Foundations of tensor induction well-logging

Abstract

One of the most challenging problems in the field of electromagnetic well logging is the development of interpretation methods for the characterization of conductivity anisotropy in an earth formation. We examine the response of a triaxial electromagnetic induction well-logging instrument in an unbounded, homogeneous, transversely isotropic conductive medium. This instrument detects three components of magnetic field due to each of three transmitters for a total of nine signals. These can be mathematically organized as a tensor array which we call the magnetic induction tensor. The magnetic induction tensor components provide a general description of the electromagnetic field in a transversely isotropic medium. By theoretically analyzing the triaxial induction instrument for its response to the magnetic field components induced in the conductive medium, we derive low frequency approximations for the quadrature components of our induction tensor. Based on this analysis, we find that by measuring the quadrature components of the induction tensor in a deviated borehole, the conductivity anisotropy of the media can be resolved from the instrument response. This information includes not only the vertical and horizontal conductivities, but also the orientation of the logging instrument axis with respect to the tensor principal axes. We introduce the formulas for the apparent horizontal and vertical conductivities σha, σva, the apparent anisotropy coefficient λa, and the apparent relative deviation angle αa. These can be used as the basis for a tensor logging instrument response interpretation in unbounded, homogeneous, anisotropic media. The theory is illustrated by numerical examples of induction tensor calculations.