Mixing laws and causality in high frequency induction log applications

Abstract

High frequency electromagnetic technologies for subsurface formation evaluation provide high spatial resolution and new opportunities for petrophysical interpretation of data. Dispersion of rock properties and up-scaling from pore to reservoir scale (homogenization) represent the two most challenging problems. In electrodynamics of porous media, various mixing and dispersion laws are used to homogenize rock properties and describe their frequency behavior. Mixing laws and dispersion have a close link to the fundamental physical principle of causality and therefore cannot be introduced arbitrarily. For any mixing/dispersion law, we need to prove that causality holds. For testing causality, we use Titchmarsh’s theorem and, particularly, one of its modifications—Kramers–Kronig relations. Causality is discussed for Debye, Cole–Cole, Havriliak–Negami, and CRIM models. Dispersion is closely related to wave propagation. Evaluation of phase and group velocities shed new light on the physics of phase and amplitude measurements in lossy media. We evaluated various definitions of both velocities and their dependence on spatial spectra or, in other words, on the arrangement of transmitting and receiving elements. To illustrate theoretical findings, we use dielectric logging as an exemplary technology. Usually, in modern dielectric tools, amplitude and phase data are acquired, for various frequencies and sensor positions. The measured phase is discontinuous at high frequencies and requires detection of discontinuity as well as unwrapping. Remarkably, one can determine formation attenuation and loss angle based on multifrequency/multisensor amplitude data and transform them into dielectric permittivity, resistivity, and true continuous phase. Transformations of exemplary tool data used in this paper are suitable for a conceptual study and are specific for a uniform formation. We intentionally do not address the accuracy of measurements and propagation of errors in the inversion process, since they are tool- and processing-specific. Different tools require joint analysis of all available data and special noise reduction techniques associated with the structure of the acquisition system.