The Forward Modeling Of Multi-component Induction Logging Response In Anisotropic Formations

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.