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在使用最小曲率法推导有关公式的过程中,涉及了大量的三角函数变换,过程有时很复杂,极易出现错误。为了解决这个问题,分析了井眼轨迹切线与井斜角和方位角之间的数学关系,利用井眼方向矢量给出了最小曲率法的矢量形式。这种矢量形式深刻地揭示了圆弧井段弯曲角、井眼曲率、坐标增量、井斜角和方位角之间的关系的数学实质,具有形式简洁和几何直观性。使用矢量形式的最小曲率法对井眼轨迹内插、完钻井段设计等问题进行了研究。结果表明,有关公式的推导过程非常简单,完全避免了复杂的三角函数变换,而且最终得到了计算公式更加简单。同时,还用实际算例进行了验算,证实了本文新公式的正确性。本文提出的矢量形式的最小曲率法新公式可以应用于与圆弧井段有关的井眼轨迹设计和计算问题中。
In the process of deriving the formulas using the minimum curvature method, a great deal of trigonometric function transformation is involved, and the process is sometimes complicated and prone to errors. In order to solve this problem, the mathematical relationship between well trajectory tangent and well inclination angle and azimuth angle is analyzed, and the vector form of minimum curvature method is given by borehole orientation vector. This vector form profoundly reveals the mathematical essence of the relationship between arc wellbore bending angle, wellbore curvature, coordinate increment, well inclination and azimuth, and has the form conciseness and geometric intuition. The minimum curvature method in vector form is used to study the interpolation of borehole trajectory and the design of the complete drilling section. The results show that the derivation of the formula is very simple, completely avoiding the complicated trigonometric transformation, and finally the formula is simpler. At the same time, the actual calculation is also used to verify the correctness of the new formula in this paper. The new formula of minimum curvature method in vector form proposed in this paper can be applied to the wellbore trajectory design and calculation problems related to circular arc sections.