Abstract: Spatial arc is a typical component of wellbore trajectory widely used in 3D wellbore trajectory planning, drilled trajectory monitoring, and slide steering drilling trajectory control. The shape and posture of spatial-arc trajectory are characterized by using the borehole curvature with initial tool face angle and the basic trajectory parameters at two ends of a well section, respectively. Two methods for describing spatial-arc trajectory (i.e., coordinate conversion and vector analysis) are established, and two sets of formulas for calculating the parameters of spatial-arc trajectory are proposed. These formulas are mathematically exact solutions, which can be used to calculate the inclination angle, azimuth angle, spatial coordinate, tool face angle, orientation azimuth, and other parameters at any point on the spatial-arc trajectory; additionally, they are single-valued analytic functions of well depth or bending angle. The spatial-arc trajectory model is systematically described and universal methods for interpolation and extrapolation calculation of borehole trajectory are provided. The proposed methods, which are concise and practical, can simplify the design and calculation methods of 3D wellbore trajectory and anti-collision scan between adjacent wells; their rationality and applicability are validated in an example study.

Key words: drilling theory, directional drilling, wellbore trajectory, mathematical model, calculation method

摘要： 空间圆弧作为一种典型的井眼轨迹组件，广泛应用于三维井眼轨道设计、实钻轨迹监测和滑动导向钻井轨迹控制等领域。分别基于井眼曲率及初始工具面角和井段两端点的基本轨迹参数来表征空间圆弧轨迹的形状和摆放姿态，建立了空间圆弧轨迹的两种描述方法即坐标转换法和矢量分析法，提出了两套空间圆弧轨迹参数的计算公式。这些计算公式在数学上是精确解，可计算空间圆弧轨迹上任一点的井斜角、方位角、空间坐标、工具面角、定向方位角等参数，并且它们都是井深或弯曲角的单值解析函数。对空间圆弧轨迹模型进行了系统地描述，形成了普遍适用的井眼轨迹内、外插值计算方法，可直接用于计算井眼轨迹的分点参数、外推预测井眼轨迹等，具有形式简明、方法实用等特点。此外，应用此研究成果，还可简化三维井眼轨道设计、邻井防碰扫描等设计与计算方法。应用实例验证了研究结果的科学性和适用性。

关键词: 钻井理论, 定向钻井, 井眼轨迹, 数学模型, 计算方法