Modeling of the dielectric logging tool at high frequencies: theory

The high-frequency dielectric logging tool is analyzed, using first-principle electromagnetic field analysis, to determine characteristics such as tool behavior during measurement in a well logging environment. The theoretical model, a composite boundary value problem, consists of a source backed by an infinite ground plane (the measurement tool) in front of a two-dimensional inhomogeneity (geologic formation) which is divided into regions. The solution in each region is treated analytically in two dimensions, and numerically in one dimension. The one-dimensional problem is solved using the finite-element method, resulting in a conventional eigenvalue problem. This allows the eigenmodes of each region to be found systematically. Once the eigenmodes are found in each region, the solution for the two-dimensional inhomogeneity is obtained by matching boundary conditions at a discontinuity as in the method of mode matching. This gives the reflection and transmission operators characterizing each discontinuity