Ki Ha Lee

A new approach to modeling the electromagnetic response of conductive media

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‐diffe...

Inequality constraint in least-squares inversion of geophysical data

This paper presents a simple, generalized parameter constraint using a priori information to obtain a stable inverse of geophysical data. In the constraint the a priori information can be expressed by two limits: lower and upper bounds. This is a kind of inequality constraint, which is usually employed in linear programming. In this paper, we have derived this parameter constraint as a generalized version of positiveness constraint of parameter, which is routinely used in the inversion of electrical and EM data. However, the two bounds are not restricted to positive values. The width of two bounds reflects the reliability of ground information, which is obtained through well logging and surface geology survey. The effectiveness and convenience of this inequality constraint is demonstrated through the smoothness-constrained inversion of synthetic magnetotelluric data.

A theoretical study of surface‐to‐borehole electromagnetic logging in cased holes

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 conduct...

A numerical solution for the electromagnetic scattering by a two‐dimensional inhomogeneity

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.

Crosshole electromagnetic tomography: A new technology for oil field characterization

With the advent of crosshole seismic technology in the 1980s, a new generation of high resolution geophysical tools has become available for reservoir characterization. The chief improvement is simply that the tools are deployed in boreholes so measurements take place much closer to the region of interest.

3D interpretation of electromagnetic data using a modified extended Born approximation

We present a new method, dubbed the modified extended Born approximation (MEBA), for efficient three-dimensional (3D) simulation and inversion of geophysical frequency-domain electromagnetic (EM) data for a targeted object lodged in a layered half-space. Based on the integral equation method and modified from an extended Born approximation technique, the MEBA method calculates the total electric field in an electrical conductivity inhomogeneity without any need for solving a huge matrix equation. This is done by multiplying the background electric field by a depolarization tensor. The Fourier transform and the convolution theorem are used to dramatically increase the computational efficiency. Comparisons of MEBA-generated numerical data for tabular targets with data generated by other means are used to verify the scheme and check its range of validity. The results indicate that the MEBA technique yields better accuracy when current channeling in the conductivity anomaly dominates over the induction process. The MEBA algorithm has been incorporated into a least-squares inversion scheme which is used to interpret borehole-to-surface EM tomography field data. The survey served to monitor the subsurface conductivity change associated with the extraction of a volume of saltwater previously injected into a known aquifer.

High-frequency electromagnetic impedance method for subsurface imaging

This paper presents a high-frequency electromagnetic (EM) impedance method that extends the utility of conventional controlled-source audio-frequency magnetotelluric (CSAMT) method to the frequency range from 100 kHz to 100 MHz. In this frequency range diffusion and wave propagation must be considered together. In principle, both the electrical conductivity and the permittivity of the shallow subsurface can be imaged using impedance data gathered on the surface of the earth. The impedance approach has a distinct advantage in that coupling with the source is unnecessary, provided that the source can be positioned far enough away to yield plane waves at the receiver positions.At high frequencies the EM impedance is a function of the angle of incidence or the horizontal wavenumber, so the electrical properties cannot be readily extracted without eliminating the effect of horizontal wavenumber on the impedance. For this purpose, this paper considers two independent methods for accurately determining the horiz...

Effect of a Steel Casing On Crosshole EM Measurement

Crosshole electromagnetic (EM) methods have become important tools for detailed evaluation of the underground conductivity distribution. However, when a drill hole is cased with steel, we usually abandon the idea of applying the crosshole EM methods assuming that the casing would absorb most of the energy. In order to examine the effect of the casing on EM signals, we developed a computer program capable of calculating EM fields outside a steel casing due to a current filament source seated inside the casing. We tested this code with some numerical experiments. The frequency range considered here is from dc to IOkHz. Although the energy absorption of the EM signals due to the casing is significant, especially at higher frequencies, the ratio and the phase difference between the EM fields with and without the casing remain constant at any receiver position for a fixed frequency, provided the receiver is some distance away from the casing. This implies that we may be able to overcome the casing effect, and to recover the whole-space EM fields by correcting the observed data with the constant. away from the source borehole. The implication of this finding is very significant because it will allow us to apply cmsshole EM methods to imaging conductivity distributions even if boreholes are cased with steel pipes, provided that at least one receiving borehole that is not cased with steel pipes is available.

Electromagnetic traveltime tomography using an approximate wavefield transform

We present an imaging scheme for mapping cross‐hole electrical conductivity using nonlinear traveltime tomography. Data used are peak arrival time estimates based on an approximate wavefield transform of the synthetic frequency‐domain electromagnetic (EM) field. Direct transformation of frequency‐domain EM fields to wavefields is known to be an ill‐posed problem because the kernel of integral transform is highly damped. In this study, instead of solving such an unstable problem, we approximate the wavefield in the transformed domain via a ray series expansion. If reflected and refracted energy is weak compared to that of direct wave, picking of the peak arrival time may be reduced to estimating the coefficients of the leading term in the ray series expansion. This simplification is valid when the conductivity contrast between background medium and the target anomalous body is small. The first three terms in the expansion are identical to the closed‐form solution for the vertical magnetic field caused by a...

Crosshole EM in steel-cased boreholes

The application of crosshole EM methods through steel well-casing was investigated in theoretical, laboratory and field studies. A numerical code was developed that calculates the attenuation and phase delay of an EM dipole signal propagated through a steel well casing lodged in a homogeneous medium. The code was validated with a scale model and used for sensitivity studies of casing and formation properties. Finally, field measurements were made in an oil field undergoing waterflooding. Our most important findings are that (1) crosshole surveys are feasible using a well pair with one metallic and one non-metallic casing. (2) The casing effect seems be localized within the pipe section that includes the sensor. (3) The effects of the casing can be corrected using simple means and (4) crosshole field data that are sensitive to both formation and casing were acquired in a working environment.