Frequency dispersion of rock properties in equations of electromagnetics

The frequency/time dispersion of the electrical properties of rock is intensively investigated in geo-electromagnetics. The frequency dispersion of conductivity is known as induced polarization (IP). Strong IP effects in transient geo-electromagnetics, such as non-monotonous transient response and sign reversals in combined transmitter-receiver loops, have been reported by many authors. The aim of this paper is to describe the electromagnetic field in dispersive/polarizable media in the most general form. The validity of replacing constant parameters with frequency/time variable parameters is also examined. It is emphasized, first of all, that the system of Maxwell's equations written in their canonical form is a universal mean to describe the electromagnetic (EM) field both in non-dispersive and dispersive media, provided that material correlations are presented by Duhamel integrals. It is shown that in the time domain, additional conduction and displacement currents appear due to changes of conductivity and permittivity parameters over time. Equations of the second order are found to be different from Telegrapher's or heat-flow equations in the time domain. If parameters change very slowly with time, the format of equations remains the same, but they still include the summary admittivity consisting of normal conductivity and the time derivative of permittivity, which means that approximate methods of changing constant parameters with time variable parameters are, in the time domain, principally incorrect. Equation of the second order in the frequency domain is the same Helmholtz equation, and the method of substituting constant parameters with frequency variable parameters is correct. It is shown, however, that a separate analysis of conductivity or permittivity dispersions is impossible. The EM field defined by complex admittivity, which includes both frequency dependent conductivity and frequency dependent permittivity, must be analyzed. One practical example of dispersion of a simple physical nature is used to illustrate the above approach. Similar to the system of Maxwell's equations, the suggested theory of the EM field in dispersive/polarizable rocks is phenomenological theory. The nature of EM rock properties, such as conductivity or permittivity and magnetic permeability, as well as the nature of their dispersion/polarization, lies outside the scope of the study, which determines both its limitations (no information about the nature of different IP types) and advantages (the possibility of describing the diverse IP types with the help of a small number of parameters).