A New Algorithm for Borehole Electromagnetic Measurements Employing Lambert W Function

Abstract

The Lambert W function is defined to be the multivalued inverse of the function w → we ^w. It has been widely used in many fields. In this paper, we developed an accurate algorithm for calculating apparent formation conductivity and dielectric constant from electromagnetic logs by making use of Lambert W function. Transforming the complicated nonlinear system into a linear system of much simpler format, this new algorithm can achieve a much higher accuracy of data processing and provide physically meaningful results over the full frequency band of electromagnetic logs. In the frequency band of induction logs (<20 MHz) this algorithm eliminates the complicated procedure for correcting for the skin effects. It also removes the conventional limitation of ωε ≪ σ for induction logs, and therefore, makes accuracy of induction log measurements no longer dependent on the true formation conductivity. In the frequency band of 20–100 MHz, this method can obtain accurate formation conductivities and dielectric constant simultaneously. Theoretical analysis, numerical modeling, and field data processing cases of Oklahoma formation demonstrate the advantages of this new method over traditional methods. The algorithm for this method is easy to implement and it can be used with electronic hardware to produce accurate formation conductivity and dielectric logs directly.