Three-dimensional simulation of eccentric LWD tool response in boreholes through dipping formations

We simulate the response of logging-while-drilling (LWD) tools in complex thee-dimensional (3-D) borehole environments using a finite-difference time-domain (FDTD) scheme in cylindrical coordinates. Several techniques are applied to the FDTD algorithm to improve the computational efficiency and the modeling accuracy of more arbitrary geometries/media in well-logging problems: (1) a 3-D FDTD cylindrical grid to avoid staircasing discretization errors in the transmitter, receiver, and mandrel geometries; (2) an anisotropic-medium (unsplit) perfectly matched layer (PML) absorbing boundary condition in cylindrical coordinates is applied to the FDTD algorithm, leading to more compact grids and reduced memory requirements; (3) a simple and efficient algorithm is employed to extract frequency-domain data (phase and amplitude) from early-time FDTD data; (4) permittivity scaling is applied to overcome the Courant limit of FDTD and allow faster simulations of lower frequency tool; and (5) two locally conformal FDTD (LC-FDTD) techniques are applied to better simulate the response of logging tools in eccentric boreholes. We validate the FDTD results against the numerical mode matching method for problems where the latter is applicable, and against pseudoanalytical results for eccentric borehole problems. The comparisons show very good agreement. Results from 3-D borehole problems involving eccentric tools and dipping beds simultaneously are also included to demonstrate the robustness of the method.     

双绞传输线电磁兼容特性的FDTD分析

将曲线坐标系中的时域有限差分法（FDTD）拓展到空间为任意媒质情形，然后针对双绞传输线的特殊结构综合运用曲线坐标系、圆柱坐标系、旋转坐标系和子域连接法进行空间网格划分、FDTD建模，并通过求解双绞传输线和平行传输线的特性阻抗对该建模的准确度进行了验证，最后在空间线源激励下对双绞线和平行线的电磁兼容特性进行了比较，可以看出双绞线对空间电磁场的耦合能力远远低于平行线。     

Coupled Field Inside Shielding Enclosure With an Aperture Due to Nearby Lightning

Due to a nearby lightning return stroke, the coupled electromagnetic (EM) fields inside a rectangular shielding enclosure, on the finitely conducting ground, with an aperture in one wall are calculated numerically by using the finite-difference time-domain (FDTD) method. First, the near fields generated by the return stroke are obtained in two-dimensional (2-D) cylindrical coordinates by the FDTD method. Then, the coupled fields inside a rectangular shielding enclosure are calculated in three-dimensional (3-D) rectangular coordinates through the total field-scattered field connecting boundary, with the sources obtained by coordinates transformations of the return stroke near fields     

Analysis of cylindrical transmission lines with thefinite-difference time-domain method

In this paper, the finite-difference time-domain (FDTD) method is used to calculate the propagation characteristics of cylindrical transmission lines. An efficient two-dimensional FDTD algorithm is developed by projecting the three-dimensional FDTD cell in the cylindrical coordinates onto the r-φ plane. An effective absorbing boundary condition is employed to truncate the mesh at its outer radial boundary. Numerical results are derived for different cylindrical transmission lines and compared to data available in the literature. Specifically, the newly proposed cylindrical coplanar waveguide is studied both theoretically and experimentally     

Cylindrical FDTD Analysis of LWD Tools Through Anisotropic Dipping-Layered Earth Media

Electrical logging-while-drilling (LWD) tools are commonly used in oil and gas exploration to estimate the conductivity (resistivity) of adjacent Earth media. In general, Earth media exhibit anisotropic conductivities. This implies that when LWD tools are used for deviated and horizontal drilling, the resulting borehole problem may include dipping-layered media with dipping beds having full 3 times 3 conductivity tensors. To model this problem, we describe a 3-D cylindrical finite-difference time-domain (FDTD) algorithm extended to fully anisotropic conductive media and implemented with cylindrical perfectly matched layers to mimic open-domain problems. The 3-D FDTD algorithm is validated against analytical results in simple formations, showing good agreement, and used to simulate the response of LWD tools through anisotropic dipping beds for various values of anisotropic conductivities and dipping angles     

基于FDTD方法的同轴电缆孔缝辐射效应研究

利用时域有限差分法(FDTD)研究了同轴电缆传输数字脉冲信号时的孔缝辐射效应。文中首先推导了三维柱坐标系下完全匹配层(PML)的差分表达式，并给出保证其解稳定性的更为严格的时间步长公式。然后利用FDTD法分析了数字信号经同轴电缆孔缝的辐射效应，对孔缝内外的电磁场及其远场进行了数值分析和计算。处理孔缝边界时，用级数展开和积分方程得到金属拐角处电磁场的近似解。最后对结果进行了分析。     

FDTD algorithm in curvilinear coordinates [EM scattering]

The finite-difference-time-domain (FDTD) algorithm for the solution of electromagnetic scattering problems is formulated in generalized coordinates in two dimensions and implemented in a code with the lowest-order Bayliss-Turkel radiation boundary condition expressed in cylindrical coordinates. It is shown that, for a perfect conductor, such a formulation leads to a stable, well-posed algorithm and that, in regions where the curvature of the coordinate lines is not great, the dispersion and anisotropy effects are negligible. Such effects become more pronounced in regions of high curvature, leading to unphysical phase shifts. The magnitude of such shifts and the amount of wavefront distortion is studied via numerical experiments using a cylindrical mesh. Near-field results are given for two canonical shapes for each polarization: the circular cylinder and cylinders of square and rectangular cross sections. These results are compared with those obtained by exact eigenfunction expansion techniques, with method-of-moments (MM) solutions, and with solutions obtained from an alternate FDTD approach. In each case, agreement is excellent. The propagation of a plane wave through a polar space in the absence of a scatterer is also examined, and it is shown that the FDTD algorithm is capable of tracking the incident wave closely     

On the modeling of conducting media with the unconditionally stable ADI-FDTD method

The Courant-Friedrich-Levy stability condition has prevented the conventional finite-difference time-domain (FDTD) method from being effectively applied to conductive materials because of the fine mesh required for the conducting regions. In this paper, the recently developed unconditionally stable alternating-direction-implicit (ADI) FDTD is employed because of its capability in handling a fine mesh with a relatively large time step. The results show that the unconditionally alternating-direction-implicit-finite-difference time-domain (ADI-FDTD) method can be used as an effective universal tool in modeling a medium regardless of its conductivity. In addition, the unsplit perfectly matched layer combined with the ADI-FDTD method is implemented in the cylindrical coordinates and is proven to be very effective even with the cylindrical structures that contain open conducting media.     

A three-dimensional unconditionally stable ADI-FDTD method in the cylindrical coordinate system

An unconditionally stable finite-difference time-domain (FDTD) method in a cylindrical coordinate system is presented in this paper. The alternating-direction-implicit (ADI) method is applied, leading to a cylindrical ADI-FDTD scheme where the time step is no longer restricted by the stability condition, but by the modeling accuracy. In contrast to the conventional ADI method, in which the alternation is applied in each coordinate direction, the ADI scheme here performs alternations in mixed coordinates so that only two alternations in solution matching are required at each time step in the three-dimensional formulation. Different from its counterpart in the Cartesian coordinate system, the cylindrical ADI-FDTD includes an additional special treatment along the vertical axis of the cylindrical coordinates to overcome singularity. A theoretical proof of the unconditional stability is shown and numerical results are presented to demonstrate the effectiveness of the cylindrical algorithm in solving electromagnetic-field problems.     

Dispersion analysis of multilayer cylindrical transmission linescontaining magnetized ferrite substrates

Recently, there has been a growing interest in using cylindrical transmission lines that contain magnetized ferrite material in a variety of applications. In this paper, the finite-difference time-domain (FDTD) method (in cylindrical coordinates) and the spectral-domain analysis (SDA) are used to calculate the propagation characteristics of cylindrical transmission lines that contain magnetized ferrite material. The magnetization can be either in the longitudinal or azimuthal directions. Specifically, the cylindrical microstrip line, and the cylindrical coplanar waveguide printed on magnetized ferrite substrate are analyzed. Both the FDTD and SDA results are in very good agreement. In addition, the results are compared to those of planar structures by taking the radius of the substrate to be large enough such that the curvature effect is negligible     

用高阶叠层矢量有限元法计算随钻测井的三维电磁响应

用矢量有限元方法对随钻测井的三维地层电磁响应进行了数值仿真。为了提高随钻测井响应的计算效率和精度,采用了高阶叠层矢量基函数,根据场变化快慢的情况,在不同的剖分单元和同一单元的不同方向采用了不同阶数的基函数,有助于减少剖分网格和未知量。采用了基于圆柱坐标下六面体剖分,从而提高了发射、接收线圈、钻铤和井眼边界的几何建模精度。用解析法和数值模式匹配法对有限元计算的结果进行了验证,显示了非常好的吻合,最后仿真了地层倾角对随钻测井响应的影响。     

毫米波有源柱面共形微带阵列天线的理论与实验研究

本文介绍了一种新型Ｋａ波段毫米波有源柱面共形微带阵列天线,实现了振荡器与天馈系统的单片集成,并应用圆柱坐系下的时域有限差分法对该阵列天线进行了计算和理论分析,得到了令人满意的理论和实验结果。     

An FDTD model for calculation of gradient-induced eddy currents in MRI system

In most magnetic resonance imaging (MRI) systems, pulsed magnetic gradient fields induce eddy currents in the conducting structures of the superconducting magnet. The eddy currents induced in structures within the cryostat are particularly problematic as they are characterized by long time constants by virtue of the low resistivity of the conductors. This paper presents a three-dimensional (3-D) finite-difference time-domain (FDTD) scheme in cylindrical coordinates for eddy-current calculation in conductors. This model is intended to be part of a complete FDTD model of an MRI system including all RF and low-frequency field generating units and electrical models of the patient. The singularity apparent in the governing equations is removed by using a series expansion method and the conductor-air boundary condition is handled using a variant of the surface impedance concept. The numerical difficulty due to the "asymmetry" of Maxwell equations for low-frequency eddy-current problems is circumvented by taking advantage of the known penetration behavior of the eddy-current fields. A perfectly matched layer absorbing boundary condition in 3-D cylindrical coordinates is also incorporated. The numerical method has been verified against analytical solutions for simple cases. Finally, the algorithm is illustrated by modeling a pulsed field gradient coil system within an MRI magnet system. The results demonstrate that the proposed FDTD scheme can be used to calculate large-scale eddy-current problems in materials with high conductivity at low frequencies.     

An FDTD model for low and high altitude lightning-generated EM fields

To explore lightning-generated electromagnetic wave behavior and lightning-related ionospheric phenomena, a full-wave two-dimensional cylindrical finite-difference time-domain (FDTD) model was developed to simulate lightning-generated electromagnetic wave propagation in the ionosphere with high altitude and long distance capabilities. This FDTD model removes the approximations made in other similar models to extend its applicability, and incorporates a variety of existing methods and new techniques. A dispersive and anisotropic realization of the nearly perfectly matched layer (NPML) absorbing boundary condition is adopted in this numerical model for ease of implementation. Earth curvature is included in the model through the modified refractive index method. The surface impedance boundary condition is adopted to treat arbitrary but homogeneous ground parameters. We quantify the errors through dispersion relations, and the solution convergence is analyzed. Comparisons between our simulation, numerical waveguide mode theory, and experimental data validate this model and show its capabilities compared to other methods. Although this FDTD model was developed for the lightning-generated electromagnetic field simulation, it is also applicable for other very low frequency (VLF, 3-30 kHz) and extremely low frequency (ELF, 3-3000 Hz) wave propagation problems.     

柱坐标系中的驻波-行波边界条件

时域有限差分法模拟中所用的驻波-行波边界条件(STWBC)具有较高的计算效率,且非常易于实现.但此吸收边界条件最初仅用于直角坐标的情形.文章将STWBC扩展到柱坐标的情形与直角坐标的情形一样,柱坐标中的STWBC也在计算域外附加理想导电(磁)壁进行截断,并将边界处的驻波转化为行波,从而模拟行波在无限大空间的传播.文中给出的数值算例证明了此种吸收边界条件的有效性.     

An Improved Implicit Split-Step Algorithm for Dispersive Band-Limited FDTD Applications

An improved unconditionally stable envelope finite difference time domain (FDTD) algorithm is presented for modeling open region dispersive band-limited electromagnetic applications. The algorithm is based on incorporating the split-step implicit FDTD scheme into the stretched coordinates perfectly matched layer absorbing boundary conditions. Numerical examples carried out in 2-D domains are included to show the validity of the proposed formulations.      

Novel FDTD simulation method using multiple-analysis-space for electromagnetic far field

Wide-band undesired electromagnetic noise near electronic systems, which includes small noise source like the printed circuit board (PCB), is a current problem in the field of electromagnetic interference. However, the estimation method for the electromagnetic noise near a system under test has not been established. This paper proposes a newly developed estimation method of the electromagnetic noise for a wide area, from near to far field, using the finite difference time-domain (FDTD) method. The proposed FDTD simulation method is an estimation technique for near to far field with multiple analysis spaces (MAS). The MAS has an internal analysis space (IAS) and an external analysis space (EAS). The analysis near a radiation source can be calculated in the IAS. The EAS is the outside space from IAS, which is for calculation of the far field. It is expected that the proposed FDTD method by MAS (FDTD-MAS) decrease in the calculation cost in terms of computational time and memory costs, especially for estimation of radiation from PCB. The principle procedure of the FDTD-MAS method is described in the first part of this paper. As example of advantages of the calculation and confirmation of the calculation accuracy, the electric field distributions radiated from a 1-GHz half-wavelength dipole antenna in an IAS of 0.3/spl times/0.3 m/sup 2/ area and an EAS of 7/spl times/7 m/sup 2/ area are used as examples. When the cell size ratio of IAS to EAS is changed from 6 to 20, the FDTD and theoretical values show good agreement. It is indicated that the FDTD-MAS simulation method is one of the most powerful tools for the estimation of electromagnetic noise from near field to far field from small and thin source.     

利用分离变量法计算电磁波电阻率仪器的测井响应

研究随钻电磁波电阻率仪器中线圈系电磁响应的计算方法,对于测井数据的快速反演具有重要的意义。从Maxwell方程出发,导出了修正磁矢位满足的泊松方程,并采用分离变量法和电磁场边界条件导出了圆电流环在径向分层介质中的磁矢位的解析解,该方法形式简洁,易于编程,精度高。利用上述方法对随钻电磁波电阻率测量仪器进行了建模和计算,并分析了钻铤刻度曲线和井眼校正方法,所得到的结论对于随钻电磁波电阻率测量仪器的研制具有重要的理论价值和实际意义。     

Z 变换法改进PML-FDTD 吸收边界条件

时域有限差分（FDTD）方法是计算时域电磁散射和辐射的一种简单有效的方法,被广泛应用于求解电磁场问题中,但由于计算机容量的限制,FDTD 计算只能在有限区域进行,为了能模拟开域电磁过程,在计算区域的截断边界处必须给出吸收边界条件,完全匹配层(PML)是一种行之有效的吸收边界条件。在PML 中应用Z 变换,和传统的引入PML 的方法相比,得到的迭代公式的程序更方便、更简单;考虑到要模拟的FDTD 计算区域的虚拟物质属性,采用了特殊的处理方法;数值实验验证了这种方法的有效性和吸收边界的吸收效果。