Fractal Geometry, porosity and complex resistivity: from rough pore interfaces to hand specimens

Abstract

We propose a new model to interpret the electrical behavior of rocks containing metallic or clay particles. This new model encompasses some of the other commonly used models as special cases.This model is a generalization of two models, one developed by Dias (Journal of Geophysical Research 77, 4945-4956, 1972) and another by Pelton et at. (Geophysics 43, 589-609, 1978). Its circuit analog includes an impedance K(iw,) -'~ which simulates the effects of the fractal rough pore interfaces between the conductive grains (metallic or clay minerals which are blocking the pore paths) and the electrolyte. This generalized Warburg impedance is in series with the resistance of the blocking grains and both are shunted by the double layer capacitance. This combination is in series with the resistance of the electrolyte in the blocked pore passages. The unblocked pore paths are represented by a resistance which corresponds to the normal DC resistivity of the rock. The parallel combination of this resistance with the 'bulk' sample capacitance is finally connected in parallel to the rest of the above-mentioned circuit. The model was tested over a wide range of frequencies against experimental data obtained for amplitude and phase of resistivity or conductivity as well as for the complex dielectric constant. The samples studied are those of sedimentary, metamorphic and igneous rocks.

The Influence of the Model Fractal Parameters on the Electromagnetic Response in Environment with Cylindrical Layers

“…However, the phase angle response was only slightly affected. This is similar to the results observed by [2] [3], which demon- strates the fractal nature of the complex resistivity, since the scale variation in the measurements did not change the phase angle response of the cylindrical environment. In addition, the fractal exponent parameter η, which dominates the response phase, is not dependent on the electrical properties of the fluids filling the empty spaces of the rocks present in the environment, depending only on their mineralogical composition.…”

“…As in the case of horizontal layers [11], the fractal parameters η, δ r e τ f , particularly the fractal exponent η dominates the phase angle response of the apparent complex resistivity, mainly at low frequency. According to [2] [11], this feature is very important because at low frequency the parameters carry information about the roughness of the pores of rocks. Thus, it becomes possible to investigate, from …”

“…Several relaxation models have been proposed to describe the electrical polarization of rocks, in the works of Debye [7], Cole-Cole [8], Davidson and Cole [9] and Dias [10], each taking into account a certain specific characteristic for a given frequency range, limited to 10 2 Hz. Relaxation models demonstrate the general behavior of the amplitude spectrum and the complex resistivity phase (conductivity) at different frequencies for different types of materials.…”

Characterization of Rock Wettability Though Dielectric Measurements

“…In short, the models we have referred to can provide a qualitative picture of the phenomena, but they are not able to give a satisfactory quantitative description. Accordingly, a more complex model should be tested [20].…”

High Pressure Electromagnetic Fractal Behaviour of Sedimentary Rocks

“…A fractal model for the complex resistivity of rocks was first introduced by Rocha and Habashy [1995], using an analogue circuit including the diffusivity ( K ) of ions in the vicinity of the electrode/electrolyte interface and a fractal parameter ( η ). Later, Rocha [1995] proposed the adoption of a fractal time to substitute for the diffusivity ( K ) of ions.…”