【作者】 胡平；

【导师】 汪宏年；

【作者基本信息】 吉林大学 ， 理论物理， 2011， 博士

【摘要】 各向异性地层中多分量感应测井响应的数值模拟算法,是当前地球物理测井中非常重要的研究课题,通过数值模拟技术考察复杂地层条件下多分量感应测井仪器的响应特征,对于仪器设计过程中参数选取和测量结果中有用信号提取以及建立更有效的数据处理和反演方法等均有重要意义。本论文深入研究了多分量感应测井仪器的几何因子理论、柱状各向异性分层介质中电磁场的解析算法以及水平层状非均质各向异性地层中电磁场的轴向混合算法等,取得了一些较有意义的理论和数值结果,具体有如下几部分内容。第一章,对于多分量感应测井理论的研究意义以及发展现状进行了系统总结,并概述本论文的主要研究内容以及创新部分。第二章,首先,利用Fourier变换理论、横电波（TE）和横磁波（TM）分解技术,给出均匀各向异性地层中频率-波数域磁流源并矢Green函数表达式,并结合Fourier逆变换和Bessel函数的积分公式,得到了频率空间域中磁流源磁场并矢Green函数以及磁流源电场并矢Green函数解析表达式。在此基础上,通过摄动理论与一阶Born近似,建立均匀各向异性地层中电导率微小摄动与磁流源并矢Green函数变化的摄动方程,利用其积分核函数的空间分布确定多分量感应测井仪器的空间响应函数。并通过数值计算结果,具体考察当空间各点上水平电导率和垂直电导率变化时,视电导率张量的三个主分量σ_a,xx、σ_a,yy和σ_a,zz的空间响应的灵敏度。并进一步推导了纵向微分几何因子和积分几何因子,径向微分几何因子和积分几何因子的计算方法。通过数值计算,系统考察了当水平电导率和垂直电导率变化时对视电导率张量的三个主分量σ_a,xx、σ_a,yy和σ_a,zz的纵向分辨率和探测深度的影响。数值结果表明,共面线圈系的空间响应与垂直电阻率和水平电阻率均密切相关,而共轴线圈系的空间响应只与水平方向的电阻率密切相关,且σ_a,xx,σ_a,yy对应的空间响应,对于空间各点上水平电导率和垂直电导率变化灵敏度较高;各向异性介质中,当垂直电导率变化时,与σ_a,xx,σ_a,yy对应的响应函数值相对各向同性介质有所减小,与σ_a,zz对应的响应函数值相对各向同性介质而言是不变的。第三章,利用混合势理论研究建立柱状各向异性介质中多分量感应测井响应的解析算法。首先从Maxwell方程出发,通过Fourier变换给出柱状分层介质中TE波和TM波的不同谐变分量在频域-波数域中的解析解,以及电场和磁场的其它谐变分量与横电和横磁谐变分量间的确切关系。在此基础上利用电磁场在柱状界面上的连续性边界条件,推导出反射系数矩阵、透射系数矩阵以及广义反射系数矩阵,得到频率-波数域中磁流源并矢Green函数的解析解。并进一步利用二维Fourier逆变换公式将频率-波数域中并矢Green函数转化到频率-空间域中,得到Sommerfeld积分形式的解析解。最后,利用三次样条插值与数值积分给出快速确定柱状各向异性介质中多分量感应测井响应的有效算法,通过数值结果研究考察仪器偏心、侵入深度、地层电阻率、频率变化、井眼泥浆、各向异性系数等变化对测井响应的影响。数值结果证明,相对于共轴线圈系,共面线圈系响应存在着非常强的非线性特征。其测量结果对泥浆电阻率、仪器长度、仪器偏心、频率、侵入半径、各向异性系数等变化很敏感,并且在图中我们可以出现负响应的情况非常多。另外,测量结果显示,地层纵向电阻率的灵敏度也与仪器长度等有很大关系。这些结果充分说明,需要研究一些新的迭代反演算法,以便从多分量感应资料中提取出地层纵向电阻率。第四章,利用轴向混合方法（Axial Hybrid Method, AHM）研究了水平层状非均质各向异性介质中多分量感应测井响应。从Maxwell方程出发,推导出水平层状各向异性介质中磁偶极子源电磁场的TE波（（?））和TM波（（?））满足的偏微分方程,并利用傅里叶级数将非轴对称问题转化为一系列轴对称问题,再应用轴向混合方法求解。整个过程包括:通过分离变量法将轴对称问题转化为本征值和本征向量问题,利用有限元法确定纵向（方向）上的微分方程本质值和本征向量、获得纵向本征解,再用解析法求解径向微分方程、确定每个本征值对应的径向解析解,将所有本征解叠加得到TE波（（?））和TM波（（?））的半解析解;在此基础上,利用电磁场水平分量与垂直分量转换矩阵以及柱状界面上电磁HzzEzzH Ez场的边界条件,给出柱状分层界面上的反射系数、透射系数、广义反射系数矩阵以及不同柱状分层地层中TE波和TM波振幅的递推公式,最后得到水平层状非均质地层（含井眼和侵入带）各向异性的电磁场的半解析解。该算法的基本思想,就是在轴向上形成数值的本征模式解,而在径向上应用广义反射矩阵、透射矩阵,描述各模式在界面上的相互藕合,用解析递推方法来计算纵向各层中的场强。AHM算法将2.5维问题降为二维数值问题,把二维数值问题转化为一维数值解和一维解析解的结合,这样既保证了计算结果的精度又提高了计算效率。对于复杂的多层地层模型,可以深入理解电磁场的分布以及其变化规律和物理意义。它可灵活应用于纵向有任意多层平面分层径向又具有任意多层柱面分层的非均匀介质中的模型中。此外,利用电导率导数的奇异性,在纵向微分方程中引入了一个附加奇异微分算子,用于解释水平层状分界面上积累面电荷对电磁场的影响,取得了有效的结果。更多还原

【Abstract】 The numerical simulation study of multi-component induction logging response in anisotropic formations is a very important geophysical logging research topic currently. By numerical simulation, we can investigate the multi-component induction tool response characteristic in the complex formation conditions, which can help to select proper tool parameters in instrument design, extract of the useful signal, and establish more efficiency inversion algorithm. This thesis deeply study the geometric factor theory of multi-component induction logging instrument , the analytic solution in electromagnetic field from cylindrical anisotropic layered media and the axial mixing algorithm of electromagnetic field from the horizontal layered non-homogeneous anisotropic layer. Some interesting theory and numerical results are shown finally. Specific parts are arranged as following.In the first chapter, we systematically summarize the study process of the multi-component induction logging theory and the current development status. Besides, the main contents and innovative parts of this paper are presented in end of the chapter.In chapter II, Using Fourier transform theory and the transverse electric wave （TE） and transverse magnetic （TM） decomposition technique, we give the dyadic green function expression of magnetic current source in homogeneous anisotropic formation. And then combining inverse Fourier transform with the Bessel functions‘integral formula, we get the analytical expression of the magnetic current source dyadic Green function in magnetic field and electric field .On this basis, through perturbation theory and first order born approximation we establish the perturbed equations between the small perturbation of conductivity and the changes of the dyadic Green function by magnetic current source in homogeneous anisotropic formation and use its spatial distribution of integral kernel function to determine the space response function of multi-component induction logging tool. Through numerical results, we investigated the sensitivity of the spatial response of the three principal componentsσ_a,xxσ_a,yy, andσ_a,zz about the horizontal and vertical conductivities in the formation. Besides, we further derived the calculation method about the differential and integral geometry factors on the longitudinal and radial directions through the calculation of the multi-component induction logging response. Numerical results show that the coplanar system spatial response is closely related to the vertical resistivity and the coaxial system response is closely related to the horizontal space resistivity. Moreover, the corresponding ofσ_a,xx,σ_a,yy to the spatial response is high sensitivity to the changes of each point on the horizontal and vertical electrical conductivity in space. In anisotropic media, when the vertical conductivity changes, relatively to isotropic media, the response function value corresponding toσ_a,xxσ_a,yy has been reduced, and the response function value corresponding toσ_a,zz is relatively constant.In chapter III, we establish the analytic algorithm of multi-component induction log responses by mixed potential theory in cylindrical anisotropic formation. Firstly, we use Fourier transform to extract the electromagnetic field’s analytical solution of different Harmonic components about TE wave and TM wave in the frequency - wave number domain, and then we give the precise relationship between the electric and magnetic components’other harmonic component and the transverse magnetic and transverse electric harmonic component. On this basis, using the electromagnetic field boundary continuity conditions of the columnar interface, we derived reflection coefficient matrix, transmission coefficient matrix, the generalized reflection coefficient matrix in each bed. Finally, we get the dyadic Green function’analytical solution of magnetic current source in frequency-wavenumber domain. Using two-dimensional inverse Fourier transform formula, we can further obtain the dyadic Green function in the frequency-space domain. The Somerfield integral generated in above process can be resoved by the cubic spline interpolation and numerical integration. Finnaly, we build the fast and efficient algrothm of multi-component induction logging response and system study the influence of the multi-component induction logging response on the instrument bias, invasion depth, formation resistivity, frequency changes, mud hole, and anisotropy coefficient. Numerical results show that compared to the axis coil system, the response of the coplanar coil system has a very strong nonlinear characteristic. Its logging response is sensitive to the changes of mud resistivity, instrument length, eccentric instrument, frequency, invasion radius and anisotropy coefficient of resistivity. Besides, some negative respond appears in figures. In addition, the numerical results show that the sensitivity of the vertical apparent resistivity also have many relationships to the instrument length and frequency .These results fully demonstrated some new iterative inversion algorithms are needed to extract the formation resistivity from the multi-component induction logging data.In chapter IV, Using axial mixing method （Axial Hybrid Method, AHM） we study the multi-component induction logging response in horizontally layered heterogeneous transversely isotropic medium.First; we consider the horizontally layered media model without cylindrical interface. Applying the Maxwell equation, we can obtain two partial differential equations which are produced by magnetic dipole source in a horizontal layered anisotropic medium, describing TE and TM waves separately. And further, the non-axisymmetric problem is transformed into a series of axisymmetric problem with the help of the Fourier series, and then calculates the problem applying the axial mixing method.The whole process including the following: transform the axisymmetric problem into the eigenvalues and eigenvectors problems using the method of separation variables, then use the finite element method to determine the eigenvalues and eigenvectors of the partial differential equations in longitudinal direction, use the analytic method to determine the analytic solution of the partial differential equations in radial direction corresponding to each eigenvalue. Sum all the eigensolutions; we can get the semi-analytical solution of the TE and TM waves. Based on the above, we can obtain the reflection coefficient, transmission coefficient, the generalized reflection coefficient matrix in columnar interface and amplitude recurrence formula of TE and TM waves in different columnar layered medium, using the horizontal component and vertical component of electromagnetic field transformation matrix and the columnar interface boundary conditions, finally we get the semi-analytical solution of electromagnetic fields in horizontal layered non-homogeneous （including the borehole and invasion zone） anisotropic medium. The basic idea of the AHM is obtaining the numerical Eigen mode solution in the axial and deriving the analytic solution by the application of the generalized reflection matrix, transmission matrix in radial direction. Both of them coupling with each other at the interface, so the analytical solution can be obtained by anaylitical recursive method. 2.5 dimensional calculation is decrease to 2 dimension throuth the AHM algorithm, then the 2 dimensional calculation is further transform into one dimensional analytical solution combing with one dimensional numerical solution.It not only ensures the accuracy of calculation but also improves the computational efficiency. For complex multi-layer formation model, we can deeply understand the variation of the electromagnetic field distribution and the physical meaning. It can be flexibly applied to the non-homogeneous medium model which has arbitrary multi-layered planer in vertical and radial directrions. Besides, when there are presenting multilayers, using the current density continuity condition, it can be proved that the accumulated surface charge will be produced at layer boundary.In this chapter, using the singularity of the conductivity derivative, introduced differential equations an additional singular differential operator in the longitudinal, which used to explain accumulatied charge from the level of the surface boundary layer on the impact of electromagnetic fields.更多还原