Over the last 40 years several resistivity relaxation models have been developed, which provided quantitative characterization of the electric charging phenomena, including the empirical Cole-Cole model [22] - [27] , electrochemical model [2] [28] , the GEMTIP model [29] - [31] , based on generalized effective-medium theory of induced polarization.

One of the most widely used models is the Cole-Cole resistivity relaxation model introduced in the pioneering work of Pelton et al. [23] . According to this model, the complex resistivity can be presented as following:

where is the DC resistivity (ohm-m), ω is the frequency (rad/sec), τ is the time constant, C is the relaxation parameter, and is the intrinsic chargeability [32] .

We should note that, the Cole-Cole model uses empirical parameters, while the GEMTIP model uses the effective medium theory to describe the complex resistivity of heterogeneous rocks. At the same time, it was shown by Zhdanov [29] - [31] that, in the case of spherical inclusions, the GEMTIP model reduces exactly to the Cole-Cole model. The GEMTIP resistivity model incorporates the physical and electrical characteristics of rocks at the porous/grain scale and translates them into an analytic expression for the effective complex resistivity. These characteristics include grain size, porous space shape, fluid and host rock conductivities, porosity, anisotropy, polaizability, etc. [29] - [31] . Note that, in a simple case of spherical grains in the homogeneous host medium, GEMTIP model reduces to the Cole-Cole model [29] .

In the framework of the GEMTIP model, we represent a complex heterogeneous rock formation as a composite model formed by a homogeneous host medium of a volume V with a complex conductivity tensor (where r is an observation point) filled with grains of arbitrary shape and conductivity. In a general case, the rock is composed of a set of N different types of grains, the l^th grain type having complex tensor conductivity. The grains of the l^th type have a volume fraction f_l in the medium and a particular shape and orientation. Following [29] - [31] , we can write the following expression for the effective conductivity of the polarized inhomogeneous medium:

where is an effective-medium conductivity tensor; is an anomalous conductivity tensor; is the polarized anomalous conductivity; is a surface polarizability tensor; is a volume depolarization tensor; and index l corresponds to the grain of the l^th type. The last formula provides a general solution of the effective conductivity problem for an arbitrary multiphase composite polarized medium. This formula allows us to find the effective conductivity for inclusions with arbitrary shape and electrical properties. That is why the new composite geoelectrical model of the IP effect may be used to construct the effective conductivity for realistic rock formations typical for mineralization zones and/or HC reservoirs.

For this study, we have developed the three-phase ellipsoidal GEMTIP model for a medium with randomly oriented ellipsoidal inclusions. The expression for GEMTIP model in this case takes the following form:

where

The [Ohm-m] is the resulting effective resistivity, [Ohm-m] is the matrix resistivity of the rock being modeled, is a volume fraction of the l-th grain, [Hz] is an angular frequency, [second] is a time constant, is a relaxation parameter, [m] is an average value of the equatorial and polar radii of the ellipsoidal grains. The coefficients and are the structural parameters defined by geometrical characteristics of the ellipsoidal inclusions [29] [33] , and they are functions of ellipticity e_l. If all the grains are oriented in one specific direction, the effective conductivity of this medium will become anisotropic. Thus, the effective conductivity may be a tensor in spite of the fact that the background medium and all the grains are electrically isotropic.

The terms “two-phase” and “three-phase” model are related to structural model, which is characterized by one type of inclusions in the host medium or two types of inclusions, respectively. In the first case, for example, we may have pyrite in the host rock, while in the second case we may have the pyrite grains and pores in the host reservoir rock. The classic Cole-Cole model corresponds to a two-phase system with the host rock and one type of grains (e.g. pyrite). Equation (3) describes a three-phase model, which represents the host rock and two types of inclusions—grains of pyrite and pores filled with HC. Note that,a three-phase Cole-Cole model [26] was proposed as well.

GEMTIP model, Equation (3), can be formed by more than three phases as well. The exact choice of thenumber of phases depends on the structure and composition of the rock sample. Each phase should produce the IP effect, observable in the resistivity relaxation curve. It was demonstrated [29] that a two phase GEMTIP model can be reduced to the two-phase empiric Cole-Cole model, if the inclusions have a spherical shape. Following the same derivations [29] [30] , it can be shown that three phase GEMTIP model (Equation (3)) can be reduced to three phase Cole-Cole model for spherical inclusions. We should note that, the GEMTIP theory and model are based on a direct solution of Maxwell’s equations for EM field propagation in the complex model of reservoir rocks, which accounts for the relaxation of the resistivity with the frequency caused by the IP effect. In a general case, GEMTIP model can be simplified in order to reduce the number of unknown parameters, shown in Equation (1). For example three-phase GEMTIP model of Equation (3) is characterized by seven parameters only. The geoelectrical parameters of the GEMTIP model are determined by the intrinsic petrophysical and geometrical characteristics of the composite medium. Therefore, effective resistivity can serve as a basis for determining the intrinsic characteristics of the reservoir rock (the porosity, hydrocarbon saturation, etc.).

We introduce a vector, m, of the unknown model parameters:, and a vector, d, of the observed data:. Thus, we have the following GEMTIP inverse problem:

where A_IP is a forward modeling operator described by the corresponding analytical equations of GEMTIP model, Equation (3).

We can solve the inverse problem described by Equation (4) by using the regularized conjugate gradient method as follows [34] :

where is the residual, F is the Fréchet derivative matrix of operator A_IP at iteration n, which can be found analytically by calculating the first variation of A_IP, and is some a priori estimate of the GEMTIP model parameters.

The viability of the GEMTIP conductivity model was tested with multifrequency EM measurements acquired for shale-oil, laminated shale gas, and shale gas samples. Shale-oil, laminated shale gas, and shale gas samples (#8, #33, and #45) were provided by TerraTek. The shale samples and their thin sections are shown in Figure 1. All of the samples are rich in clay minerals, and they also contain disseminated pyrite. The samples were cut along less fragile direction, which were along the core axis for sample #8 and orthogonal to the core axis for samples 45 and 33. The X-ray diffraction data for whole rock, clay mineralogy by weight (%), and their core analysis were provided by TerraTek as well.

The shale samples were examined using Quantitative Evaluation of Minerals by Scanning Electron Microscopy (QEMSCan) for structural analysis and phase evaluation [35] . The color-coded maps of mineral composition were created as well as a quantitative measurement of mineral abundance and inclusion size was conducted using QEMSCan 4300. Energy-dispersive X-ray spectral analysis (EDX) involves interpretation of secondary X-ray spectra to determine elemental composition and, ultimately, mineralogy. The rock samples were prepared as standard thin polished sections, then carbon coated and submitted for the QEMSCan measurement and analysis. The right panels in Figure 1 show samples of the thin sections, prepared for the QEMSCan measurement.

We have conducted complex resistivity measurements for each sample at 27 frequencies over a range from 0.005 Hz to 1000 Hz using the TechnoImaging’s experimental lab. The details of CR measurement system were reported [16] . The CR system is operating in frequency domain in order to avoid errors related to the conversion from time to frequency domain. The statistical standard error (SE) for the CR measurement was defined by measuring the artificial rock samples [32] and the SE does not exceed 1.2% for any frequency. The phase error at the 5 kHz was within 1 mrad for RC circuit, which had impedance of about 10 kΩ [36] .

Figures 2-4 present the results of QEMSCan analysis for shale samples. Note that, according to the QEMSCan results, sample #8 contains 6.64% of the disseminated pyrite, sample #33 has 1.47% of the pyrite, and sample #45contains 3.53% of the pyrite, respectively.

Figures 5-7 show imaginary parts of the complex resistivity spectra measured for samples # 8, # 33, and # 45, respectively. We have inverted the observed complex resistivity data for the parameters of the GEMTIP models using two-phase and three-phase models. Both two- and three-phase models produce good misfits, which were 5% for two-phase and 3.2% for three-phase models, respectively, for sample #8; 5% and 3.5% for sample #33; and 7% and 6.2% for sample #45, respectively.

Figures 8-10 show the maps of the misfit functional for three-phase models for samples #8, #33, and #45, respectively, plotted as a function of the relaxation parameter (C) and the time constant (τ). The shaded isolines in this plot signify the direction of decreasing misfit. The paths of the conjugate gradient inversions are shown by the red lines. The final models are represented by red triangles. One can see that, for all three rock samples, the inversion converges rapidly to the specific parameters of the GEMTIP models. The corresponding GEMTIP parameters, produced by the inversion, are shown in the tables provided below.

Table 1 summarizes the results of GEMTIP inversion for rock sample #8. The recovered values of the volume fraction of inclusions with different electrical properties for both the two-phase and three-phase models are about 10%, which corresponds approximately to the combined volume fraction of pyrite (6.64%), determined by QEMSCan analysis, and gas-filled porosity (p_gas = 6.34%) determined by the core analysis.

Table 2 presents similar results for rock sample # 33. The recovered values of the volume fraction of inclusions with different electrical properties for both the two-phase and three-phase models are about 13%, which corresponds approximately to the combined volume fraction of pyrite (2.4%), determined by QEMSCan analysis, and gas-filled porosity (p_gas = 8.11%) determined by the core analysis.

Table 3 presents similar results for rock sample #45. The recovered values of the volume fraction of inclusions with different electrical properties for both the two-phase and three-phase models are about 13%, which corresponds approximately to the combined volume fraction of pyrite (4.4%), determined by QEMSCan analysis, and gas-filled porosity (p_gas = 1%) determined by the core analysis.

It is important to note that, the three-phase inversion makes it possible to estimate separately the volume fraction of one phase in the GEMTIP model representing the polarization caused by disseminated pyrite, and that of another phase representing induced polarization caused by the presence of HC in the porous space of the sample. For example in sample #8 the dissiminated pyrite show a volume fraction 6.35% and 4% corresponds to the HC in the porous space. The time constant of pyrite is larger than the time constant of HC, because the capacitance of the mineral carrying the charge is much stronger than that of the HC.

We should note that, the p_gas value in turn is directly associated with TOC in shale gas deposits. It was demonstrated [37] that in core shale samples the ratio between TOC and p_gas is in 0.5 - 1 range. It was also reported close to 0.5 ratio between TOC and p_gas in shale gas deposits [38] . This ratio was equal to 0.5 for shale samples used in our study, according to TerraTek core analysis. The same analysis between the internal structure of the

samples and their complex resistivity spectra was conducted for the shale-gas and laminated shale-gas samples. The results of the study for all three samples are summarized in Table 4.

The result of QEMSCan analysis have demonstrated that the shale rock samples have very complex mineral composition. For example, Figure 2 and Figure 4 show that, the percentage of the clay minerals (kaolinite and chloride) reaches 4.5% for sample #8 and 1.43% for sample #45. It is known that the clay minerals may cause a non-linearity in CR spectra, especially at higher frequencies (or lower time constants). The question may arise if the IP effect in the shale samples could be caused by these clay minerals. In addressing this question we should note that, according to published data, a significant IP effect in clay-containing samples has been observed at

high frequency only (>1 kHz) [39] - [41] . The low frequency IP effect in clays was not observed or it was caused by other factors than the presence of the pure clay minerals [6] [10] [19] [42] -[49] . In addition, in our study the kaolinite and chloride minerals appear in two samples only (#8, Figure 2, and #45, Figure 4). Sample #33 (Figure 3) does not contain any of these clay minerals, however we observe a strong frequency variations of the resistivity in this sample and GEMTIP modeling recovers rather accurately the shale porosity for this sample. Thus, the results of GEMTIP modeling of sample #33, which does not contain any clay minerals, confirm that the IP effect in our experiments is attributed to pyrite and HC fractions.