A physical interpretation of Cole-Cole equations and their ambiguous time constants for Induced Polarization models

Summary The Cole-Cole (CC) equation was introduced as an empirical formula that in many cases matched laboratory measurements of dispersive frequency-dependent dielectric, conductive or resistive physical properties. The CC formula has four parameters, a high and low frequency limit, one of which is often normalised into a polarizability η or a chargeability m, a time constant τ and a frequency dependence c. The chargeability can be directly related to the volume fraction of polarizable material in a uniform conductive background. In an electrically resistive host, the chargeability is given by the fraction of pores blocked by sulphides or other electronically conductive polarizers in the Pelton conceptual model. Fundamentally, an electrochemical model predicts that uniformly sized, non-reactive polarizers would produce exponential or Debye decays during the return to equilibrium after excitation, while uniformly-sized, chemically-reactive materials such as sulphides should exhibit Warburg decays. Numerically, the frequency dependence c can be matched and predicted from the standard deviation of a log-normal size distribution of polarizable material. Using a simple circuit analogy, excitation by a Voltage or a Current source can produce very different decay time-constants. This analogy and mathematical analysis predict that the time constants from fitting a CC conductivity model (τ _σ) can be very different from a CC resistivity model (τρ). A published laboratory sulphide measurement example with τ _ρ = 10 days, m = 0.96 and c = 0.2 corresponds to τ _σ = 88 ms, or 7 orders of magnitude shorter in time. Further, the physical circuit analogy confirms the electrochemical model that while the different time constants are mathematically independent of m and c, τ _σ is an intrinsic time-constant fundamentally related to the grain size of polarizable material and as such a far better parameter than τρ to use for mineral discrimination studies. Published tabulations of fitted parameters need to specify c as well as the chargeability and CC time-constant to allow for transformation between conductivity and resistivity time-constants. The maximum phase time constant τ _φ = √ τ _σ τ _ρ is a fitting rather than experimentally measurable parameter, that is still dependent on m and c, but less dependent than τ _ρ as previously discussed in the literature.