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In many applications, electromagnetic scattering from inhomogeneous objects embedded in multiple layers needs to be simulated numerically. The straightforward solution by the method of moments (MoM) for the volume integral equation method is computationally expensive. Due to the shift-invariance and correlation properties of the layered-medium Green’s functions, the stabilized-biconjugate gradient fast Fourier transform (BCGS-FFT) has been developed to greatly reduce the computational complexity of the MoM, but so far this method is limited to objects located in a homogeneous background or in the same layer of a layered medium background. For those problems with objects located in different layers, FFT cannot be applied directly in the direction normal to the layer interfaces, thus the MoM solution requires huge computer memory and CPU time. To overcome these difficulties, the BCGS-FFT method combined with the domain decomposition method (DDM) is proposed in this article. With the BCGS-FFT-DDM, the objects or different parts of an object are first treated separately in several subdomains, each of which satisfies the 3-D shift-invariance and correlation properties; the couplings among the different objects/parts are then taken into account, where the coupling matrices can be built to satisfy the 2-D shift-invariance property if the objects/subdomains have the same mesh size on the $xy$ plane. Hence, 3-D FFT and 2-D FFT can respectively be applied to accelerate the self- and mutual-coupling matrix-vector multiplications. By doing so, the impedance matrix is explicitly formed as one including both the self- and mutual-coupling parts, and the solver converges well for problems with considerable conductivity contrasts. The computational complexity in memory and CPU time for self-coupling matrix-vector multiplication are $O(N_{{z}}^{q} N_{x} N_{y})$ and $O(N_{{z}} N_{x} N_{y} \log (N_{{z}} N_{x} N_{y}))$ respectively, and for mutual-coupling matrix-vector multiplication are $O(N_{{z}}^{p} N_{{z}}^{q} N_{x} N_{y})$ and $O(N_{{z}}^{p} N_{{z}}^{q} N_{x} N_{y} \log (N_{x} N_{y}))$ , respectively, for the proposed method, where $N_{x}$ and $N_{y}$ are the cell numbers of all the subdomains in the $x$ - and $y$ -directions, and $N_{{z}}^{p}$ and $N_{{z}}^{q}$ the cell numbers of different subdomains in the ${z}$ -direction. Several results of different subsurface sensing scenarios are presented to show the capabilities of this method. [ABSTRACT FROM AUTHOR]

Hydraulic fracturing is critical for unconventional oil and gas production. Recently, galvanic methods, including borehole-to-surface and surface-to-borehole electromagnetic methods, have been applied in hydraulic fracture evaluation. However, conventional computational electromagnetic methods require huge computational cost to simulate the common galvanic fracture model, which consists of a long cased borehole, 3-D thin fractures, and an underground stratified medium. More seriously, modeling both vertical and horizontal energized cased boreholes in layered media is exceptionally difficult. In this article, we present a hybrid numerical mode matching method with stabilized bi-conjugate gradient fast Fourier transform method (NMM-BCGS-FFT) as the forward solver to efficiently model 3-D fractures in the galvanic survey. Both vertical and horizontal energized cased boreholes are modeled in the forward solver. The numerical results show the accuracy and efficiency of the hybrid forward solver. The realistic field galvanic detection model analysis will show the practicability of the forward solver. [ABSTRACT FROM AUTHOR]

This article presents an efficient method for calculating the scattered electromagnetic (EM) fields from anisotropic objects embedded in general anisotropic multilayers, both with arbitrary anisotropy. The electric field integral equation (EFIE) is derived based on the electric dyadic Green’s functions for general anisotropic multilayers. The volume integral equation (VIE) involves a convolution form between the dyadic Green’s function and equivalent sources. To accelerate the integral computation, the fast Fourier transform technique (FFT) is adopted which reduces the computational cost from $O(N^{2})$ to $O (N \log N)$. The biconjugate gradient stabilized method (BiCGSTAB) is then applied to solve the EFIE efficiently to calculate the total electric fields in the computational domain. In order to validate the algorithm, several examples are modeled for anisotropic objects embedded in both isotropic and anisotropic multilayers. The modeled scattered fields are compared with the finite-element method (FEM) results. The good agreement between the two results shows the algorithm works properly. [ABSTRACT FROM AUTHOR]