#### Abstract

This paper investigated fractal characteristics of microscale and nanoscale pore structures in carbonates using High-Pressure Mercury Intrusion (HPMI). Firstly, four different fractal models, i.e., 2D capillary tube model, 3D capillary tube model, geometry model, and thermodynamic model, were used to calculate fractal dimensions of carbonate core samples from HPMI curves. Afterwards, the relationships between the calculated fractal dimensions and carbonate petrophysical properties were analysed. Finally, fractal permeability model was used to predict carbonate permeability and then compared with Winland permeability model. The research results demonstrate that the calculated fractal dimensions strongly depend on the fractal models used. Compared with the other three fractal models, 3D capillary tube model can effectively reflect the fractal characteristics of carbonate microscale and nanoscale pores. Fractal dimensions of microscale pores positively correlate with fractal dimensions of the entire carbonate pores, yet negatively correlate with fractal dimensions of nanoscale pores. Although nanoscale pores widely develop in carbonates, microscale pores have greater impact on the fractal characteristics of the entire pores. Fractal permeability model is applicable in predicting carbonate permeability, and compared with the Winland permeability model, its calculation errors are acceptable.

#### 1. Introduction

Compared with sandstones, carbonate pore structures are usually more complex. There are many diverse types of pore space in carbonates such as intergranular pores, intraparticle pores, moldic pores, fractures, and vugs. Generally, carbonate reservoirs are more heterogeneous and flow mechanisms are more intricate, making carbonate reservoir development problematic. The Y oil reservoir is a porous carbonate reservoir in the Middle East[1, 2]. Even though fractures and vugs develop in carbonates, grain pores are the main oil storage space. In this paper, HPMI technique was utilized to describe the sizes and distributions of carbonate microscale and nanoscale pores, and the fractal characteristics of microscale and nanoscale pores were also investigated.

Since Mandelbrot first proposed the fractal theory in 1980s [3], its application has been widely used for rock pore structure characterization, due to its effectiveness in describing complex and irregular structures with self-similar characteristics. The microstructures of rock pore space have been proven to be fractal [4–6]. With scanning electron microscopy (SEM), Katz and Thompson [7] demonstrated that sandstone pore space is fractal and self-similar with over 3 to 4 orders of magnitude. Angulo et al. [8] calculated fractal dimensions of sandstone using mercury intrusion test. Ever since, mercury intrusion method has been extensively used to study fractal characteristics of pore structures in sandstones [9–11] and coals [12–14]. On the other hand, nitrogen adsorption technique is widely used to analyse the fractal characteristics of nanoscale pores in shales [15–17]. The fractal characteristics of carbonate pores have been expansively studied by many scholars. Krohn [18] studied fractal characteristics of carbonate pores with SEM. Xie [19] calculated fractal dimensions of carbonate pores using box-counting method from environmental scanning electron microscope (ESEM) images and indicated that carbonate pores are multifractal. The previous studies have demonstrated that fractal dimension can effectively reflect roughness of pore surface and characterize heterogeneity of pore structures.

Accurately predicting carbonate permeability is challenging due to high heterogeneity in carbonate pore structures. Several models proposed have been commonly used to estimate carbonate permeability, including Purcell model [20], Winland model [21], Swanson model [22], and Pittman model [22]. Besides pore structure analysis, fractal theory has also been used to predict porous media permeability [23, 24]. One of the most broadly used methods was proposed by Yu and Cheng [23]; their model was developed based on the tortuous fractal capillary tube model, although no empirical parameters were introduced in their model. Similarly, based on the tortuous and fractal tubular bundle model, Buiting and Clerke [25] developed a different fractal permeability model, which had been verified with more than 500 carbonate core samples, but empirical parameters were introduced. With more than 200 carbonate core samples, Nooruddin et al. [26] compared nine different permeability models using mercury intrusion capillary pressure data, and they found that Swanson and Winland permeability models gave the best carbonate permeability prediction results. However, Yu and Cheng’s permeability model was not evaluated in their studies. In this paper, the adaptability of Yu and Cheng’s permeability model for carbonate permeability prediction was evaluated and compared with Winland model.

#### 2. HPMI

16 core samples collected from the Y carbonate reservoir were used for this study. The pore space of these carbonate core samples excludes visible fractures and vugs. The measured air permeability and porosity of core samples are shown in Table 1. The porosity of core samples varies from 12.485% to 31.805%, with an average of 17.526%, and air permeability varies from 0.591 mD to 70.556 mD with an average of 12.380 mD. As shown in Figure 1, porosity-permeability correlation is very weak, indicating high heterogeneity of carbonate core samples.

Pore structures and pore size distributions of carbonate core samples were characterized with HPMI technique. In the experiment, the Auto Pore III-9420 was used, and the maximum mercury intrusion capillary pressure is approximately 276 MPa. Figure 2 presents mercury intrusion capillary pressure curves of all the core samples. Mercury is nonwetting to rock surface and has a high surface tension with air. With mercury intrusion pressure increasing, nonwetting mercury enters the small pores of the core samples, and the pore radius can be calculated from the Washburn equation [28]:where is the mercury intrusion capillary pressure; is pore radius; is the interfacial tension between mercury and air; and is mercury and rock contact angle. Under the maximum intrusion pressure, the corresponding minimum pore radius that can be detected is 2.7 nm. The structures and distributions of almost the entire pores, from nanoscale pores to microscale pores, were tested during the HPMI test, and then the parameters of pore structure were calculated from HPMI curves as shown in Table 2.

#### 3. Methodology Description

According to fractal geometry theory, when pore structures are fractal, the relationship between the pore number and pore radius can be presented as [3] where is the pore radius (characteristic length); is pore number with pore radius larger than ; and is fractal dimension. There are several different methods to calculate pore number based on different assumptions. In this paper, four different fractal models were used to calculate fractal dimensions from mercury intrusion capillary pressure curves, and the most suitable fractal model was selected to analyse the fractal characteristics of microscale and nanoscale pore structures.

##### 3.1. I-2D Capillary Tube Model Method

In this method, pore networks of core samples are made up of a bundle of tortuous capillary tubes. The numbers of capillary tubes with tube radius can be calculated with the volume of mercury intrusion as given bywhere is the number of capillary tubes with radius ; is the single volume of mercury intrusion at radius ; and is the core length. Then the cumulative number can be calculated asAccording to (2), fractal dimension can be determined from the slope of the line of and in a log-log plot.

##### 3.2. II-3D Capillary Tube Model Method

Instead of using single volume of mercury intrusion , Li [9] used the cumulative volume of mercury intrusion to calculate the number of filled capillary tubes as given bySubstituting (5) into (2),The core length is constant, and therefore (6) can be simplified asAs can be expressed with mercury saturation , and pore radius can be expressed with mercury intrusion capillary pressure , (7) can be expressed asThen fractal dimension can be calculated from the plot. With the same datapoints, the calculated fractal dimensions using (6) and (8) should be the same.

Although methods I and II are based on the capillary tube model, they are essentially different. Method I reflects the fractal characteristics of the capillary tube distribution in the cross-section of core samples, which is in two-dimensional space, and the range of the calculated fractal dimension using method I is . Method II obtains fractal dimension by filling pore space of core samples with different size of capillary tubes, which can reflect the fractal characteristics of pore space in three-dimensional space, and generally the calculated fractal dimension using method II is .

##### 3.3. III-Geometry Model Method

Based on the geometry fractal characteristics of coals, Friesen and Mikula [29] proposed an equation to calculate fractal dimension which can be expressed asThe fractal dimension was calculated by plotting verse in a log-log plot.

##### 3.4. IV-Thermodynamic Model Method

Based on the thermodynamic analysis of mercury intrusion process and fractal characteristics of pore surface, Zhang and Li [30] proposed a thermodynamic model, and the simplified equation for calculating fractal dimension can be expressed as [31]where is the cumulative surface energy during the mercury intrusion process; is pore radius; and is the intrusion volume of mercury at stage .

The calculated fractal dimensions using method III and IV reflect the volume heterogeneity and surface heterogeneity of carbonate pores, respectively [12, 32].

#### 4. Fractal Dimension Calculation

The calculated fractal dimensions using methods I, II, III, and IV were expressed as , , and , respectively. The no. 2 core sample was selected as an example to demonstrate the process of fractal dimension calculation using four methods presented above. As depicted in Figure 3, fractal dimensions were calculated from the slope of straight line in log-log plots. All carbonate pores, from the maximum pore radius to the minimum pore radius, were selected for linear fitting, and therefore the calculated fractal dimensions reflect the average fractal characteristics of entire carbonate pores, including microscale pores and nanoscale pores. Method I, method III, and method IV have good linear fitting results, and the determination coefficients are larger than 0.96. However, for method II, the determination coefficient of the linear fitting is less than 0.60, the curve of breaks into two segments, and the linear fitting result with one straight line is poor.

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Table 3 shows the calculated fractal dimensions with the four methods. The results show that the calculated fractal dimensions depend on the methods used. The calculated fractal dimensions with method I vary from 1.267 to 1.853, indicating that reflects fractal characteristics of pore distribution in two-dimensional space. The calculated fractal dimensions with methods II, III, and IV are between 2 and 3, which means , , and reflect fractal characteristics of pore structures in three-dimensional space. Overall, and are close to each other, but larger than .

To study the fractal characteristics of microscale pores and nanoscale pores, respectively, method II was used to calculate the fractal dimensions of microscale pores and nanoscale pores by fitting the curve of with two straight lines. As shown in Figure 4, the determination coefficients of linear fitting with two straight lines are much larger than that of linear fitting with one straight line, which indicates that microscale pores and nanoscale pores have different fractal characteristics.

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Table 4 presents the calculated fractal dimensions of microscale pores and nanoscale pores . The fractal dimensions of microscale pores vary from 3.006 to 5.044 and are much larger than the fractal dimensions of nanoscale pores , which vary from 2.028 to 2.122, indicating that pore structures of microscale pores are complex than those of nanoscale pores. According to fractal theory, the calculated fractal dimension in three-dimensional space should be less than 3. The reason for being greater than 3 may be the oversimplification of cylinder shape of microscale pores [10]. When (8) was derived from (7), the assumption was that that the shape of the pores is cylindrical, hence rendering (1) valid. However, for the microscale pores, especially for the carbonates, fractures and large pores with complex shapes may exist, which can lead to the value of beyond 3.

Figure 5 shows the relationship between the fractal dimensions of the entire microscale and nanoscale pores. The fractal dimensions of microscale pores strongly agree with the fractal dimensions of the entire pores , but poorly correlate with the fractal dimensions of nanoscale pores , and with increasing, decreases. It can be concluded that fractal characteristics of nanoscale pores and microscales pores are different, and microscale pores have greater impact on the fractal characteristics of the entire pores compared with nanoscale pores.

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Lai et al. [10] also used plot to study the fractal characteristics of pore structures in tight gas sandstones. They found that nanoscale pores of tight sandstone can represent the fractal characteristics of entire pores much better than the microscale pores, which is opposite to the results derived from carbonates studied in this paper. The different deductions may be caused by the difference in pore networks between carbonates and tight sandstones. The porosity of the carbonate core samples studied in this paper is much higher than that of tight sandstones. The nanoscale pores of these carbonates are not well connected, and carbonate pore networks are dominated by microscale pores, such as nonvisible microfractures, which have better connectivity.

#### 5. Fractal Characteristic Analysis

The fractal dimensions calculated with the four methods are not the same. The relationships between the carbonate petrophysical properties and the fractal dimensions calculated with the four different methods, , , , and , were evaluated. Figure 6 shows the correlations between the air permeability of core samples and the fractal dimensions calculated from the four different methods. In contrast, with increasing, decreases logarithmically; however, , , and strongly agree with . According to Liu et al. [33], for the fractal capillary tube model, the poor correlation between permeability and fractal dimension is reasonable, which means is more suitable for fractal analysis of carbonate pore structures and petrophysical properties. As depicted in Figure 7, with increasing, displacement pressure and tortuosity increase and pore radius and decrease, indicating pore structures become more complex.

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Hydraulic Flow Unit (HFU) identification is imperative in reservoir characterization. Amaefule et al. [33] first introduced the reservoir quality index/flow zone indicator (RQI/FZI) method to identify HFU. Figure 8 shows the relationship between fractal dimensions of the entire pores and RQI and FZI. The results after the investigation showed that reservoir quality becomes poorer when RQI and FZI decrease with increasing . Using the improved HFU identification method proposed by Mirzaei-Paiaman et al. [27], HFUs were identified with the carbonate core samples studied in this paper. As indicated in Figure 9, the 16 carbonate core samples belong to two different HFUs, i.e., HFU-1 and HFU-2. The average fractal dimensions of HFU-1 and HFU-2 were calculated and are 2.34 and 2.24 respectively. The higher the quality of the HFU is, the lower the average fractal dimension will be.

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The fractal characteristics of the microscale pores and nanoscale pores were analysed. Figures 10 and 11 present the fractal analysis of carbonate petrophysical properties using and , respectively. Comparatively, fractal analysis using in Figure 7 and provided similar results, unlike which gave different result, and it can be inferred that fractal dimensions of nanoscale pores cannot be used for fractal analysis.

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The distributions of pore volume and permeability contribution for microscale pores and nanoscale pores were calculated. Figure 12 presents the result of the no. 2 core sample. The pore radius distribution was from several nanometers to dozens of micrometers with bimodal distribution. The first and second peaks appeared around 0.1 *μ*m and 1 *μ*m, respectively. Although nanoscale pores were widely predominant, microscale pores contributed to most of the permeability. The distribution of permeability contribution is unimodal, and the peak appears at the same location of the maximum volume of microscale pores.

Table 5 illustrates the total pore volume and permeability contribution of microscale pores and nanoscale pores. Although nanoscale pores averagely occupy more than 38% of total pore volume, the average permeability contribution is less than 4%. Even for the no. 2 core sample, nanoscale pores only contributed 0.79% of the total permeability. Contrarily, microscale pores averagely contribute more than 96% of the total permeability, though the average volumes of microscale pores are only 55.14%. It could be concluded that though nanoscale pores are predominant in the Y carbonate reservoir, petrophysical properties are mainly dominated by microscale pores. Figure 13 presents the relationship between the permeability contributions and fractal dimensions of microscale and nanoscale pores. As shown in the figure, the permeability contribution of microscale pores and nanoscale pores decreases with increasing fractal dimensions. It can be explained that the increasing fractal dimensions indicated much complexity in the pore structures, which can in turn result in reduced permeability contribution.

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#### 6. Carbonate Permeability Modelling

Winland permeability model is an empirical equation considering the effect of pore size distribution on permeability, and it has been commonly used for predicting carbonate permeability. The Winland permeability model is expressed as [21] where is the pore radius with 35% mercury saturation and is porosity. Figure 14 presents the calculated permeability using Winland permeability model compared with the measured air permeability. Overall, the calculated permeability is smaller than the measured air permeability. It may be due to the natural microfractures existing in carbonate core samples, which can enhance the permeability of the core samples.

The fractal permeability model proposed by Yu and Cheng is based on the tortuous fractal capillary tube model which can be expressed as [23]where is the representative length; is the total section area and ; is the tortuosity fractal dimension, and it can be calculated with the average tortuosity which can be measured from HPMI; and is the maximum pore diameter. As (12) is based on the tortuous capillary tube model, the two-dimensional fractal dimension should be used. Therefore, the fractal dimensions calculated with method I were used to predict carbonate permeability. Figure 15 shows the comparison between the calculated permeability using (12) and the measured permeability. Considering how heterogeneous carbonate pore structures are, the calculation errors are acceptable. The calculation errors may be due to the different fractal characteristics of microscale pores and nanoscale pores. In general, the fractal permeability model can be applied to model the permeability of porous carbonates without visible fractures and vugs.

#### 7. Conclusion

Fractal theory was used to analyse microscale and nanoscale pore structures and petrophysical properties of carbonates with HPMI. The following conclusions can be drawn from this study:(i)The calculated fractal dimensions strongly depend on the fractal models used. When carbonate pore space is assumed to be made up of bundles of capillary tubes, 3D capillary tube model can effectively analyse fractal characteristics of carbonate pore structures. Strong and correct correlations between carbonate petrophysical properties and fractal dimension calculated from 3D capillary tube model have been observed.(ii)Microscale pores instead of nanoscale pores can represent the fractal characteristics of the entire pores of the carbonates in the Y oil reservoir. With the fractal dimensions increasing, the permeability contribution of microscale pores and nanoscale pores can both be reduced.(iii)Fractal permeability model proposed by Yu and Cheng is applicable in predicting the permeability of porous carbonates without visible fractures and vugs, and compared with Winland permeability model, the calculation results are acceptable.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 51604285), Beijing Municipal Natural Science Foundation (no. 3164048), Scientific Research Foundation of China University of Petroleum, Beijing (no. 2462017BJB11 and no. 2462016YQ1101), and the National Science Foundation for Young Scientists of China (Grant no. 41702359).