An Analytic Method for Producing Multiarray Induction Logs That Are Free of Dip Effect
- T.D. Barber (Schlumberger) | G.N. Minerbo (Schlumberger)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- October 2003
- Document Type
- Journal Paper
- 342 - 350
- 2003. Society of Petroleum Engineers
- 3.3.2 Borehole Imaging and Wellbore Seismic, 1.11 Drilling Fluids and Materials, 5.6.1 Open hole/cased hole log analysis, 5.1.2 Faults and Fracture Characterisation, 1.6 Drilling Operations, 4.1.5 Processing Equipment, 1.12.2 Logging While Drilling, 4.1.2 Separation and Treating
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Dip effect on induction-tool response is a well-documented and well-understood environmental effect that produces increased shoulder effect at moderate dip and a complete scrambling of multiarray induction logs at high dip. Although techniques such as maximum-entropy inversion allow a full correction of the logs, these remain computer-center products that are somewhat time-consuming. Wellsite inversion to correct for dip effect has remained elusive, waiting on more computer power.
In the mid-1960s, Pierre Grimaldi discovered an induction measurement technique that had the remarkable property of returning the same reading in a "thin" bed that it gave in an infinite, homogeneous medium of the same conductivity. The only requirement to satisfy this condition was that the instrument sensors were all in the thin bed. The instrument configuration was such that the voltage measured was a direct solution to Maxwell's equations. Although this was a breakthrough, the instrument configuration was difficult to realize in hardware, particularly with the well-logging technology of the 1960s, and the idea was never published.
The Grimaldi configuration was independently rediscovered in the early 1980s and studied with new modeling codes. It was found that the "remarkable property" of having no shoulder effect was also approximately true at any dip angle. Again, the hardware was deemed to be impractical.
More recently, it was discovered that the array configuration of the AIT* Array Induction Imager family of tools was such that the raw array-measurements data could be processed to be the same as those from a Grimaldi configuration. This led to a real-time algorithm, and a set of Grimaldi curves is now available at the wellsite. This algorithm produces synthetic arrays equivalent to a Grimaldi coil configuration that solve the inverse problem without having to solve Maxwell's equations iteratively.
The Grimaldi configuration is not without its drawbacks - nature is not so kind. The logs have a shallower depth of investigation than those from a conventional induction array of the same spacing. For this reason, the Grimaldi logs supplement, but do not replace, the standard AIT log sets.
The response of induction logs in dipping formations, or, equivalently, in deviated wells, is documented extensively in the literature.1,2 The main effect of the resulting relative dip between formation and borehole is to introduce shoulder effect and to produce horns or spikes at bed boundaries at high relative dip. Two methods have been reported as successful in removing the dip effect - filter-based methods,3,4 which work up to moderate relative dip angles, and maximum-entropy-based methods,5,6 which work at any angle.
However, these methods require knowledge of the relative dip angle, which is often poor. Deviated wells are usually drilled in known reservoirs in which borehole-image logs are not run. Although the well deviation is known accurately, local dip can only be derived loosely from seismic images or estimated. In addition, local dip can change considerably over the reach of the well. For this reason, although the filter-based methods are fast enough, wellsite dip correction has not been practical. The maximum-entropy method produces recognizable signatures6 on the logs when the input relative dip is too high or too low, but making the necessary multiple runs at different relative dips and deciding which is appropriate is time-consuming.
Recently, a method of processing AIT logs was developed that has very little sensitivity to dip angle. It is based on an analytic solution to Maxwell's equations in a layered medium, and it can be run in real time at the wellsite. Without the need to know the relative dip, wellsite processing is practical.
As one may suspect, the insensitivity to relative dip does not come without a price. Logs processed in this manner have a much shallower depth of investigation than the standard AIT logs. For this reason, the new processing does not replace the standard AIT processing, nor does it replace the maximum-entropy processing at the computer center. However, because most deviated wells are drilled with oil-based mud (OBM), we may expect shallow invasion in this case, and the dip-invariant processing can return useful logs for petrophysical analysis.
Grimaldi's Remarkable Relation
In 1966 Pierre Grimaldi discovered - but never published - a solution to Maxwell's equations for the response of a two-coil induction array (Fig. 1 ) that had the remarkable property that
where zR and zT are the positions of the transmitter and receiver of a two-coil induction sonde, and v(zR ,zT) is the voltage on the receiver coil. The constant is independent of zR+zT and has the same value in a thick bed as in an infinite homogeneous medium. The equations leading to Eq. 1 are given in the Appendix.
Although these results were interesting, there seemed to be no way to realize a measurement that would satisfy Eq. 1, and the work was forgotten. In the early 1980s, one of the authors rediscovered the relation and developed coil configurations that would satisfy Eq. 1. One of these is shown in Fig. 2 . If V1 is the voltage on receiver R1 from transmitter T, and V2 is the voltage on receiver R2 from transmitter T, and zR1-zR2 is small, then the difference V1 -V2 is an approximation to the derivatives in Eq. 1, and V1-V2 is constant and is proportional to the conductivity of the formation. The measured conductivity in a bed with thickness greater than the zT-zR1 distance will be the same as the measured conductivity in an infinite homogeneous medium with the same conductivity. In other words, the measurement has no shoulder effect as long as the tool is within the bed.
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