## 1 Introduction

Archie’s formula uses the electrical resistivity of rocks to provide a quantitative evaluation of oil and gas saturation [1]. As a result, resistivity logging occupies an irreplaceable position in the petroleum industry. Conventional resistivity logging tools use relatively low frequencies for their measurements. It was not until Snyder et al. [2] discovered that the resistivity dispersion spectrum of a formation contains abundant useful information that scientists began to pay attention to the complex resistivity spectra of formations. Klein and Sill [3] manufactured artificial shaly sandstones and explored the resistivity dispersion characteristics of the samples in the frequency range of (1/128)–1,024 Hz. They found that the resistivity dispersion of the samples in this frequency band was consistent with the Cole–Davidson model, and the time constant *τ* had a strong correlation with the pore size distribution of the sample. The complex resistivity spectra of natural shaly sandstone samples in the frequency range of 1–1,000 Hz were measured, and a dispersion model that accounted for the shale content, water salinity, and oil saturation was established in ref. [4]. Khairy and Harith [5] studied the effects of pore geometry, confining pressure, and water saturation on the complex resistivity of argillaceous sandstones with frequencies ranging from 1 Hz to 0.2 MHz. Kavian et al. [6] measured the complex resistivity of natural cores at four frequencies (105 kHz, 332 kHz, 665 kHz, and 1.05 MHz) and found that the relationships between their real components of each frequency, porosity, and water saturation follow Archie’s law. Zisser et al. [7,8] studied the complex resistivity characteristics of sandstones and tried to use this information to predict their permeability. The resistivity and dielectric dispersion of natural cores in the frequency range of 0.1 Hz to 1 kHz were studied by Liu et al., who defined the degree of frequency dispersion of the permittivity or resistivity values to evaluate the water saturation of the cores [9]. Norbisrath et al. [10] estimated the permeability of a dolomite formation based on its complex resistivity spectra in the frequency range of 0.1 Hz to 100 kHz. Most similar studies used induced polarization exploration, so the frequency bands in their research are relatively low and are not suitable for well logging. Few studies have focused on the complex resistivity dispersion of the high-frequency band. Tong and Tao [11] measured the complex resistivity spectra of natural shaly cores from 100 to 20 MHz, and they found that the slope of the real component curve of the complex resistivity spectrum is strongly correlated with porosity and permeability. Li et al. [12] produced artificial pure sandstones and shaly sandstones and measured their complex resistivity spectra in the frequency range of 40–15 MHz. The experimental data were fitted with a second-order Cole–Cole equation, and a saturation evaluation model was established based on the time constant. Jiang et al. [13] designed a set of logging tools to obtain the complex resistivity spectrum of a formation in the frequency range of 1–500 kHz, and they used the frequency corresponding to the bottom of the imaginary component to evaluate the water-filled porosity.

The information extracted from the complex resistivity spectra in the previous research is immensely important to geophysical exploration. However, the existing literature has not considered all characteristics related to reservoir parameters contained in the complex resistivity spectra. In addition, previous research used complex resistivity spectra to calculate reservoir parameters mostly based on various empirical resistivity dispersion models, such as the Cole–Cole model. While such models can accurately describe the resistivity dispersion of formation in most cases, they are not universal and sometimes yield results that diverge from the measured data. Therefore, the accuracy of their evaluation method used depends largely on the applicability of the dispersion model that is selected. In this study, the spectra of 15 pure sandstone cores from western China in the frequency range of 100–15 MHz were analyzed, and some useful information were discovered from the spectra.

## 2 Spectrum measurement

### 2.1 Core features

We collected 15 cores from western China (Table 1). All core contained less than 10% shale, so they can be considered as pure sandstone cores. The depths of these cores ranged from 4731.25 to 4780.48 m, with porosity of 4.8–26.5% and permeability of 2.9–669.12 mD.

Parameters of core samples

Sample number | Depth (m) | Shale contents (%) | Porosity (%) | Permeability (mD) | Group |
---|---|---|---|---|---|

YD-12 | 4738.5 | 1.90 | 5.10 | 2.9 | A |

YD-38 | 4750.68 | 3.60 | 7.70 | 15.49 | |

YD-18 | 4741.39 | 8.60 | 22.80 | 23.16 | |

YD-43 | 4752.93 | 8.30 | 18.50 | 112.25 | |

YD-50 | 4759.18 | 5.20 | 23.80 | 221.36 | |

YD-49 | 4758.6 | 6.10 | 23.90 | 468.66 | |

YD-2 | 4731.25 | 2.90 | 22.60 | 550.23 | |

YD-37 | 4750.05 | 5.60 | 20.30 | 669.12 | |

YD-47 | 4757.73 | 3.00 | 4.80 | 10.81 | B |

YD-67 | 4769.38 | 2.00 | 6.70 | 17.64 | |

YD-65 | 4768.24 | 3.10 | 10.30 | 24.12 | |

YD-84 | 4780.48 | 5.00 | 21.80 | 179.38 | |

YD-35 | 4749.56 | 4.70 | 22.90 | 433.56 | |

YD-23 | 4743.43 | 2.40 | 24.50 | 505.7 | |

YD-61 | 4766.33 | 3.40 | 24.00 | 585.12 |

### 2.2 Experimental methodology

We used the two-electrode method to measure the complex resistivities of the core samples with a 4194A impedance analyzer (HP, USA) operating with measurement frequencies from 100 to 15 MHz (Figure 1). The cores were saturated with NaCl solution with a salinity of 1,00,000 ppm, and then measurements were made at room temperature and at a constant confining pressure of 10 MPa. The solution in the core was displaced with kerosene by a pump during the measurement to change the water saturation of the core. The metering tube was used to collect the liquid displaced from the core, which allowed us to calculate the water saturation of the core at different moments. The complex resistivity spectra of each core were collected at different water saturation levels by the impedance analyzer.

## 3 Results

### 3.1 Spectrum characteristics

Figure 2 shows the complex resistivity spectra of core YD-43, with different levels of water saturation. When the measurement frequency was lower than 10 kHz, both the imaginary part and the real part (absolute value) of the complex resistivity decreased slowly as the frequency used to make the measurements increased, but were relatively stable overall. When the measurement frequency reached about 10 kHz, the imaginary resistivity began to increase gradually with the frequency until it reached an extreme value and then began to decrease again, a pattern reflected in the local minimum, labeled

Because the impedance of the rock matrix is much greater than that of aqueous pore solution, higher water saturation will result in a smaller real resistivity measurement in the low-frequency band. Similarly, the impedance of kerosene is much larger than that of aqueous solution, and displacement of water will cause an intensification of interfacial polarization in the pores because of the more complex surface morphology of aqueous pore solution. The migration path of ions becomes complicated, and it takes longer to complete the polarization process, so smaller water saturation will cause higher *R*_{xb} and smaller corresponding frequencies. A power exponential relationship was identified between the maxima of the imaginary resistivity of each core and the saturation values (Figure 3a). Figure 3b shows the maxima of the imaginary resistivity of different cores with water saturation of 100% and the power exponential relationship between these values and porosity.

The phenomenon can be explained qualitatively by means of the polarization mechanism. When an alternating current is applied to water-bearing rocks, various polarization phenomena occur in the rocks, which will cause the amplitude of the current to decay and the phase to shift [14]. When this happens, complex models are needed to characterize the impedance of rocks [15]:

*ρ**(

*ω*),

*ρ*′(

*ω*), and

*ω*is the angle frequency;

*σ*is the conductivity of the core;

*ε*is the dielectric constant of the core; and

The intensities of various polarization phenomena change in response to the frequency of the alternating current, so the complex resistivity of the core changes with the frequency used to make the measurement. When the frequency is lower than 1 MHz, the interfacial polarization phenomenon (Maxwell–Wagner effect) in the core is most obvious [10]. This is mainly caused by the accumulation of ions on the inner surface of pores under the action of the electric field.

The interface polarization leads to the generation of the imaginary component of the rock’s impedance. The position of the maximum value of the imaginary resistivity corresponds to the frequency at which the interface polarization is the strongest. When the porosity of the core is small, or after the kerosene enters the pore space, the interface morphology of the solution with other components becomes more complicated, and it takes longer for the ions to accumulate at the interface. Thus, as water saturation and porosity decrease, interface polarization is more intense, and rocks have smaller *R*_{xb} at a smaller frequency.

A capacitance model can also be used to understand the effects of porosity and water saturation on the imaginary part of the complex resistivity spectrum. A pore of rock can be imagined as a capacitor, where the fluid in the pore is the dielectric and the pore inner surface constitutes the capacitive plates. The phases of the voltage and the current across the capacitor are always 90° different, so the reactance of the capacitor contributes to the imaginary component of the complex resistivity. The larger the porosity of the core is, the larger the distance between the plates of the capacitor is and the smaller the capacitive reactance of the capacitor will be, resulting in a smaller maximum in imaginary resistivity. The dielectric constant of water is larger than that of hydrocarbons and matrix, and smaller water saturation will also result in great imaginary resistivity maximum values just like the porosity is getting smaller.

### 3.2 New characterization parameters

Our measurement results revealed good power exponential relationships between the maxima of imaginary resistivity, porosity, and the water saturation of the core. These are similar to the relationships between porosity, water saturation, and the low-frequency resistivity of the formation obtained by traditional resistivity logging tools. Therefore, the real resistivity corresponding to 1 kHz was extracted from the complex resistivity spectrum to investigate the relationship with the maximum of imaginary resistivity. Surprisingly, there was a nearly perfect linear relationship between the maxima of imaginary resistivity and the real resistivity at 1 kHz, for a core with different water saturation levels, as shown in Figure 4. In other words, when the porosity and the permeability of the core are held constant, the ratio between the real resistivity at 1 kHz and the maxima of the imaginary resistivity is constant and independent of the water saturation of the core. This slope of fitted line is expressed as *β*_{RX}.

It should be noted that the frequencies corresponding to *R*_{xb} and *R*_{1k} is, by definition, always at 1 kHz, whereas *R*_{xb} corresponds to the frequency with the strongest interface polarization. The real resistivity is relatively stable in the frequency domain centered on 1 kHz, whereas *R*_{xb} is not fixed and depends on the porosity of the rock and the nature of the fluid in the pore. Therefore, *β*_{RX} is different from the traditional phase angle of the complex resistivity.

Figure 5 shows the relationship between *β*_{RX} and porosity in these natural cores, revealing a good linear relationship between porosity and *β*_{RX}. When the porosity is high, the distribution of data points in the figure is disorderly. The inset figure shows the relationship between permeability and *β*_{RX} for the seven cores with porosities greater than 22%, showing another good linear relationship. Therefore, the disorderly distribution of data points when the porosity is greater than 22% can be attributed to the differences in their permeability values.

There is also a good linear relationship between *β*_{RX} and the permeability of the cores, as shown in Figure 6. The distribution of data points in the figure deviates from the fitting line when the permeability is less than 100 mD. The linear relationship between porosity and *β*_{RX} of the six cores with permeability less than 100 mD is also strong, as shown in the inset (Figure 6). Therefore, this deviation when the permeability is less than 100 mD is driven by their different porosities.

## 4 Discussion

The experimental results of this study confirm strong linear relationships between porosity, permeability, and *β*_{RX}. These relationships provide a new idea for estimating permeability based on complex resistivity spectra. Porosity and permeability of cores can be treated as independent variables, with *β*_{RX} as a dependent variable for binary linear fitting. Many logging curves can provide the porosity of a formation, such as acoustic logging, neutron-density and NMR logging, so permeability can be estimated based on the fitting result when *β*_{RX} is determined from the complex resistivity spectra.

Since this study did not use actual logging data, half of the cores were selected to build the model and the remaining cores were used to verify the model. The 15 cores were ranked by permeability from smallest to largest, and the eight cores with an odd ranking (group A in Table 1) were selected to fit the relationship between the three parameters. The resulting equation is given as follows:

*K*and

*ϕ*are the permeability and the porosity, respectively.

A new slope (*β*_{RXa}) for the remaining seven cores (group B in Table 1) was obtained by plugging their porosity and permeability values into equation (2). As shown in Figure 7, the relative error between *β*_{RXa} and *β*_{RX}, which was extracted directly from the complex resistivity spectra, is small. The data points in Figure 7 are roughly distributed on the angular bisector of the first quadrant, which proves that the relationships between porosity, permeability, and *β*_{RX} can be described by a binary first-order equation.

Figure 8 shows the relationship between the measured permeability (*K*) and the calculated permeability (*K*_{a}) of the seven cores in group B. While these two permeability metrics were derived from very different sources, the data points still distribute along the angular bisector of the first quadrant. This demonstrates the feasibility of estimating permeability with equation (2).

There should be a strong positive correlation between porosity and permeability for sedimentary rocks from the same sedimentary environment. However, the cores used in this study were collected from a range of depths and exhibited a poor correlation between porosity and permeability (Figure 9). This may impact the accuracy of the permeability estimation method based on *β*_{RX}.

Estimating permeability has always been a difficult problem in well-logging interpretation, and the various existing permeability estimation methods struggle to achieve the desired accuracy. Although the accuracy of the permeability estimates made with the method developed here is not perfect in this example, the results are sufficient to show a strong correlation between porosity, permeability, and *β*_{RX} in these sandstone cores. This approach may encourage other researchers and industry scientists to pay attention to *β*_{RX}, to obtain more accurate permeability estimates, and to further refine these methods.

### 4.1 Saturation estimation

The maxima of imaginary resistivity have a power exponential relationship similar to that of porosity and permeability in cores like the real resistivity of low frequency. Therefore, we can try to calculate the water saturation based on the maxima of the imaginary resistivity spectra. Imitating the formation factor and resistivity increase factor, the following definitions were made:

*R*

_{xb}is the maximum of the imaginary part of core’s complex resistivity spectrum,

*F*′ is the formation factor related to

*R*

_{xb},

*I*′ is the increase factor related to

*R*

_{xb0}is the maximum of the imaginary part of the complex resistivity when the core is saturated with solution,

*m*′ is the cementation exponent related to

*R*

_{xb},

*n*′ is the saturation exponent related to

*R*

_{xb}, and

*a*′ and

*b*′ are lithology-related coefficients related to

*R*

_{xb}.

To obtain the imaginary resistivity of the solution, a plastic pipe with an inner diameter of 3 mm and a length of 2.4 m was filled with NaCl solution, and the complex resistivity spectrum of the solution was measured by an impedance analyzer with the electrodes inserted into the nozzles of the plastic pipe. The *R*_{xbw} value was calculated from the complex resistivity spectrum.

By fitting the relationship between *I*′ and *S*_{w}with a power function, the values of *n*′ and *b*′ were obtained. Figure 10 shows the relationship between *I*′ and *S*_{w} of core YD-43. According to the definition, *R*_{xb} = *R*_{xb0} and *I*′ *=* 1 when water saturation reaches 100%. Therefore, before fitting the relationship between *I*′ and *S*_{w} for each core, ten additional points with coordinates (1,1) were added to the measured datasets to bring the curve closer to the theoretical relationship.

The eight cores of group A were selected to build the model, and the seven cores of group B were used to verify the model. Figure 11 shows the relationship and the fitting result between *F*′ and *ϕ* for the cores of group A. In an approach similar to that used for *I*′ and *S*_{w}, ten additional points with coordinates (1,1) were added to the datasets because *R*_{xbw} = *R*_{xb0} and *F*′ = 1 when *ϕ* is equal to 1.

We then used a model similar to Archie’s formula to calculate the water saturation of the seven cores of group B by using the imaginary part of complex resistivity:

*S*

_{wa}) and the water saturation recorded during the experiment (

*S*

_{w}). The correlation between the two datasets is very high, and the data points are distributed along the angular bisector of the first quadrant. This model can be used to effectively calculate the water saturation sandstone cores.

Although the structure of the new model is similar to Archie’s formula, it is different in several essential ways. For example, Archie’s formula uses the real resistivity of low frequencies to evaluate the water saturation, whereas the model in this article is based on imaginary resistivity. Real resistivity is largely affected by the conductivity of the medium being analyzed, while imaginary resistivity is mainly affected by the dielectric constant. In addition, the real resistivity used in Archie’s formula comes from a constant, predetermined determined frequency, whereas the frequency corresponding to the maximum value of imaginary resistivity varies, and it is related to the water saturation of the medium. The frequency corresponding to the maximum value of imaginary resistivity varies with the water saturation and other parameters of the medium, so the model proposed in this study is based on a dimension of the frequency domain that is different from that used by Archie’s formula.

## 5 Conclusion

The complex resistivity spectra of 15 natural sandstone cores in western China were measured with a frequency range of 100–15 MHz. Our analysis revealed a nearly perfect linear relationship between the real resistivity at 1 kHz and the maxima of imaginary resistivity. Moreover, the slopes of the linear fitting lines had strong linear correlations with both the porosities and permeabilities of the cores. We developed a permeability estimation model based on these slopes, and it proved effective at estimating the permeability of sandstone samples, even the samples with poor correlations between porosity and permeability.

As with low-frequency resistivity, the maximum values of imaginary resistivity had power relationships with porosity and water saturation. The saturation evaluation model demonstrated high accuracy and reliability in its saturation predictions by using the maxima of imaginary resistivity. These models and the conceptual understanding that underlies them should help popularize the use of complex resistivity spectra in the field of geophysics.

This research is supported by the National Oil and Gas Major Projects (No. 2011ZX05020-009). The authors would like to thank Dr. Ming Jiang and Bing Zhang for their guidance. In addition, special thanks are given to Qining Zhao for her continued support.

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