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The induced polarization response of an environment presenting cylindrical layers was obtained. The fractal model for complex resistivity was employed as an intrinsic property of the polarizable layers. The influence of the model fractal parameters on the electromagnetic response was investigated. The results demonstrated that the fractal parameters dominate the apparent resistivity phase response; measurements of the induced polarization data allow for the determination of the fractal properties of the environment without noticeable electromagnetic coupling effects at frequencies below 10
^{4} Hz.

The induced polarization effect has an electrochemical origin and is usually associated with geological and biological environments [

In geophysics, the induced polarization method uses fact that the constituent parameters of rocks (conductivity and permittivity) are frequency-dependent to carry out prospecting activities. This method was originally applied to the prospecting of disseminated ores, and has gradually evolved over the years, currently being used in mineral discrimination [

The quantitative interpretation of field induced polarization data is a difficult task due to the fractal nature of geological environments and the inductive coupling caused by electromagnetic interactions between the environment and the electrode arrays used for current injection and potential measurements. The interpretation of these kinds of data requires a physical model to explain the behavior of a polarizable environment in an ample frequency range.

Several relaxation models have been proposed to describe the electrical polarization of rocks, in the works of Debye [^{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. The most widely used model is the Cole-Cole model, which does not, however, consider the fractal nature of the environment.

Rocha [

Rocha [

The fact that the fractal exponent is independent from the electrical resistivity of the percolating solutions avoids any influence of the invasion zone in the electric profiling of wells. Thus, it is interesting to investigate the response of a polarizable medium in well environments by applying the fractal model for complex resistivity.

Farias et al. [

The main aim of electrical well profiling is to estimate the electrical resistivity of the geological formation where the well is inserted. However, the response of the resistivity profiling is influenced by the resistivity of the formation itself, as well as by the invaded zone, which is generated during the drilling process. Therefore, the effects of this invaded area should be avoided.

In the present study, the fractal model for complex resistivity [

Representing the time dependence of the electric field as e^{−iωt}, the expression proposed by [

where _{r} is the parameter that relates the resistivity of the conductive grains blocking the pores of the geologic environment to the DC resistivity of the rock matrix;_{o} is the relaxation time constant associated with the material as a whole; τ_{f} is the time of fractal relaxation, related to the time involved in the charge and energy transfer in the rough interfaces; and η is the parameter directly related to the fractal geometry of the environment, determined by the type and distribution of the mineral that causes the polarization at low frequencies.

Some typical values of the fractal model parameters for complex resistivity cited by [

Normally a four-electrode configuration is used to measure the complex resistivity of a geological environment. An electric current is introduced into the environment via an electrode pair (A and B) and the voltage is measured by the other electrode pair (M and N).

To calculate the potential measured by the receiver electrode pair, the electromagnetic problem for a four-electrode configuration must be solved. From Maxwell equations and assuming a time dependence of the

where

and (3) can be denoted as:

When applying the divergence operator to (2) and (5), the following equations are obtained:

where the left side of (7) is a result of the charge accumulation caused by the injected current. When observing (6), the Maxwell equations can be displayed in terms of a vector potential A and scalar potential

observing (4), (5), (8) and (9), and considering the condition

where

with

where u is a Heaviside function, I is the current intensity, z_{1} and z_{2} are the positions of the current electrodes (A e B) and e_{z} is the unit vector in the z direction.

Considering (8), (9) and (10) and the fact that the vector potential is of the (0,0,A_{z}) form, the following equations are obtained:

With

is valid for

is valid for

for

and

thus, by using (14) and (15) the boundary conditions to solve (16), (17) and (18) are obtained.

In order to solve Equations (16), (17) and (18) subject to the boundary Equations (19), (20), (21) and (22), it is convenient to introduce the Green

subject to the boundary conditions below:

The solutions to Equations (23), (24) and (25) are given by:

Since

Using the identity (15) [

in Equation (31), the following equation is obtained:

valid for

valid for

in

The vector potential function

Imposing boundary conditions (26)-(29) in Equations (32), (33) and (34), functions B, C, D and E are determined. In this way, the following system of equation is obtained:

to calculate the induced potential difference measured by electrodes M and N (

When combining Equations (32) and (35),

With

with

thus,

The potential

where

A similar result was found by [

The induced polarization responses of the two geological two situations were obtained by applying Equation (38): 1) the environment presenting two cylindrical layers (the well and the formation); 2) the environment presenting three cylindrical layers (the well, the invaded zone and the formation). The resistivity of the mud for the two geometries was of 1 Ω・m when disregarding the polarization effect. The distances between the electrodes, in meters, were of 0.41, 6.1, 20.9 and 26.59 for AM, NA, BN and BM, respectively. The default value for the well radius was of 10 cm .

In order to analyze the influence of parameters η, m, δ_{r}, τ and τ_{f} of the fractal model, the simulations were carried out for three different values for each of these parameters, and when variations in a certain parameter occurred the others assumed the typical values described above.

_{r}; 10^{−9} s, 10^{−6} s and 10^{−3} s 3 for the time constant τ and 10^{−4} s, 10^{−3} s and 10^{−2} for the fractal time constant τ_{f}.

It is observed from _{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 [

data of induced polarization in the frequency domain in the well, the transport properties of the geological environment.

Three thicknesses of the invaded zone were considered when analyzing the induced polarization response in an environment with three cylindrical layers (mud, invaded zone and formation): one, two and five times the radius of the well. The DC resistivity of the invaded area was presumed equal to 10 Ω・m. Figures 3-7 shows the response of the induced polarization when varying η, m, δ_{r}, τ and τ_{f}, respectively:

The amplitude response of the apparent complex resistivity was affected by the variation of the invaded zone. However, the phase angle response was only slightly affected. This is similar to the results observed by [

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. Thus, the influence of the invaded zone is attenuated in the phase response.

The induced polarization response of a cylindrical stratified environment was obtained and the fractal model for complex resistivity was applied as an intrinsic electrical property of a polarizable environment presenting cylindrical layers. The influence of the model parameters on the induced polarization response was

investigated. The results demonstrate that, as in the case of an environment presenting horizontal layers, the parameters of the fractal model dominate the response phase of the complex apparent resistivity of the environment at low frequencies, with particular emphasis on the fractal exponent parameter^{4} Hz.

Farias, V.J.C., da Rocha, B.R.P. and Mores, A.N. (2017) The Influence of the Model Fractal Parameters on the Electromagnetic Response in Environment with Cylindrical Layers. International Journal of Geosciences, 8, 349-363. https://doi.org/10.4236/ijg.2017.83018