calculation scheme is simple and robust. It has application value in the three-dimensional well trajectory design and directional control of intelligent drilling.