It is well established that phase shift and attenuation measurements acquired by an electromagnetic propagation tool come with different depths of investigation (DOI). The attenuation measurement sees deeper into the formation than the phase shift measurement. This difference has been reported not only for the 2 MHz propagation resistivity tool, but also for the deep propagation tool that operates at 25 MHz.
Although the difference has been demonstrated with modeling, test tank experiments and logs, a complete physical explanation has been notably absent since the introduction of the MHz-frequency propagation logging in 1980s. The question is so intriguing that it has been raised repeatedly over the past decades: what drives the difference of DOI for the two measurements that are acquired with the same electromagnetic field?
In this paper, we revisit this problem with an aim of providing a physical insight to bridge the gap between theory and application. This is an extension of our recent work on the theory of apparent conductivity for propagation measurements.
We address the problem by applying high-order geometric theory for low-frequency electromagnetic problems in lossy media in conjunction with the Taylor series expansion for the voltage ratio measured by a propagation tool. In so doing, we find that in a resistive formation where the dielectric effect is small: 1) the phase shift measurement is primarily due to the first-order eddy current induced in the formation; 2) in contrast, the leading source of the attenuation measurement is the second-order eddy current.
Since the second-order eddy current is more spread out than the first-order eddy current, this explains why the DOI of attenuation resistivity is larger than that of phase shift resistivity. The difference in spatial distribution of two eddy currents is also the reason for the difference of vertical resolution between the two.
The same root cause for the difference of DOI and vertical resolution also holds when comparing R-signal and X-signal from induction resistivity logging. Other properties shared by propagation and induction resistivity logging will be discussed, such as skin effect and dielectric effect, as well as their asymptotic properties in high-resistivity formations. We conclude that propagation and induction resistivity logging are essentially similar, even though the two measurement principles may seem rather different.