Ning Yuan

Exact Solution to Electromagnetic Scattering by an Impedance Sphere Coated With a Uniaxial Anisotropic Layer

Electromagnetic fields in uniaxial anisotropic media can be obtained based on spherical vector wave functions in isotropic media. Applying the boundary conditions of electromagnetic fields on each interface of an impedance sphere coated with a uniaxial anisotropic layer, the expansion coefficients of electromagnetic fields in the uniaxial anisotropic medium and free space can be derived. The present method is very general, which can be reduced to many isotropic subcases. The present method has been validated in those subcases. Scattering by an impedance sphere with an anisotropic coating is presented and discussed.

Fast Analysis of RCS Over a Frequency Band Using Pre-Corrected FFT/AIM and Asymptotic Waveform Evaluation Technique

The pre-corrected fast Fourier transform (PFFT)/adaptive integral method (AIM) is combined with the asymptotic waveform evaluation (AWE) technique to present fast RCS calculation for arbitrarily shaped three-dimensional PEC objects over a frequency band. The electric field integral equation (EFIE) is used to formulate the problem and the method of moments (MoM) is employed to solve the integral equation. By using the AWE method, the unknown equivalent current is expanded into a Taylor series around a frequency in the desired frequency band. Then, instead of solving the equivalent current at each frequency point, it is only necessary to solve for the coefficients of the Taylor series (called ldquomomentsrdquo) at each expansion point. Since the number of the expansion points is usually much smaller than that of the frequency points, the AWE can achieve fast frequency sweeping. To facilitate the analysis of large problems, in this paper, all the full matrices are stored in a sparse form and the PFFT/AIM method is employed to accelerate all the matrix-vector products on both sides of the matrix equation for the moments. Further, the incomplete LU preconditioner is used at each expansion point to improve the convergence behaviour of the matrix equation for the moments. The present method can deal with much larger problems than the conventional MoM-AWE method since the PFFT/AIM achieves considerable reduction in memory requirement and computation time. Numerical results will be presented to show the efficiency and capability of the method.

Simulation of full responses of a triaxial induction tool in a homogeneous biaxial anisotropic formation

Triaxial induction tools are used to evaluate fractured and lowresistivity reservoirs composed of thinly laminated sand-shale sequences. Thinly laminated and fractured reservoirs demonstrate transversely isotropic or fully anisotropic biaxial anisotropic electrical properties. Compared to the number of studies on transverse isotropy, relatively little work covers biaxial anisotropy because of the mathematical complexity. We have developed a theoretical analysis for the full response of a triaxial induction tool in a homogeneous biaxial anisotropic formation. The triaxial tool is composed of three mutually orthogonal transmitters and three mutually orthogonal receivers. The bucking coils are also oriented at three mutually orthogonal directions to remove direct coupling. Starting from the space-domain Maxwell’s equations, which the electromagnetic EM fields satisfied, we obtain the spectral-domain Maxwell’s equations by defining a Fourier transform pair. Solving the resultant spectraldomain vector equation, we can find the spectral-domain solution for the electric field. Then, the magnetic fields can be determined from a homogeneous form of Maxwell’s equations. The solution for the EM fields in the space domain can be expressed in terms of inverse Fourier transforms of their spectral-domain counterparts. We use modified Gauss-Laguerre quadrature and contour integration methods to evaluate the inverse Fourier transform efficiently. Our formulations are based on arbitrary relative dipping and azimuthal and tool angles; thus, we obtain the full coupling matrix connecting source excitations to magnetic field response. We have validated our formulas and investigated the effects of logging responses on factors such as relative dipping, azimuthal and tool angles, and frequency using our code. We only consider conductivity anisotropy, not anisotropy in dielectric permittivity and magnetic permeability. However, our method and formulas are straightforward enough to consider anisotropy in dielectric permittivity.

Application of OpenMP to Wireline Triaxial Induction Logging in 1D Layered Anisotropic Medium

Efficient and accurate forward modeling of logging tool responses is essential for data inversion in the log data interpretation in both real time and postprocessing. With the aggressive advancement of various high-performance computing techniques and computer hardware technology, it is possible to significantly improve the efficiency of the forward modeling. In this paper, we apply OpenMP to parallelize the simulation of triaxial induction logging tools in 1D multilayered anisotropic formation. The parallel process is explained in detail and numerical examples are presented to demonstrate the effect of the parallel programming. Comparison of the original code and the parallel code shows that the latter is much faster without loss of accuracy, which is very promising for future real-time inversion.

Electromagnetic field response of triaxial induction logging tools in 1-D multi-layered anisotropic formations

As thick-layered oil reservoirs are gradually getting exhausted, detection and exploration of thin laminated reservoirs are becoming more and more important. Thin laminated sand-shale layers can be treated as transverse isotropic in bulk, characterized by the horizontal and vertical resistivity. The recently developed triaxial induction tool can detect formation anisotropy. In a traixial tool, three orthogonal transmitter and receiver coils oriented at three directions are used. The formation anisotropy responds to different components of the transmitter–receiver combinations, therefore the tool is able to measure formation resistivities with anisotropy. The measured data must be converted to the formation resistivities by inversion. Since the inversion process requires a repeated computation of the forward modeling, a fast forward modeling is essential to the inversion procedure. In this paper, we presented an 1D analytic method for the efficient simulation of traixial induction tools in multilayered transverse isotropic formations. This work is based on the previous work [1] with appropriate modifications and improvements. The coefficient propagator method is used to obtain the generalized reflection and transmission coefficients in each layer. The fast Hankel transform integration method is implemented to evaluate the highly oscillating integrals of the Bessel functions. The developed code has been verified by comparison with published data and is proved to work stably and accurately for dipping angles up to 89.9o and larger than 90o. For lower dipping angles, we further improve the efficiency of the code by using a modified Gauss-Laguerre quadrature with less sampling points instead of the fast Hankel transform to evaluate the integration without loss of accuracy.

Simulation of Induction Logging Tools in Anisotropic Media Using a 3D Finite Difference Method

Summary In this paper, we present a 3-D finite-difference (FD) method to simulate multicomponent induction log responses in arbitrarily anisotropic media. The finitedifference method uses a staggered grid to approximate the vector equation for the scattered electric field. The resulting linear sparse matrix is solved iteratively using a generalized minimal residual (GMRES) algorithm. An incomplete LU precondition is applied to improve the convergence behavior of the equation, thus accelerating the solution. Comparisons of the results with published data validate the algorithm.

Response of a Triaxial Induction Logging Tool in a Homogeneous Biaxial Anisotropic Formation

Summary In this paper, we present a theoretical analysis for the full response of a triaxial induction logging tool in a homogeneous biaxial anisotropic formation. The triaxial tool comprises three mutually orthogonal transmitters and three mutually orthogonal receivers. The transducer axes of the tool can be arbitrarily oriented with respect to the principal axes of the conductivity tensor of the biaxial anisotropic medium. Effects of logging responses on various tool and formation parameters are investigated using the developed method.

A 3-D fast integral equation method for simulation of induction logging response in formations with large conductivity contrasts

A three-dimensional (3-D) fast integral equation method is presented for the simulation of induction logging problems in formations with large conductivity contrasts, where most existing numerical techniques fail to obtain accurate results. The volume integral equation (VIE) is used to formulate the problem and the precorrected FFT accelerated method of moments (pFFT-MoM) is used to solve the integral equation. To overcome the difficulty of high singularity and slow convergence due to large conductivity contrasts, the incomplete LU (ILU) pre-conditioner is used to significantly speed up the convergence of the matrix equation. The resultant method is efficient and flexible for 3-D simulation of induction logging, especially for problems with large conductivity contrasts.

Electromagnetic field of arbitrarily oriented coil antennas in complicated underground environment

This paper presents a numerical algorithm based on the finite-difference method to simulate the electromagnetic fields generated by arbitrarily-oriented transmitter coils in a three-dimensional (3D) fully anisotropic medium. An edge-centered, staggered-grid finite-difference method is used to solve the vector equation for the scattered electric field. The resultant matrix equation is solved iteratively using a generalized minimal residual (GMRES) algorithm and an incomplete LU precondition technique is applied to improve the convergence behavior of the linear equation. A conductivity averaging technique and optimal grid refinement are also used in the approach to obtain further efficiency. Numerical examples will be presented to demonstrate the efficiency and capability of the present method.

Dyadic green's function in a homogeneous biaxial anisotropic medium

This paper presents an analytical formulation of the dyadic Greens function in a homogeneous biaxial anisotropic medium. The spectral dyadic Greens function is first obtained by solving the spectrum-domain Maxwells equations with a delta-type source excitation in a biaxial anisotropic medium. Then the space-domain dyadic Greens function is expressed as a triple inverse Fourier transform of the spectrum-domain Greens function. Efficient numerical techniques are used to evaluate the inverse Fourier transforms of the highly oscillating integrands accurately. The derived Greens function can be used in the integral equation solution for electromagnetic problems involving biaxial anisotropic medium background or simulation of triaxial induction tool responses in a biaxial anisotropic formation.