In induction logging, both vertical and radial conductivity variations exist in the logged medium; elementary separation of variables cannot be applied directly to the partial differential equation describing the electromagnetic field in such a medium. A modal expansion method of solution can be designed using modes defined with respect to the vertical coordinate. This method breaks the original two-dimensional (2-D) problem into a set of independent one-dimensional (1-D) problems. The receiver signal is decomposed into a direct signal traveling within the borehole and a reflection signal bearing information about the conductivity distribution outside the borehole. The solution is obtained as a coupled-mode, multiple-reflection series accounting for repeated mode reflections at the vertical boundaries in the medium. Each term in the series represents the next higher order reflection in a sequence of multiple reflections; results show two terms are sufficient to reach accuracy acceptable for log modeling.

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