We introduce a new and potentially useful method for computing electromagnetic (EM) responses of arbitrary conductivity distributions in the earth. The diffusive EM field is known to have a unique integral representation in terms of a fictitious wave field that satisfies a wave equation. We show that this integral transform can be extended to include vector fields. Our algorithm takes advantage of this relationship between the wave field and the actual EM field. Specifically, numerical computation is carried out for the wave field, and the result is transformed back to the EM field in the time domain. The proposed approach has been successfully demonstrated using two‐dimensional (2‐D) models. The appropriate TE‐mode diffusion equation in the time domain for the electric field is initially transformed into a scalar wave equation in an imaginary q domain, where q is a time‐like variable. The corresponding scalar wave field is computed numerically using an explicit q‐stepping technique. Standard finite‐difference methods are used to approximate the fields, and absorbing boundary conditions are implemented. The computed wave field is then transformed back to the time domain. The result agrees fairly well with the solution computed directly in the time domain. We also present an approach for general three‐dimensional (3‐D) EM problems for future studies. In this approach, Maxwell’s equations in the time domain are first transformed into a system of coupled first‐order wave equations in the q domain. These coupled equations are slightly modified and then cast into a “symmetric” and “divergence‐free” form. We show that it is to this particular form of equations that numerical schemes developed for solving wave equations can be applied efficiently.
A new electromagnetic logging method, in which the source is a horizontal loop coaxial with a cased drill hole and the secondary axial fields are measured at depth within the casing, has been analyzed. The analysis, which is for an idealized model of an infinite pipe in a conductive whole space, has shown that the casing and formation are uncoupled at the low frequencies that would be used in field studies. The field inside the casing may be found by first finding the field in the formation and then using this field as an incident field for the pipe alone. This result permits the formation response to be recovered from the measured field in the borehole by applying a correction for the known properties of the casing. If the casing response cannot be accurately predicted, a separate logging tool employing a higher frequency transmitter could be used to determine the required casing parameters in the vicinity of the receiver. This logging technique shows excellent sensitivity to changes in formation conductivity, but it is not yet known how well horizontal stratification can be resolved. One of its most promising applications will be in monitoring, through repeated measurements, changes in formation conductivity during production or enhanced recovery operations.
We present an efficient numerical method for computing electromagnetic (EM) scattering of arbitrary three‐dimensional (3-D) local inhomogeneities buried in a uniform or two‐layered earth. In this scheme the inhomogeneity is enclosed by a volume whose conductivity is discretized by a finite‐element mesh and whose boundary is only a slight distance away from the inhomogeneity. The scheme uses two sets of independent equations. The first is a set of finite‐element equations derived from a variational integral, and the second is a mathematical expression for the fields at the boundany in terms of electric fields inside the boundary. The Green’s function is used to derive the second set of equations. An iterative algorithm has been developed to solve these two sets of equations. The solutions are the electric fields at nodes inside the finite‐element mesh. The scattered fields anywhere may then be obtained by performing volume integrations over the inhomogeneous region. The scheme is used for modeling 3-D inhomogeneities with plane‐wave and magnetic dipole sources. The results agree with earlier model analyses using the finite‐element technique.
A numerical solution for electromagnetic scattering from a two‐dimensional earth model of arbitrary conductivity distribution has been developed and compared with analog model results. A frequency‐domain variational integral is Fourier transformed in the strike direction, and a solution is obtained using the finite‐element method for each of a finite number of harmonics or wavenumbers in transform space. The solution is obtained in terms of the secondary electric fields. Principally due to the inaccuracy associated with numerical derivatives of electric fields, the secondary magnetic field is computed by integrating over the scattering currents in harmonic space and is then inverse Fourier transformed.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.