This paper introduces a novel physics-informed neural networks (PINNs) to solve Maxwell’s equations with triaxial magnetic dipole sources in transverse isotropy (TI) medium, which can be applied to simulate tool response of electromagnetic (EM) wave well logging. The physical model consists of a planar-stratified formation and some cylindrical axial symmetrical anomalous bodies embedded it. The EM field is three-dimensional (3D) distribution especially for horizontal magnetic dipoles. By analyzing the characteristics of field, the azimuth angle ϕ is extracted from the initial partial differential equations (PDEs), thus the task is transferred into a twodimensional (2D) issue about input parameters ρ and z. The singularity of dipole sources is addressed well by abstracting the background field. Each field component includes both real part and imaginary part in dissipative medium. Therefore, we train two sets of PINNs to solve for them respectively, which great increase the accuracy of output results. The lost functions related to the soft boundary and hard boundary are deeply discussed, which leads to different accuracy and efficiency. Finally, we simulate the magnetic field signals for some typical well logging scenarios in vertical and deviated wells. The results are validated by comparing with mature analytical solutions and finite element method (FEM). The proposed PINNs method provides an innovate approach for forward modeling of EM wave logging for dealing with local anomalous body (2.5D).

In this work, we develop a novel analytical solution for the electromagnetic (EM) field in a 2-D cylinder-plane model that is utilized to calibrate the measured data from airhang experiments with an ultradeep azimuthal resistivity (UDAR) tool. The tool adopts a set of tilted coil antennas wound around the drill collar and is lifted horizontally above the sea water for signal calibration. The EM fields are first solved in the cylindrical and rectangular coordinate systems to match the boundary conditions of the cylindrical and plane interfaces, respectively. After the mutual transformation relationships between two types of waves, cylindrical waves and plane waves, are derived, the whole-space field is expressed exactly in the spectral domain. The voltage expression for a tilted receiving coil is further obtained via a 1-D integral. These formulations ensure the high accuracy and efficiency of forward modeling for UDAR tools with multiscale measurements and high conductivity contrast. Finally, we propose a preliminary calibration formulation based on data fitting and apply it to simulate airhang experiments to investigate the effects of sea water.

In this article, the electromagnetic (EM) field excited by the triaxial electrical/magnetic dipole sources is presented in a cylindrical multilayer transverse isotropic (TI) medium. A set of modified generalized reflection and transmission coefficients is introduced to describe the EM field propagation, which avoids the overflow problem in numerical computation. Those coefficients consist of Bessel and Hankel functions with complex argument. The divergent factors of those cylindrical functions are extracted and rearranged in the recursive formulations, which ensure that the field attenuates along its propagating direction. The centered dipole sources are in-deep discussed in our model to simplify previously published formulations and improve the computational efficiency, as there are only up to two harmonics are considered due to axial symmetry. The singularity issue due to the Bessel and Hankel functions with zero argument is also well handled, while the field point is located along the central axis. Those formulations can be widely used for forward modeling for EM well logging. Finally, the radial geometrical factors are simulated and analyzed for ultradeep azimuthal resistivity logging (UARL).

The Earth’s interior consists of multiscale structures that range from micrometer-scale mineral assemblages to 1000-km-scale heterogeneities. Mantle plumes are one such mega-scale structure that connects the core–mantle boundary with Earth’s surface. Reconstructing these structures can provide insights into mantle material and energy convection, as well as Earth’s long-term evolution. However, mantle plume has not yet been convincingly reconstructed by electromagnetic (EM) induction; even they have significantly high electrical conductivity compared with the surrounding mantle. Here, we numerically reconstruct mantle plumes by employing a deep Earth EM induction method—geomagnetic depth sounding (GDS). We build the electrical structure of mantle plumes and conduct inversion tests to investigate how different station coverage areas, station spacings, noise levels, and response period ranges influence the construction. The test results indicate that the reconstruction of a broad 10° diameter plume head near the mantle transition zone (MTZ) requires a station coverage area of at least 10$^{\circ }\,\,\times10^{\circ }$and a 2° station spacing; the station spacing can be increased to 5° for a 20$^{\circ }\,\,\times20^{\circ }$coverage area. A continuous two-year record with ~5% noise is sufficient to recover the electrical structure of the plume head. Plumes with different types and roots can be distinguished by images near the MTZ, while reconstruction of the narrow tail in the deep lower mantle seems to be difficult due to the limited resolution of GDS. A mantle plume beneath South China is discovered by GDS from the field geomagnetic data. GDS is expected to be used for reconstructing the mantle plume beneath important locations and contributing to the study of Earth’s dynamics.

A nonoverflow representation of the electromagnetic (EM) field radiated by dipole sources in the cylindrical multilayered anisotropic media is presented. The sources include both magnetic and electric triaxial dipoles. The permeability, permittivity, and conductivity of each layer are all uniaxial anisotropic. A set of normalized reflection/transmission coefficients is defined and utilized to describe the propagation of EM wave. The recursive algorithm of EM field in each layer is with respect to the ratios of the outgoing and standing waves. In the expressions, all the Hankel or Bessel functions are in the form of their ratios, which avoid the overflow problem in numerical integral. To improve the stability and accuracy of numerical calculation, we subtract the background field from the total field in spectral domain. The background field is calculated by the algebraic solution in spatial domain rather than its integral representation. The rest of the field, namely the reflection field, has a smaller convergence region compared with its total field, and is calculated using the cubic spline interpolation method. The parities of the integrand are discussed by using numerical simulations, which yield the corresponding folded expressions of the field components to accelerate the computational efficiency. This work can be widely used in the applications of geophysical exploration.

An ultradeep boundary detection method based on the characteristics of the electric field in a single well is proposed. Unlike any conventional azimuthal electromagnetic (EM) wave logging approach, the new design employs an open-loop half-circle antenna other than the tilted or transverse closed-loop coil antenna as the receiver. The voltage on the half-circle antenna varies periodically with the tool-face angle, which provides the azimuthal information of the boundary. To explore how this new measuring method works, we investigate the electric field characteristics around the receiving antenna. The voltage is calculated by the integral of the electric field. By analyzing the voltages on each segment and their serial connection, the reason is revealed for the significantly enhanced voltage on the half-circle antenna compared to the tilted/transverse closed-loop coil. It lays a new foundation of ultradeep boundary detection based on the electric field characteristic. The voltage response behaviors are further investigated with respect to the tool offset, frequency, distance to the boundary (DTB), and conductivity anomaly. The monotonous changing of the voltage is observed within a wide range of those parameters. The results show that the half-circle antenna has great potential in realizing ultradeep boundary detection with a small offset.

This article presents an efficient analytic algorithm to calculate the electromagnetic (EM) fields radiated from orthogonal magnetic and electric dipoles in the fully anisotropic planar-stratified media. The EM fields are transformed into the frequency–wavenumber domain by the double Fourier transform, which yields a set of ordinary differential equations, consisting of a system matrix, a field vector, and a source vector. The eigenvalues of the system matrix are uniquely determined, but the corresponding eigenvectors can be multiplied by an arbitrary constant or be normalized by energy. With these eigenvalues and eigenvectors, the field vectors are expressed as mode fields consisting of the up-going and down-going fields. The propagation of the mode fields in each layer is described by the generalized reflection/transmission method. To accelerate the evaluation of the inverse Fourier transform of EM field from spectral domain to spatial domain, an approximated primary field was subtracted from total field. Numerical results from a finite element method (FEM) validate the proposed method. Then, we apply this new algorithm to a variety of real applications in geophysical prospecting.

An analytic algorithm is developed to solve the electromagnetic (EM) field excited by orthogonal magnetic dipoles in the multilayered biaxial anisotropic medium. It allows forward modeling and inversion of EM well logging in the planar-stratified formation without borehole and invasion zones. First, the transverse components of the EM fields are divided into up/down-going waves in the frequency wavenumber domain. A novel concept of reflection/transmission is introduced to describe the propagation of the EM waves through the boundaries. Different from previous work, those reflection/transmission coefficients are concerning the propagation of up/down-going waves directly rather than the amplitude of the total EM field in each layer, which makes the derivation process clearer and easier to understand. Then, we adopt the cubic spline interpolation to calculate the 2-D Fourier integral that transforms the field from the spectral domain to the spatial domain, and an efficient sampling rule is also provided to achieve high accuracy and stability in numerical calculation. For avoiding the singularity when the field locations are close to the source along the vertical direction, the primary field is subtracted in the spectral domain to make the integrand decay fast. Finally, we validate our new formulas compared with another numerical method and analyze the responses of the triaxial induction tool in biaxial anisotropic formations.

In this letter, we present a set of compact and no-overflow formulations to calculate the electromagnetic (EM) fields from coaxial coil antennas in a concentric cylindrically multilayered medium. It can be applied to fast forward computation and the inverse problem for EM well logging. The derivation is performed by using the tangential component of electric field. In contrast with previous formulations, the adopted novel reflection and transmission coefficients are scalars rather than matrices, which make it easy to get an accurate and efficient Jacobian matrix for inversion problem. The basic unit of the formulations is the ratio of the cylindrical functions so that the notorious overflow problem for numerical computation can be obviated. Numerical results in comparison with those from other approaches have demonstrated the validity and stability of our new formulations for forward modeling. To show the potential of the proposed formulations, an inverse result from a simple formation model is also presented.

In this paper, we present a set of compact formulations to model the response of the electromagnetic well-logging tools in eccentric multilayered medium. Different from previous models, the coil antennas are imbedded in an insulating protection layer (IPL) rather than expose to the borehole directly, which agrees much better with some real tools, e.g., the array induction logging tool. The effects of both metal mandrel and an IPL are considered in eccentric scenarios. Our formulations are derived on the basis of the ratios of cylindrical functions, which obviate the notorious overflow issues during numerical computation. Numerical results show that the effect of IPL gradually appears with the increase of the eccentric distance.

This paper presents a novel accurate and stable current and charge integral equation (CCIE) solver for the low-frequency scattering of penetrable objects in layered medium (LM). To the best of our knowledge, this is the first time to extend CCIE into LM simulations at low frequency. In order to integrate the matrix-friendly LM Green’s functions (LMGFs) into CCIE, we have rederived them to define new quasi-vector/scalar potentials, which are able to annihilate the frequency singularity in the original LMGFs. Moreover, an effective preconditioner is adopted to improve the conditioning of impedance matrices. In comparison with other perconditioners, this method performs much better. The excellent performance of this new CCIE solver is then demonstrated by numerical experiments.

A set of compact and stable formulations is presented to calculate the response of logging tools employing tilted-coil antennas in anisotropic concentrically cylindrical multilayered formation. The presented formulations can be used to investigate the effect of mandrel, borehole, and invasion. Our formulations are different from previous ones in three aspects. First, the tool mandrel can be cylindrically multilayered, and electrical parameters of each layer are arbitrary. That is to say, the mandrel is not limited to being perfectly metallic. Second, the formation can be either anisotropic or isotropic. Finally, the propagation of electromagnetic wave in an inhomogeneous cylindrical medium is described by the generalized reflection and transmission coefficients without the numerical overflow issue. Numerical results in comparison with those from other approaches have demonstrated the validity and stability of our new formulations.

We present a set of stable and efficient formulas to compute the electromagnetic radiation of arbitrary magnetic dipoles in a cylindrically layered anisotropic medium. This set of formulas is derived on the basis of the generalized reflection and transmission coefficients. By expressing our new formulas by the ratios of Bessel or Hankel functions, rather than directly by these special functions, the notorious overflow problem for numerical computation is successfully obviated. Our formulas can be easily implemented for any number of layers, as well as for arbitrary locations and orientations of the transmitter and the receiver. Numerical experiments have demonstrated their correctness and stability. They are then utilized to investigate the influence of the eccentricity distance, the eccentricity angle, and the formation anisotropy on the response of a multicomponent induction logging tool.

In this letter, we present new insight into the response behaviors of triaxial induction tools. The symmetrical measurements by two sets of receiving coils fixed on both sides of the transmitting coils are simulated in deviated wells. Theoretical analysis and numerical simulations show that some cross-components of two symmetrical receivers satisfy reciprocity principle, while other corresponding components have the same response behavior with both the shoulder-bed effects and borehole effects. The reciprocity can be applied to data processing of existing triaxial induction tools with unilateral configuration. A rule is provided to determine a triaxial receiver as Up-receiver or Down-receiver before data processing.

This letter describes a method to extract the unknown parameters of the formation using triaxial induction data based on the whole-space model. Two single-parameter coupled functions are constructed to determine the horizontal conductivity and the relative dip. After that, the vertical conductivity is solved analytically using another single-parameter function. The method is extremely fast and can be implemented for real-time processing and interpretation to allow decision making on the well site. The results can be further refined via postprocessing while the relative dip is required to be constant within an interval.

There are two nonzero cross-components of a triaxial induction logging tool in the medium coordinate system in a transversely isotropic (TI) formation. The sum of the cross-components holds dependence on only the horizontal conductivities with little nonlinearity and small shoulder bed effect in a horizontally layered formation. Based on the generalized reflection coefficient method, simplified formulas of the sum are derived to analyze its response characteristics. Numerical results show that the apparent conductivity obtained from the sum is a square-looking log with high vertical resolution. Moreover, the readings almost equal to the true horizontal conductivity of the formation beds. These characteristics can be applied to reconstruct the horizontal conductivity and the formation boundaries with triaxial induction logging data.