Studying On Forward And Inversion Algorithms Of Multicomponent Electromagnetic Induction Logging Data In An Anisotropically Layered Formation

In this paper, we systematically and deeply study algorithm for forward and inversion of multicomponent electromagnetic induction well logging data in an anisotropically layered formation. The main content includes the 3 dimensional numerical modeling of multicomponent electromagnetic induction well logging response in an anisotropically layered formation by finite difference time domain (FDTD) algorithm, an analytic solution of magnetic current dyadic Green function in 1 dimensional anisotropically layered media by transmission line theory, and full parameter iterative inversion of the logging data. The corresponding computer codes have been independently developed. Many useful results have been obtained through a lot investigation of the response characteristic of the tool in differently theoretical models and many execution inversions of the synthetic well logging data.For investigation of 3 dimensional responses of the tool in the anisotropic formation, it is set up the algorithms of 3 dimensional finite difference time domain to solve Maxwell equation in time domain and that of fast extracting the amplitude and phase of electromagnetic field during modeling. Finite difference in space and time domains and linear interpolation formula are used to approximate Maxwell equation in space and time domain to obtain recursive relation of electromagnetic field on Yee's staggered grids. The area weighted average and rotation matrix are used to compute effective conductivity on the grid with different conductivities to enhance precision of the approximation. Besides, an absorbing boundary condition based on uniaxial anisotropic perfectly matched layer (UPML) is executed to reduce the reflection of outer-boundary and reduce numbers of grids to improve efficiency of numerical modeling. Furthermore, it is applied three mutually orthogonal transmitter coils with a single-frequency sinusoidal current to excite EM fields in time domain. Then a special two-equation and two-unknown (2E2U) approach is used to extract the amplitude and phase of EM from the EM field in time domain and obtain the responses of multicomponent EM induction logging tool. The computer code has been developed and it is executed a series of numerical tests on many 3 dimensional models with varying horizontal and vertical conductivities, varying invasion zone conductivities, varying bed thicknesses, varying dipping angles and varying invasion radius to achieve different responses of the tool in these different models. Through comparison of these numerical results in the different models, the characteristics of multicomponent electromagnetic induction well logging tool in complex 3 dimensional models have been more completely and more deeply understood. The numerical results show that coplanar transmitter–receiver pair has higher vertical resolution than coaxial transmitter–receiver pair while the former has shallower investigation depth. In dipping borehole, dipping angle will have great influence on the responses of the tool and large great horns will appear near interfaces when dipping angle is large.At present, the 3D numerical calculation has the characteristic of larger workload and lower calculation efficiency, and it can't satisfy the requirement of the inversion, so the analytical and semi-analytical forward algorithms are still the popular research topic. In order to fast calculate the multi-component induction logging response, we present an analytical algorithm of magnetic current dyadic Green's function in layered anisotropic formation using the transmission line method. We decompose the EM field into two independent transverse electric wave (TE) and transverse magnetic wave (TM) using Fourier transform, and give the analytical expression of the magnetic current magnetic field dyadic Green's function in frequency domain due to the transmission line and wave propagation theory. Then, Sommerfeld integral expression of the magnetic current magnetic field dyadic Green's function in space domain is obtained by inverse Fourier transform. Considering the formation usually having more layers in electromagnetic induction logging, we use cubic spline interpolation on the fixed discrete node combined with Gauss-Legendre quadratrue to calculate the Sommerfeld integral. The numerical solution of the magnetic current dyadic Green's function in the space domain not only solves the modeling of the multicomponent induction logging response in the tilted or horizontal anisotropic formation, but also provides a very effective means for fast determining Frèchet derivative matrix in multicomponent induction logging data inversion. Compared to the FDTD algorithm, the TLM algorithm has more dominance in calculation precision and efficiency in the simple layered anisotropic formation neglecting well hole. We investigate the influence of the dipping angle, anisotropic coefficient of the resistivity, work frequency and the tool length on the multicomponent logging response. The numerical results show that along with the increase of the dipping angle, the apparent conductivity of the coplanar transmitter–receiver pair approaches the value of the horizontal conductivity, while that of the coaxial transmitter–receiver pair approaches the value of the vertical conductivity. The increase of frequency leads to all of the apparent conductivity decline and the vertical resolution increase. The fewer distance between transmission coil array and the receiving coil array, the higher resolution the apparent conductivity has.Based on 1D and 3D numerical simulation, the calculation of Frèchet derivative is transformed to the double integral of product of the two transmission line Green functions ( I i p, Vip),( I vp , Vvp) on z and kρfor multicomponent induction logging using perturbation theory and analytical solution of magnetic current dyadic Green's function in frequency domain. We first confirm the integral on variable z by analytical method, and then determine the integral on kρby Sommerfeld integral algorithm. Consequently, we investigate and establish the fast algorithm of Frèchet derivative matrix of multicomponent induction logging response. By investigating the Frèchet derivative matrix containing the partial derivatives of the entire model vector, we find the Frèchet derivative to bed boundary is maximal, while the one to vertical conductivity is minimal. These cases explain the future inversion results will be sensitive to the interface per bed, and the vertical conductivity can be obtained more easily.To reconstruct the resistivity and the geometric parameter of the formation, the iterative inversion theory of differential equation must be used to explain and evaluate the logging data. We obtain the fast full parameter iterative inversion technique of multicomponent logging data in layered anisotropic formation using the fast algorithm of Frèchet derivative with normalized processing and singular value decomposition technique. The horizontal and vertical resistivity, the bed boundary all can be obtained simultaneously by the fast inversion algorithm. The inversion results of the theoretical bed model validate the algorithm and the anti-noise ability. From the inversion results of several theoretical bed models, we find that the inversion results are sensitive to bed boundary, and the relative error of the vertical resistivity is larger than the horizontal resistivity. For the more layers formation, the initial bed model will affect the final inversion results. So, when invert the actual logging data, the interactive delamination technique between the human and the computer can be adopted to improve the inversion precision.