径向成层介质的Green函数及其在随钻电磁波电阻率测量中的应用

Green s function for radial stratified media and its application to electromagnetic wave resistivity measurement while drilling tool

采用递推矩阵方法计算径向成层介质的Green函数,根据圆柱形层界面处电场和磁场的连续性条件得到确定待定系数的矩阵方程组,并通过递推方法快速求解。采用将变型Bessel函数的指数项单独列出的方式有效地解决了Green函数计算中的上溢问题。用得到的Green函数对随钻电磁波电阻率测量仪器进行数值模拟,分析钻铤的影响规律、井眼校正方法和仪器的径向探测深度。结果表明,只须改变方程组中源项元素的位置,可以方便地得到源点和场点在任意层时的Green函数,形式简洁,易于编程。在仪器测量范围内金属钻铤对随钻电磁波电阻率测量仪器的影响可近似视为恒定,在井眼直径较小且钻井液电导率较低的情况下无须对随钻电磁波电阻率测量仪器进行井眼校正,仪器的径向探测深度并非只取决于仪器参数,也与侵入带电导率、地层电导率以及井眼因素有关。所得到的结论可为随钻电磁波电阻率测量仪器的研制和使用提供理论指导。

A recursive matrix method was adopted for the computation of Green＇s function in radial stratified media. According to the continuity condition of electric and magnetic fields on the cylindrical interface, a matrix equation group for computing the coefficients was obtained. And it can be quickly solved via the recursive method. The problem of overflow during the computation of Green＇s function can be successfully solved by subtracting the exponential term from the modified Bessel functions. The obtained Green＇s function was applied to numerical simulation of electromagnetic wave resistivity MWD tool. The results show that the Green＇s function with source point and field point in any layer can be conveniently obtained by merely changing the position of the elements within the source term of the equation group, and the obtained Green＇s function is terse in form and is easy to be programmed. The influence of the metal drill collar can be approximately taken as constant within the measurement range, and there is no need for borehole correction when the borehole＇s diameter is small and the mud＇s conductivity is low. The radial depth of investigation is determined not only by the tool＇s parameters but also by the conductivity of the invasion zone, the conductivity of the formation and the borehole＇s parameters. The obtained conclusions can provide theoretical guide for the manufacture and application of electromagnetic wave resistivity MWD tool.