Modeling of the Dielectric Logging Tool at High Frequencies: Theory

We present a theoretical analysis of the high-frequency dielectric logging tool, using first principle electromagnetic field analysis, to determine the characteristics such as measurement in a well logging environment. The theoretical model, a composite boundary value problem, consists of the measurement tool in front of a two-dimensional inhomogeneity. The two-dimensional inhomogeneity 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 and easily. 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 us the reflection and transmission operators characterizing each discontinuity. When the reflection and transmission operators for a single discontinuity are known, the case of more than one discontinuity can be easily derived.