用传播矩阵法研究层状正交各向异性地层中多分量感应测井响应

本文采用传播矩阵技术研究并建立了层状正交各向异性地层中多分量感应测井响应的有效算法.首先通过Fourier变换将频率空间域中的Maxwell方程组求解问题转化为频率波数域中关于电磁场水平分量常微分方程组的定解问题.利用该方程组系数矩阵的本征值和归一化本征向量将电磁场分解成上行波和下行波模式的组合,推导出均匀正交各向异性介质中由任意方向磁偶极子产生的电磁波模式解析表达式;在此基础上,利用叠加原理和边界条件研究了电磁波在层状正交各向异性地层中的反射和透射,给出各个界面上的广义反射系数和不同地层中电磁波振幅的递推公式,进而得到电磁波模式的解析解.为了有效确定频率空间域中的电磁场,采用二维Patterson自适应求积算法结合有限连分式展开技术计算傅氏逆变换.最后通过数值模拟结果证明了该算法的有效性,考察了不同各向异性系数、不同井眼倾角以及仪器长度和工作频率变化等情况下的多分量感应测井响应特征.

Study on the responses of multi-component induction logging tool in layered orthorhombic anisotropy formations by using propagator matrix method

In this paper, we present an efficient algorithm to numerically simulate the responses of multi-component induction logging tool (MCIL) in layered orthorhombic anisotropy formations by using propagator matrix method (PMM). First, the problem of solving Maxwell equations in frequency-spatial domain is transformed into a definite solution problem of ordinary differential systems of horizontal components of electromagnetic (EM) field in frequency-wavenumber domain by Fourier transform. The EM field can be decomposed into upgoing and downgoing wave mode by using eigenvalues and normalized eigenvectors of the system matrix. And an analytical expression of electromagnetic wave emitted by a magnetic dipole of arbitrary direction in homogeneous orthorhombic anisotropy media is derived. On this basis, we derive the recurrence formulas of generalized reflection coefficient per interface and the amplitude of electromagnetic wave in each bed by studying the reflection and transmission of electromagnetic wave in layered orthorhombic anisotropy formations based on superposition principle and interface conditions. Then the analytical expression of electromagnetic wave mode in layered orthorhombic anisotropy formations is achieved. In order to determine numerical solution of EM field in frequency-space domain, a 2D adaptive Patterson integral formula and the limitedly continued fraction expansion technique are used to compute the 2D inverse Fourier transform. Finally, numerical tests were carried out to validate the new algorithm and investigate the influence of changes in anisotropy coefficients, borehole dipping angles, tool lengths and work frequencies etc. on the response characteristics of MCIL tool.