A Fourier method for borehole electromagnetic problems

Abstract

The induction logging problem is solved for a two‐coil sonde in a borehole surrounded by a bedding sequence which varies arbitrarily in the vertical direction. The solution is based on parallel computation of 500–[Formula: see text] m of log data. Both the R and X components of the voltage log are modeled. The problem is solved in the spatial‐frequency domain in which a transfer function relates the anomalous borehole voltage for a two‐coil sonde to the conductivity spectrum of the formation, resulting in a natural extension of Doll’s geometric factor theory. This computationally compact formulation does not require matrices and is designed as a computer aid for log analysts. The mathematical method (with proper conditioning) can efficiently solve the inverse problem of transforming the voltage induced in the receiving coil to a formation conductivity log. This paper establishes the forward method. Numerical results are given for beds of finite thickness for both brine‐ and oil‐base muds. The Fourier method is in good agreement with both a finite‐element method and a semi‐analytic method.