Introduction to the Phasor Dual Induction Tool
Authors: Barber T.D. 
Published: December 2007

Barber, T.D., Schlumberger Well Services

Summary
Automatic environmental corrections to the induction log have been an elusive goal for many years. Recent work in induction response has been combined with modern signal processing theory to derive an automatic shoulder-effect correction algorithm, but the method requires measurement of the induction quadrature or X-signals. The Phasor* dual induction tool incorporates these Phasor* dual induction tool incorporates these measurements and makes automatic shoulder corrections for induction a reality.

Introduction
The dual induction-electrode logging tool has evolved into the primary logging tool for openhole formation evaluation. The induction log, however, must be corrected by hand for environmental distortions, such as shoulder effect and borehole effect. Although the environmental distortions are fully understood and predictable, automatic correction algorithms have not been successful. Recent work on computing the response of the induction tool in realistic formation models, combined with modern signal processing theory, has allowed the development of a nonlinear deconvolution technique that corrects the induction log for shoulder effect over the full range of formation conductivities. This algorithm, however, requires additional measurements that are not made with the present commercial tools. These measurements are the induction quadrature signals. or X-signals. The introduction of digital telemetry to well logging has opened up new data channels, and the Phasor dual induction tool has incorporated new measurements that use these new channels. The most important new measurements are the X-signals on both deep induction (ID) and medium induction (IM) arrays, which are required by the new shoulder-effect algorithm. In addition, quality control of the log is improved by continuous calibration along with independent measurements of the calibration signals.

Shoulder Effect and Deconvolution
Shoulder effect is the response of an induction tool to distant conductive beds when in a relatively nonconductive bed. This is illustrated in Fig. 1, which is a computed simulation of an Oklahoma field log using Anderson's method. In the resistive zones, the ID raw log reads too low, even in the thicker beds. This fact implies that, for the ID array, a "thick bed" is one much thicker than any shown on this log. The traditional processing, labeled in Fig. I as "panel," squares up the corners of the raw log and leaves the center-bed readings unchanged. Development of deconvolution filters has been simplified by modern signal processing theory. A well-known technique is to tailor the response in the frequency domain after Fourier-transforming the spatial response function. The deconvolution filter is computed with the Remez algorithm. Such a filter was derived for the ID array using its response in low-conductivity formations, with emphasis on low spatial frequency response to correct the shoulder effect of the array. The use of this filter is illustrated in Fig. 1 by the curve labeled "deconvolved. "The log reading in the high-resistivity beds is much closer to the true resistivity than either the raw or panel logs. panel logs. During the 1950's, a lot of work was done on the induction deconvolution problem, with Doll and others making significant contributions. The commercial result of this work was the three-point filter shown as the panel log in Fig. 1. Why is a more sophisticated filter not used for induction? The answer to this is in the nature of the induction response itself. The induction signal does not change linearly with formation conductivity, nor is the response function that maps the formation conductivity into the measured signal a constant function. In fact, Gianzero and Anderson demonstrate that the response function is different at every point in the log. This variation of the response with formation conductivity is referred to as "skin effect" or "propagation effect."Because the induction response is not constant, at higher conductivities the deconvolution filter that worked well at low conductivities no longer suffices. The induction response function has changed at higher conductivities from the one for which the filter was computed. Fig. 2 shows a computed log using the same resistivity contrasts and bed thickness as the formation of Fig. 1, but with all magnitudes shifted down one decade in resistivity (or up one decade in conductivity). Clearly, the deconvolution filter no longer works at these resistivities.

Induction Response
The response of an induction tool in a layered formation can be computed by solving Maxwell's equations for the fields from an induction transmitter coil in the given formation geometry. The usual methods of solution lead to an equation involving a volume integral over all regions of space in which induced currents are flowing. Gianzero and Anderson show that the resulting integrand, suitably normalized, is in the form of a mapping function. They write the kernel of the integration in the form g(z, a) and call it the vertical response function.
JPT
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