An effective medium derivation of the Cole-Cole relation for electric conductivity

Abstract

We provided a derivation of the Cole-Cole relation for the conductivity of fluid-saturated, mineralized rocks. This was done via an upscaling procedure that assumed that each element in the system was itself described by the Cole-Cole relation. The purpose of this was to give the parameters of the Cole-Cole relation, most notably the exponent and time constant, an interpretation in terms of the variability of the underlying pore-scale variables. In this treatment, the exponent known as frequency dependence was fixed to [Formula: see text] on the pore level. However, many field observations gave [Formula: see text], and we indeed found this range of [Formula: see text]-values when we calculated the conductivity of a network of elements. The conductivity of this network, which represented the upscaled system, was calculated via an effective medium theory. In this way, we found the upscaled system to be described by the Cole-Cole relation with [Formula: see text] over three decades of frequencies. The underlying condition for this to happen was that the individual bonds have a sufficiently broad distribution of time constants. For very small, or very large, frequencies, there was a departure from Cole-Cole behavior. However, by comparing existing network simulations and the result of an effective medium theory, we found excellent agreement between the two for all frequencies ranging over eight decades.