An iterative Debye model for a horizontal multi-layered material is found using a circuit model. A three-dimensional Finite Difference Method (3D-FDM) is developed to extract the effective permittivity and conductivity. The results obtained from the FDM and the present method agree very well, which shows the validity of the layered Debye formulas in terms of original circuit parameters. The derived iterative Debye formulas can be used to analyze the relationship between the effective electrical spectra and the electrical parameters of each layer. In addition, it can be applied for multi-layered models with extra-thin thickness geometry, which are probable models in electronic materials. Numerical results show that when the number of layers is more than two, the relaxation factor will not be a constant and more than one transition areas will appear. A thin insulation membrane in a layered model leads to the presence of a low frequency dielectric enhancement and largely decreases the effective conductivity at low frequencies. It was concluded from the numerical simulations that the dielectric enhancement is dependent on the membrane thickness, membrane conductivity, volumetric ratio, and the presence of conductivity contrast between materials.

## I. INTRODUCTION

Finding the resistivity (or conductivity) and dielectric constant of downhole formation will be of great importance in locating oil and gas reservoirs (Hizem *et al.*, 2008 and Seleznev *et al.*, 2011). Usually, formation is composed of different types of materials, including rock matrices, water, oil, and others. Hence, the concepts of effective dielectric constant and effective conductivity have been introduced to describe the overall dielectric constant and overall conductivity of mixed materials.

For simplification, the shale formation is usually considered to be represented as a thinly layered Transversely Isotropic (TI) (Liu, 2017 and He *et al.*, 2015) model consisting of mineral matrix and oil, gas, or water. Due to the electrical frequency dependency of any lossy media, the effective electrical properties (such as permittivity and conductivity) of the shale model observed through a frequency spectrum will present more information for determining the microgeometry, water content, and electrical properties of each phase of the rock model. The frequency dependency properties of the dielectric constant of solid materials have been studied by many researchers (Fricke, 1924; Hanai, 1968; and Liu and Shen, 1993). Systematic studies have been done in relation to the formation of electrical characteristics. In order to explain the frequency dispersion of conductivity and permittivity, some empirical models were developed. Therein, the Debye model (Debye, 1929), the Cole–Cole model (Cole and Cole, 1941 and Pelton *et al.*, 1978), and the Dias model (Dias, 2000) are several models used to describe the frequency dispersion of conductivity and permittivity. The Debye model uses some empirical coefficients to control the shape and position of the spectroscopy curves and can fit most of the data from experiments. However, these phenomenological models can only fit the dispersion curves of permittivity and conductivity but cannot reveal the intrinsic physics of the characteristics of the frequency dispersion. The effective dielectric constant or conductivity of a mixture is a function of the dielectric constant or conductivity of each component, the microgeometry of the mixture frame, and the volume fractions of the components. It is essential to study the relations between the effective dielectric constant or conductivity and the electrical parameters of each component. The establishment of the equivalent circuit of a mixture takes into account these factors. Equating circuit representation of the Debye model and the equivalent circuit of a mixture, Debye parameters can be found in terms of the original circuit parameters.

Numerical modeling can overcome the bias caused by lab measurements (Shen *et al.*, 1987 and Zhao and Liu, 1992) and obtain an effective dielectric constant of rocks with given geometric structures and physical properties. Several numerical methods have been developed to calculate the effective permittivity and conductivity of complicated composites, such as finite difference method (FDM) (Assami, 2006; Luo *et al.*, 2010; and Kärkkäinen *et al.*, 2001), finite element method (FEM) (Krakovský and Myroshnychenko, 2002), boundary-integral equation method (BIEM) (Ghosh and Azimi, 1994), boundary element method (BEM) (Sekine *et al.*, 2002), and FDTD method (Wu *et al.*, 2007). In this study, we compute the effective permittivity and conductivity of models by a 3D-FDM method to validate the iterative Debye formulas found by an analytical method.

In this paper, an iterative Debye model is established for a multi-layered model using original circuit parameters. To validate the present model, a 3D-FDM (Assami, 2006) method is developed to extract the effective dielectric constant and conductivity of a 3D-model. The numerical and analytical simulations turned out to be exactly the same, indicating that the analytically derived equation is correct. In Sec. III, layered models with membrane (Luo *et al.*, 2010) are analyzed. Results show that a thin membrane can lead to a large low frequency dielectric enhancement. In addition, the influence of the thickness and conductivity of the membrane on the low frequency dielectric enhancement is analyzed in detail, respectively. Finally, some conclusions are made in the last section.

## II. THEORY

### A. Debye model of a two-layer material

The Debye model is a well-known equivalent material model for a single layer lossy material. It is expressed as follows:

where *ε*′ is the real part of the complex dielectric constant, *ε*″ is the imaginary part of the complex dielectric constant, *ε*_{s} is the dielectric constant at low frequencies, *ε*_{∞} is the dielectric constant at very high frequencies, *ω* is the angular frequency, and *τ* is called the relaxation factor.

Circuit representation of the Debye model (Debye, 1929) is shown in Fig. 1. The total admittance of the equivalent circuit of the Debye model can be expressed as follows:

Let *Y*_{total} equal to that of the two-layered equivalent model, parameters of the Debye model can be obtained from He *et al.* (2019),

where *ε*_{rk}, *σ*_{k}, and *d*_{k} (*k* = 1, 2) are the relative permittivity, the conductivity, and the thickness of the *k*th layer, respectively. Based on the Debye formula for a uniform material, the effective permittivity and conductivity can be expressed as

It can be seen from the above equation that

where *σ*_{∞} is the conductivity at infinite frequency.

### B. An iterative Debye model of a multi-layer material

A layered material with three horizontal layers is considered as shown in Fig. 2(a). The relative permittivity of the three layers is *ε*_{1}, *ε*_{2}, and *ε*_{3}, respectively, and the conductivity of the three layers is *σ*_{1}, *σ*_{2}, and *σ*_{3}, respectively. The thickness of the three layers is *d*_{1}, *d*_{2}, and *d*_{3}, respectively. For the equivalent model shown in Fig. 2(b),

knowing that

where *A* is the area of the cross-section of the model, and *ε*_{reff(12)}(*ω*) and *σ*_{eff(12)}(*ω*) are the calculated effective relative dielectric constant and conductivity of the above two-layered model by formulas presented in He *et al.* (2019). Both *ε*_{reff(12)}(*ω*) and *σ*_{eff(12)}(*ω*) are the functions of frequency.

Then, let the real part and the imaginary part of Eqs. (10) and (2) equal, respectively, Debye parameters in terms of the original circuit parameters of a three-layered material can be found,

in which *d*_{total} = *d*_{1} + *d*_{2} + *d*_{3}. According to Eqs. (7) and (8), *ε*_{eff} and σ_{eff} can be obtained using parameters obtained by Eqs. (1)–(4).

The above formulas can be expanded to an *N*-layered model. Let *ε*_{r}(*i*), *σ*(*i*), and *d*(*i*) denote the permittivity, conductivity, and thickness of the *i*th (*N* > *i* ≥ 1) layer; *ε*_{eff} and σ_{eff} of an *N*-layered (*N* ≥ 2) material can be obtained through an iterative way as shown in Fig. 3.

## III. NUMERICAL RESULTS AND DISCUSSIONS

To verify the validity of the derived iterative Debye formulas, a computer program for a 3D-FDM discussed in Assami (2006) is developed. All of the results are running on a laptop computer (Intel® Core^{tm} i7-6500U CPU 2.50 GHz, RAM 12.0 GB).

Example 1: first, a model with an insulation membrane (model *A*) and a model without membrane (model *B*) shown in Fig. 4 are simulated and compared. Parameters in Fig. 4, respectively, are: *ε*_{1} = 10, *ε*_{2} = 5.0, *σ*_{1} = 0.01 S/m, *σ*_{2} = 0.1 S/m, *ε*_{m} = 7.5, *σ*_{m} = 1 × 10^{−6} S/m, *d*_{1} = *d*_{2}, *L* = 1.2 × 10^{−4} m, and *d*_{m} = 1 × 10^{−6} m.

Curves of *ε*_{eff} and σ_{eff} vs frequency by the FDM compared with the results of the iterative Debye model (solid lines) are shown in Fig. 5. The results obtained by the two methods agree very well, which shows the validity of the iterative Debye dispersion model for a horizontal multi-layered material.

From Fig. 5, *ε*_{s} of model *A* is 7616 F/m and that of model *B* is 16.9 F/m. Obviously, the low frequency dielectric enhancement occurs in the effective permittivity spectra of model *A*. From Eq. (17), *ε*_{s} (*f* < 10^{5} Hz) can be obtained approximately when *σ*_{m} ≪ *σ*_{eff(12)s},

For model *A*, *d*_{m} ≪ *d*_{total} is satisfied, which leads to the low frequency dielectric enhancement.

According to Eq. (18) and *σ*_{m} = *d*_{m}, *σ*_{s} can be simplified to the following form:

We can get *σ*_{s} < *σ*_{eff(12)s} when 1 − *d*_{m}/*d*_{total} ≈ 1, which shows that a thin insulation membrane in a layered model decreases *σ*_{s}. This conclusion can be verified by the results shown in Fig. 5(b). From Fig. 5(b), *σ*_{s} of model *A* is 0.0185 S/m, while *σ*_{s} of model *B* is 0.001 07 S/m.

In addition, using *d*_{m} ≪ *d*_{1} + *d*_{2} and *d*_{total} ≈ *d*_{1} + *d*_{2}, the following equation can be obtained from Eq. (16):

which shows that *ε*_{∞} of model *A* is almost equal to *ε*_{∞} of model *B*. In addition, *σ*_{∞} can be obtained by Eq. (9),

In this case,

In addition,

So, we have the relationship *σ*_{∞} ≈ *σ*_{eff(12)∞}, which can also be validated by Fig. 5(b).

According to Eq. (15), curves of *τ* for the two models vs frequency are shown in Fig. 6. It is shown that two transition areas appear for model *A* and only one for model *B*. *τ* of model *B* is a constant, but *τ* of model *A* is dependent on frequency. For model *A*, *τ* at low frequencies is much larger than *τ* at high frequencies, which almost equals to *τ* of model *B*. This leads to the second transition area for the two models almost coincide, which can be seen from Fig. 5.

Example 2: in the second case, model *A* shown in Fig. 4(a) is used to analyze the influence of *d*_{m} and *σ*_{m} on the spectra of *ε*_{eff} and σ_{eff}. Figure 7 shows the results of a three-layered stratified capacitor with different *d*_{m} and Fig. 9 shows the results with different *σ*_{m}. Results obtained by the two methods agree very well, which shows the adequacy of the proposed iterative Debye formulas (lines).

According to Eq. (19), *ε*_{s} is proportional to *d*_{total}/*d*_{m} when the other parameters are unchanged. With the increase of the ratio, the large dielectric enhancement is increasingly apparent, which can be seen from Fig. 7(a).

Obviously, with the increase of *d*_{m}, *σ*_{s} will decrease, which can be seen from Fig. 7(b). On the other hand, with the increase of *d*_{m}, the volumetric ratio of the insulation membrane in a layered model will increase, and thus *σ*_{s} will decrease.

*τ* for a layered model with different *d*_{m} vs frequency is shown in Fig. 8. It can be seen that *τ* at low frequencies decreases with the increase of *d*_{m}, which leads to the first transition area in the spectra of *ε*_{eff} and σ_{eff} shift to higher frequencies shown in Fig. 7.

From Fig. 9, it can be seen that for *σ*_{m} < 1 × 10^{−7} S/m, neither *ε*_{s} nor *τ* is affected. However, as the membrane becomes less of an insulator (*σ*_{m} > 1 × 10^{−7} S/m), the amplitude of the enhancement decreases clearly, but *τ* increases. As the membrane becomes no longer an insulation (*σ*_{m} > 1 × 10^{−3} S/m), dielectric enhancement at low frequencies does not appear. From Eq. (17), we can get

When *σ*_{m} > 1 × 10^{−3} S/m and *d*_{m} ≪ *d*_{total}, $\epsilon r3dm\sigma eff12s2$ in numerator and *σ*_{eff(12)s}*d*_{m} in denominator can be ignored, then *ε*_{s} ≈ *ε*_{reff(12)s} can be obtained. When *σ*_{m} < 1 × 10^{−7} S/m, *ε*_{s} ≈ *ε*_{r3} · *d*_{total}/*d*_{m}, *ε*_{s} is very large and no longer affected by *σ*_{m}. When *d*_{m} keeps unchanged, the volumetric ratios of the insulation membrane in the layered model are the same for these seven models. With the decrease of *σ*_{m}, *σ*_{s} will decrease. According to Eqs. (18) and (23), *σ*_{s} can also be expressed as

which shows that *σ*_{s} will decrease with the decrease of *σ*_{m}. This can also be validated by Fig. 9(b).

Figure 10 shows the curves of *τ* for different *σ*_{m} vs frequency. It can be seen that *τ* at high frequencies is not affected by *σ*_{m}. In addition, *τ* at low frequencies almost equals to that at high frequencies when *σ*_{m} > 1 × 10^{−3} S/m. As the membrane is an insulator (*σ*_{m} < 1 × 10^{−7} S/m), *τ* at low frequencies is not affected by *σ*_{m}. In addition, as the membrane becomes less of an insulator (*σ*_{m} > 1 × 10^{−7} S/m), *τ* at low frequencies decreases with the increase of *σ*_{m}.

For the above two examples, a cubic system with 30 × 30 × 30 grid points was used for all of the results by the 3D-FDM. 28 × 30 × 30 simultaneous equations for the potentials at all the nodes except those on the top and the bottom sides were established, which were solved by the successive over-relaxation (SOR) iteration algorithm. The comparison shows that the relative error between the results by the FDM and that by iterative Debye formulas can be guaranteed smaller than 1.5% when the threshold is set as 1 × 10^{−5} in the FDM. In addition, the iteration times are about 80. It takes more than 9 min for one frequency point by the FDM. Like other numerical methods, the more element cells, the higher computation precision of the FDM. However, the proposed Debye model is an analytical solution for extracting the effective electric parameters of horizontal multi-layered material without grid. The computation time for one frequency point using Debye formulas is less than 1 s with the same computing resources. Besides high efficiency and high precision, the proposed formulas are more suitable for analyzing the relationship between the effective electrical spectra and the electrical parameters of each layer.

## IV. CONCLUSION

In this paper, an iterative Debye model for a horizontal multi-layered material was found using original circuit parameters. The equivalent circuit considers the geometry and the electrical parameters of each component of a mixture. It can be used to analyze the frequency dependency properties of the effective permittivity and conductivity of a horizontal multi-layered material. A layered model with a membrane was analyzed. The influence of the conductivity and thickness of the membrane on the effective electrical parameters of the whole model was analyzed in detail. The study has provided theoretical background for further research on low frequency dielectric enhancement. In addition, it can be further applied to the study of the capacitance effect of the double layer. To validate the formulas, the 3D-FDM code was developed to extract the effective permittivity and conductivity of a multi-layered model. Comparisons of the results obtained by the two methods are made, which shows high precision and high efficiency of the proposed method.

From the examples given in this paper, conclusions can be drawn: (1) when the number of layers is more than 2, the relaxation factor will not be a constant and more than one transition areas will appear in the effective electrical spectra; (2) a thin insulation membrane in a layered model leads to the presence of a low frequency dielectric enhancement and largely decreases the effective conductivity at low frequencies; (3) when other parameters remain unchanged, the effective dielectric constant at low frequencies is increasingly apparent with the increase of the ratio of the total thickness of the layered model and the membrane thickness.

## ACKNOWLEDGMENTS

This work was supported, in part, by the National Natural Science Foundation of China (Grant Nos. 41704107, 61901326, and 61901336) and the Young Talent fund of University Association for Science and Technology in Shaanxi, China (Grant No. 20180107).

## REFERENCES

_{010}cylindrical cavity for dielectric measurement