# Fractal Analysis of Microscale and Nanoscale Pore Structures in Carbonates Using High-Pressure Mercury Intrusion

Authors: Fuyong Wang; Peiqing Lian; Liang Jiao; Zhichao Liu; Jiuyu Zhao; Jian Gao
Journal: Geofluids
 (2018)

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## 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.


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## Body

## 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 Table1. 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.Table 1
Air permeability and porosity of carbonate core samples.
Core No. Porosityϕ (%) Air permeabilityKair (mD) 1 31.805 70.556  2 13.752 1.220  3 18.537 1.515  4 14.579 1.164  5 19.676 1.696  6 16.481 0.591  7 18.541 1.002  8 19.145 1.814  9 15.174 1.106  10 20.059 1.103  11 22.190 18.392  12 13.787 12.844  13 14.083 24.300  14 16.089 38.170  15 14.040 13.832  16 12.485 8.772Figure 1
Porosity-permeability correlation of 16 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. Figure2 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]:(1)Pc=2σcos⁡θrwhere Pc is the mercury intrusion capillary pressure; r is pore radius; σ is the interfacial tension between mercury and air; and θ 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.Table 2
Pore structure parameters of carbonate core samples obtained from HPMI test.
CoreNo. Displacement pressurePd (MPa) Maximum pore radiusrmax (μm) r 50 (μm) r 35 (μm) Sorting coefficientSp SkewnessSkp Tortuosityτ 1 0.139 5.738 1.021 1.471 2.643 0.414 1.443  2 0.244 3.672 0.813 1.037 2.108 0.341 6.812  3 0.869 0.965 0.366 0.413 1.682 0.169 10.229  4 0.692 1.258 0.441 0.462 1.119 0.320 5.212  5 1.033 0.791 0.456 0.444 1.839 -0.029 11.880  6 0.689 1.251 0.517 0.539 1.068 0.386 5.680  7 1.033 0.790 0.345 0.339 0.846 0.340 6.307  8 1.202 0.668 0.328 0.341 2.377 -0.045 11.407  9 0.863 0.967 0.238 0.277 1.926 -0.149 10.381  10 1.202 0.667 0.327 0.324 1.081 0.449 9.658  11 0.243 3.679 1.245 1.412 1.798 0.495 2.219  12 0.251 3.629 0.850 1.293 3.005 0.035 3.774  13 0.243 3.679 1.094 1.582 2.338 0.334 2.235  14 0.156 5.065 0.686 1.364 2.660 0.190 1.638  15 0.156 5.065 0.907 1.576 2.998 0.092 3.242  16 0.087 9.588 2.295 2.903 3.431 0.298 4.394Figure 2
Mercury intrusion capillary pressure curves of 16 carbonate core samples.


## 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] (2)N>r∝r-Dfwhere r is the pore radius (characteristic length); N(>r) is pore number with pore radius larger than r; and Df is fractal dimension. There are several different methods to calculate pore number N(>r) 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 radiusr can be calculated with the volume of mercury intrusion as given by(3)nr=△VHgrπr2lwhere n(r) is the number of capillary tubes with radius r; △VHg(r) is the single volume of mercury intrusion at radius r; and l is the core length. Then the cumulative number N(>r) can be calculated as(4)N>r=∫rrmaxnrdr.According to (2), fractal dimension Df can be determined from the slope of the line of lg⁡N(>r) and lg⁡r in a log-log plot.


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

Instead of using single volume of mercury intrusion△VHg(r), Li [9] used the cumulative volume of mercury intrusion VHg to calculate the number of filled capillary tubes N(r) as given by(5)Nr=VHgπr2l.Substituting (5) into (2),(6)VHgπr2l∝r-Df.The core length l is constant, and therefore (6) can be simplified as(7)VHg∝r2-Df.As VHg can be expressed with mercury saturation SHg, and pore radius r can be expressed with mercury intrusion capillary pressure Pc, (7) can be expressed as(8)SHg∝pc-2-Df.Then fractal dimension Df can be calculated from the lg⁡SHg-lg⁡pc 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 is1<Df<2. 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 2<Df<3.


### 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 as(9)dSHgdpc∝pc-4-Df.The fractal dimension Df was calculated by plotting dSHg/dpc verse pc 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](10)ln⁡Wnrn2=Dfln⁡Vn1/3rn+Cwhere Wn is the cumulative surface energy during the mercury intrusion process; rn is pore radius; and Vn is the intrusion volume of mercury at stage n.The calculated fractal dimensions using method III and IV reflect the volume heterogeneity and surface heterogeneity of carbonate pores, respectively [12, 32].


## 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 radiusr can be calculated with the volume of mercury intrusion as given by(3)nr=△VHgrπr2lwhere n(r) is the number of capillary tubes with radius r; △VHg(r) is the single volume of mercury intrusion at radius r; and l is the core length. Then the cumulative number N(>r) can be calculated as(4)N>r=∫rrmaxnrdr.According to (2), fractal dimension Df can be determined from the slope of the line of lg⁡N(>r) and lg⁡r in a log-log plot.


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

Instead of using single volume of mercury intrusion△VHg(r), Li [9] used the cumulative volume of mercury intrusion VHg to calculate the number of filled capillary tubes N(r) as given by(5)Nr=VHgπr2l.Substituting (5) into (2),(6)VHgπr2l∝r-Df.The core length l is constant, and therefore (6) can be simplified as(7)VHg∝r2-Df.As VHg can be expressed with mercury saturation SHg, and pore radius r can be expressed with mercury intrusion capillary pressure Pc, (7) can be expressed as(8)SHg∝pc-2-Df.Then fractal dimension Df can be calculated from the lg⁡SHg-lg⁡pc 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 is1<Df<2. 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 2<Df<3.


## 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 as(9)dSHgdpc∝pc-4-Df.The fractal dimension Df was calculated by plotting dSHg/dpc verse pc 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](10)ln⁡Wnrn2=Dfln⁡Vn1/3rn+Cwhere Wn is the cumulative surface energy during the mercury intrusion process; rn is pore radius; and Vn is the intrusion volume of mercury at stage n.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 asDf1Df2, Df3, and Df4, 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 lg⁡SHg-lg⁡pc breaks into two segments, and the linear fitting result with one straight line is poor.Figure 3
Fractal dimension calculation from the slope of straight line in log-log plots with different methods ((a) method I-2D capillary tube model; (b) method II-3D capillary tube model; (c) method III-geometry model; (d) method IV-thermodynamics model).
(a) (b) (c) (d)Table3 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 Df1 vary from 1.267 to 1.853, indicating that Df1 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 Df2, Df3, and Df4 reflect fractal characteristics of pore structures in three-dimensional space. Overall, Df3 and Df4 are close to each other, but larger than Df2.Table 3
The fractal dimensions of entire pores using different fractal models.
Core No. Fractal dimensions Method IDf1 Method IIDf2 Method IIIDf3 Method IVDf4 1 1.853 2.296 2.777 2.679  2 1.640 2.253 2.428 2.574  3 1.585 2.335 2.433 2.549  4 1.441 2.367 2.274 2.451  5 1.267 2.298 2.059 2.427  6 1.480 2.338 2.356 2.430  7 1.518 2.448 2.325 2.467  8 1.559 2.281 2.413 2.537  9 1.674 2.355 2.504 2.594  10 1.614 2.367 2.503 2.547  11 1.602 2.231 2.523 2.537  12 1.562 2.213 2.678 2.592  13 1.509 2.199 2.624 2.557  14 1.772 2.276 2.858 2.683  15 1.733 2.256 2.858 2.680  16 1.774 2.215 2.763 2.689  Average 1.599 2.295 2.532 2.562To 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 oflg⁡SHg-lg⁡pc 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.Figure 4
Fractal dimensions calculation for nano- and micropores with method II using two-straight-line fitting ((a) no. 2 core sample; (b) no. 6 core sample).
(a) (b)Table4 presents the calculated fractal dimensions of microscale pores Df2m and nanoscale pores Df2n. The fractal dimensions of microscale pores Df2m vary from 3.006 to 5.044 and are much larger than the fractal dimensions of nanoscale pores Df2n, 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 Df2m 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 Df2m beyond 3.Table 4
The fractal dimensions of microscale pores and nanoscale pores calculated with method II using two-straight-line fitting.
Core sampleNo. Microscale pores Nanoscale pores Pore size range(μm) Fractal dimensionDfm Pore size range(μm) Fractal dimensionDfn 1 0.7279-5.3582 3.259 0.0027-0.6226 2.084  2 0.5412-3.0502 3.428 0.0027-0.3549 2.066  3 0.2145-0.8553 3.616 0.0027-0.1443 2.057  4 0.2716-1.0746 4.078 0.0027-0.2152 2.046  5 0.2693-0.7197 4.482 0.0027-0.2163 2.031  6 0.3598-1.0786 4.927 0.0027-0.2706 2.037  7 0.2159-0.7192 5.044 0.0027-0.1434 2.028  8 0.2153-0.6184 3.845 0.0027-0.1431 2.047  9 0.1429-0.8607 3.131 0.0027-0.1077 2.047  10 0.2156-0.6181 4.824 0.0027-0.1432 2.039  11 0.7289-3.0633 4.117 0.0027-0.6229 2.055  12 0.5388-2.9625 3.061 0.0027-0.3609 2.079  13 0.8634-3.0627 3.512 0.0027-0.7205 2.094  14 0.7162-4.7693 3.126 0.0027-0.6114 2.122  15 0.8642-4.7698 3.200 0.0027-0.7211 2.111  16 1.4593-8.5343 3.006 0.0027-1.0897 2.079Figure5 shows the relationship between the fractal dimensions of the entire microscale and nanoscale pores. The fractal dimensions of microscale pores Df2m strongly agree with the fractal dimensions of the entire pores Df2, but poorly correlate with the fractal dimensions of nanoscale pores Df2n, and with Df2m increasing, Df2n 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.Figure 5
The relationship between fractal dimensions of entire poresDf2, microscale pores Df2m, and nanoscale pores Df2n ((a) the relationship between fractal dimensions of the entire pores Df2 and fractal dimensions of microscale pores Df2m; (b) the relationship between fractal dimensions of microscale pores Df2m and fractal dimensions of nanoscale pores Df2n).
(a) (b)Lai et al. [10] also used lg⁡SHg-lg⁡pc 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,Df1, Df2, Df3, and Df4, were evaluated. Figure 6 shows the correlations between the air permeability Kair of core samples and the fractal dimensions calculated from the four different methods. In contrast, with Df2 increasing, Kair decreases logarithmically; however, Df1, Df3, and Df4 strongly agree with Kair. According to Liu et al. [33], for the fractal capillary tube model, the poor correlation between permeability and fractal dimension is reasonable, which means Df2 is more suitable for fractal analysis of carbonate pore structures and petrophysical properties. As depicted in Figure 7, with Df2 increasing, displacement pressure Pd and tortuosity τ increase and pore radius r50 and r35 decrease, indicating pore structures become more complex.Figure 6
The correlation between the air permeabilityKair and the fractal dimensions calculated from different methods.
(a) (b) (c) (d)Figure 7
Relationship between fractal dimensions of the entire poresDf2 and some pore structure parameters and petrophysical properties of carbonate core samples ((a) Df2 and displacement pressure pd; (b) Df2 and r50; (c) Df2 and r35; (d) Df2 and tortuosity τ).
(a) (b) (c) (d)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 Df2 and RQI and FZI. The results after the investigation showed that reservoir quality becomes poorer when RQI and FZI decrease with increasing Df2. 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.Figure 8
Relationship between fractal dimensions of the entire poresDf2 and RQI (a) and FZI (b).
(a) (b)Figure 9
HFU identification with the improved method proposed by Mirzaei-Paiaman et al.[27].The fractal characteristics of the microscale pores and nanoscale pores were analysed. Figures10 and 11 present the fractal analysis of carbonate petrophysical properties using Df2m and Df2n, respectively. Comparatively, fractal analysis using Df2 in Figure 7 and Df2m provided similar results, unlike Df2n which gave different result, and it can be inferred that fractal dimensions of nanoscale pores Df2n cannot be used for fractal analysis.Figure 10
Relationship between fractal dimensions of microscale poresDf2m and some pore structure parameters and petrophysical properties of carbonate core samples ((a) Df2m and air permeability Kair; (b) Df2m and displacement pressure pd; (c) Df2m and r50; (d) Df2m and r35; (e) Df2m and sorting coefficient Sp; (f) Df2m and tortuosity τ).
(a) (b) (c) (d) (e) (f)Figure 11
Relationship between fractal dimensions of nanoscale poresDf2n and some pore structure parameters and petrophysical properties of carbonate core samples ((a) Df2n and air permeability Kair; (b) Df2n and displacement pressure pd; (c) Df2n and r50; (d) Df2n and r35; (e) Df2n and sorting coefficient Sp; (f) Df2n and tortuosity τ).
(a) (b) (c) (d) (e) (f)The distributions of pore volume and permeability contribution for microscale pores and nanoscale pores were calculated. Figure12 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 μm and 1 μm, 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.Figure 12
The distribution of pore volume and permeability contribution of the no. 2 core sample.Table5 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.Table 5
The total pore volume and permeability contribution of microscale pores and nanoscale pores.
Core No. Microscale pores Nanoscale pores Total pore volumeSHgm (%) Total permeability contributionKcm (%) Total pore volumeSHgn (%) Total permeability contributionKcn (%) 1 50.00 97.93 44.69 2.07  2 55.00 99.21 39.52 0.79  3 55.81 96.34 36.55 3.66  4 60.83 95.29 35.40 4.71  5 64.73 93.41 28.16 6.60  6 64.34 94.00 33.01 6.00  7 68.93 94.85 28.65 5.15  8 54.04 93.33 35.75 6.67  9 61.16 97.71 28.97 2.29  10 61.51 92.98 34.10 7.02  11 61.45 96.49 35.10 3.51  12 49.84 98.44 41.61 1.56  13 45.31 95.96 47.54 4.04  14 41.45 98.01 52.26 1.99  15 41.77 97.04 51.20 2.96  16 46.04 98.41 43.73 1.59  Min 41.45 92.98 28.16 0.79  Max 68.93 99.21 52.26 7.02  Average 55.14 96.21 38.52 3.79Figure 13
Relationship between fractal dimension and permeability contribution ((a) microscale pores; (b) nanoscale pores).
(a) (b)


## 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] (11)log⁡r35=0.732+0.588log⁡K-0.864log⁡ϕwhere r35 is the pore radius with 35% mercury saturation and ϕ 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.Figure 14
Comparison between the measured air permeability and the calculated permeability using Winland permeability model.The fractal permeability model proposed by Yu and Cheng is based on the tortuous fractal capillary tube model which can be expressed as [23](12)K=π128L01-DTADf3+DT-Dfλmax3+DTwhere L0 is the representative length; A is the total section area and A=L02; DT is the tortuosity fractal dimension, and it can be calculated with the average tortuosity which can be measured from HPMI; and λmax 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 Df1 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.Figure 15
Comparison between the measured air permeability and the calculated permeability using fractal permeability model.


## 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.



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