# The Effect of Particle Size on the Interpretation of Pore Structure of Shale by N2 Adsorption

Authors: Chengfu Lyu; Xinmao Zhou; Xuesong Lu; Ying Zhang; Chao Li; Qianshan Zhou; Zhaotong Sun; Guojun Chen
Journal: Geofluids
 (2021)

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## Abstract

Gas adsorption experiments are becoming one of the most common methods to quantify and analyze the pore structures of shale samples in the petroleum industry. In this regard, particle size of the specimen plays an important role in the results that could ultimately affect the pore structure interpretation. Hence, in this study, five shale samples at different thermal maturity levels are picked, and all are crushed into different groups of particle sizes: less than 40 mesh (<375μm), less than 60 mesh (<250 μm), less than 80 mesh (<187.5 μm), and less than 100 mesh (<150 μm). Next, N2 adsorption is used to characterize the pore structures of the samples within different particle sizes. Furthermore, to interpret the data, several attributes such as the pore volume, surface area, fractal dimension (from the fractal analysis), and heterogeneity index (from the multifractal analysis), are studied and compared between the samples and particle size intervals to provide us with the effect that particle size could have on the pore structure analysis. The results showed that as the particle size varies, the pore structures of the shale samples could get affected. Based on the comparison of the results, it is recommended that a suitable particle size for the shale pore structure characterization in N2 adsorption experiments should be less than 60 mesh (<250 μm).


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## Body

## 1. Introduction

Understanding the pore structures is vital to assess the hydrocarbon bearing capacity of unconventional reservoirs, particularly shale formations. The nanometer to micrometer pores in such reservoirs (e.g., shale reservoirs) could act as the flow path for hydrocarbons, deciding the transportation ability during production [1–8]. Due to this, understanding the pore structures can reveal both pore space and flow abilities of the formation. The gas adsorption method especially N2 gas adsorption, which can measure the pore sizes less than 200 nm, has become one of the most popular methods to characterize the pore structures in shale formations [9–12]. Due to the low permeability nature of shale samples, using larger piece aliquots to perform the N2 adsorption could cause low diffusion of gas molecules to access the pores. Therefore, crushing the samples, to create accessibility for gas molecules to enter the pores in order to achieve the equilibrium in a reasonable time span, is recommended by the Gas Research Institute [13–16].Although the above general guide line, the appropriate range of particle size for the gas adsorption experiment, N2 adsorption in this study, is still a topic up for argument, and different particle size recommendations have been given, Kulia and Prassad [17] found that the particle size for gas adsorption should be less than 425 μm (40 mesh). Other studies have found that using a 60 mesh sieve for crushed particles would be more suitable [18–21]. Tian et al. [22] and Shao et al. [23] suggested using a particle size between 178 and 250 μm (60-80 mesh) to study the pore structures. For determining the accurate particle size necessary for gas adsorption, Chen et al. [24] compared the pore structures of samples with varying particle sizes: 4 mesh (<4760 μm), 20 mesh (<841 μm), and 60 mesh (<250 μm). Although a good range of particle sizes were studied, they did not investigate the impact of finer particle sizes on pore structures (e.g., <150 μm (100 mesh)).Mastalera et al. [25] compared the pore structures by assigning different particle sizes: from chunks to the particle size less than 0.063 mm (230 mesh). Hazra et al. [26] divided the particle sizes into five different groups and studied their pore structures in a larger range of sizes, ranging from <1 mm (18 mesh) to <53 μm (270 mesh). In their study, they regarded the particle size ranging from 212 μm (70 mesh) to 75 μm (200 mesh) as one group and compared the pore structures with the pore structures from other particle size groups (i.e., groups with particle size from 1 mm to 500 μm, 212-75 μm, and 75-53 μm, respectively). Mastalerz et al. [25] studied the effect of the particle size on the equilibrium time and equilibration pressure and found that the sample with large particle size needs more time to equilibrate. However, if the particle size is too fine, more pores could be generated; the nanopore information could be distorted [26]. Considering studies by Hazra et al. [26] and Mastalera et al.[25], the particle size intervals that were analyzed was 70-200 mesh and 60-200 mesh which is too wide. Hence, the effect of shorter ranges of particle size (e.g., 40-100 mesh) on pore structures is still ambiguous and requires further investigations.Based on what was said above, in this study, five shale samples were collected, and the effect of smaller particle size on their pore structures is analyzed. Shale samples were crushed in different particle size categories (40 mesh, 60 mesh, 80 mesh, and 100 mesh, respectively), and their pore structures were examined by N2 adsorption to reveal how the size of particles would impact the pore structure results and their interpretation.


## 2. Samples and Methods

### 2.1. Sample Preparation

Five shale samples were collected, and first were characterized for their organic matter content and mineralogy. The mineral composition shows that clay and quartz are the two major components in these shale samples where the clay mineral content ranges from 17.5 wt.% (sample 4) to 48.0 wt.% (sample 2), and the quartz content varies from 32.4 wt.% (sample 2) to 54.4 wt.% (sample 3). Feldspar, dolomite, and pyrite with smaller percentages also exist in these shale samples. The total organic content of these samples was measured from 13.21 wt.% (sample 4) to 7.50 wt.% (sample 5), explaining that these shale samples are rich in organic matter. TheTmax of sample 1-sample 5 was 435, 430, 433, 437, and 411°C, respectively, indicating that sample 2 and sample 5 are thermally immature while the rest are in the early oil window stage.


### 2.2. Nitrogen Gas Adsorption Analysis

Next, samples were homogenized into 4 separate pieces, and then, each piece was further crushed into different mesh sieves (40 mesh for part 1, 60 mesh for part 2, 80 mesh for part 3, and 100 mesh for part 4, respectively). Homogenization is the process of breaking several pieces from a single intact sample for different studies in order to limit the bias of the mineral compositions and reducing the effect of variations that could be in different representative aliquot. Next, around 2 grams from each mesh was further dried at 105°C for 8 hours before the N2 adsorption analysis. The instrument that was utilized in this study is the Micromeritics® Tristar II apparatus and the N2 adsorption volume was measured over the relative equilibrium adsorption pressure (P/P0) ranging from 0.01 to 0.99.


### 2.3. Fractal Analysis and Multifractal Analysis

The fractal theory is a powerful tool to quantify the complexity of the pore structures of porous medium including geomaterials. The FHH (Frenkel-Halsey-Hill) model is one of the most mainstream fractal models that has been commonly used for shale rock studies [27] and is expressed as:
(1)lnV=C+D−3lnln1p/p0,where V is the total volume of the adsorption, P is the equilibrium pressure, P0 is the saturated vapour pressure of the adsorption, and D is the fractal dimension. C is the constant which can be derived from curve fitting. Once the adsorption data (relative pressure P/P0 and the adsorption quantity) is quantified, through Equation (1), the fractal dimension from the slope of the linear part of the curves can be obtained.It is important to note that the multifractal method which is an extension of the fractal analysis usually uses the box counting method for dimensional analysis. In this method, a set of boxes with equal length (ε) is used to cover the pore size distribution curve. Then, through density functional theory, a power law relationship as follows is defined ([28, 29]):
(2)Piε~εαi,where I is the index for each box laid over the set for the ε and Piε is the probability of mass at I relative to the total mass for a box size. Finally, Niε is the number of mass in any box with the total mass in all boxes for this ε as below:
(3)NT=∑i=1NεNiε,Piε=NiεNT−.q is defined as an arbitrary range of values (in this study, q is from -10 to 10) to use as exponents for distorting the data set. For the box size ε, the sum of all mass probabilities distorted can be calculated as [30]:
(4)Iqε=∑i=1NεPiεq.For the box sizeε, the magnitude of distorted mass probability at a specific box compared to the distorted sum of all boxes can be quantified as:
(5)uiq,ε=PiεqIqε,where Iqε can be expressed as [31]:
(6)Iqε~ετq.Then,τq can be calculated as:
(7)τq=limε→0logIqεlogε.Dq will be:
(8)Dq=τqq−1=limε→0logIqε/logεq−1.Ifq=1,
(9)D1=limε→0∑i=1Nεpiεlnpiεlnε.


## 2.1. Sample Preparation

Five shale samples were collected, and first were characterized for their organic matter content and mineralogy. The mineral composition shows that clay and quartz are the two major components in these shale samples where the clay mineral content ranges from 17.5 wt.% (sample 4) to 48.0 wt.% (sample 2), and the quartz content varies from 32.4 wt.% (sample 2) to 54.4 wt.% (sample 3). Feldspar, dolomite, and pyrite with smaller percentages also exist in these shale samples. The total organic content of these samples was measured from 13.21 wt.% (sample 4) to 7.50 wt.% (sample 5), explaining that these shale samples are rich in organic matter. TheTmax of sample 1-sample 5 was 435, 430, 433, 437, and 411°C, respectively, indicating that sample 2 and sample 5 are thermally immature while the rest are in the early oil window stage.


## 2.2. Nitrogen Gas Adsorption Analysis

Next, samples were homogenized into 4 separate pieces, and then, each piece was further crushed into different mesh sieves (40 mesh for part 1, 60 mesh for part 2, 80 mesh for part 3, and 100 mesh for part 4, respectively). Homogenization is the process of breaking several pieces from a single intact sample for different studies in order to limit the bias of the mineral compositions and reducing the effect of variations that could be in different representative aliquot. Next, around 2 grams from each mesh was further dried at 105°C for 8 hours before the N2 adsorption analysis. The instrument that was utilized in this study is the Micromeritics® Tristar II apparatus and the N2 adsorption volume was measured over the relative equilibrium adsorption pressure (P/P0) ranging from 0.01 to 0.99.


## 2.3. Fractal Analysis and Multifractal Analysis

The fractal theory is a powerful tool to quantify the complexity of the pore structures of porous medium including geomaterials. The FHH (Frenkel-Halsey-Hill) model is one of the most mainstream fractal models that has been commonly used for shale rock studies [27] and is expressed as:
(1)lnV=C+D−3lnln1p/p0,where V is the total volume of the adsorption, P is the equilibrium pressure, P0 is the saturated vapour pressure of the adsorption, and D is the fractal dimension. C is the constant which can be derived from curve fitting. Once the adsorption data (relative pressure P/P0 and the adsorption quantity) is quantified, through Equation (1), the fractal dimension from the slope of the linear part of the curves can be obtained.It is important to note that the multifractal method which is an extension of the fractal analysis usually uses the box counting method for dimensional analysis. In this method, a set of boxes with equal length (ε) is used to cover the pore size distribution curve. Then, through density functional theory, a power law relationship as follows is defined ([28, 29]):
(2)Piε~εαi,where I is the index for each box laid over the set for the ε and Piε is the probability of mass at I relative to the total mass for a box size. Finally, Niε is the number of mass in any box with the total mass in all boxes for this ε as below:
(3)NT=∑i=1NεNiε,Piε=NiεNT−.q is defined as an arbitrary range of values (in this study, q is from -10 to 10) to use as exponents for distorting the data set. For the box size ε, the sum of all mass probabilities distorted can be calculated as [30]:
(4)Iqε=∑i=1NεPiεq.For the box sizeε, the magnitude of distorted mass probability at a specific box compared to the distorted sum of all boxes can be quantified as:
(5)uiq,ε=PiεqIqε,where Iqε can be expressed as [31]:
(6)Iqε~ετq.Then,τq can be calculated as:
(7)τq=limε→0logIqεlogε.Dq will be:
(8)Dq=τqq−1=limε→0logIqε/logεq−1.Ifq=1,
(9)D1=limε→0∑i=1Nεpiεlnpiεlnε.


## 3. Results and Discussions

### 3.1. The Adsorption Isotherms of the Shale Samples with Different Particle Sizes

The adsorption isotherms that are obtained from the shale samples crushed into different particle sizes can be seen in Figure1. Although these shale samples are at different thermal maturity levels, the adsorption isotherms still shows some similarities considering their difference in particle size. For instance, in samples with smaller particle size, under the same relative pressure, the adsorption quantity is recorded higher. This indicates that the adsorption capacity of the sample has increased. Considering sample 1 as an example, as the particle size decreased from 40 mesh (<375 μm) to 100 mesh (<150 μm), the adsorption volume has increased from 4.04 cm3/g to 8.04 cm3/g. Based on the recommendations by the International Union of Pure and Applied Chemistry (IUPAC), the shape of the adsorption isotherms of the samples with different particle sizes all belong to type IV, showing the existence of mesopores (2-50 nm) and macropores (>50 nm) [32]. Moreover, hysteresis loop is observed between the adsorption and the desorption isotherm branch in all samples. The shape of the hysteresis loop points out that there is an abundance of slit-shape pores in these shale samples although being different in particle size [32]. Ultimately, as the particle size has changed, the adsorption isotherms are somehow affected which can be regarded as alteration in sample microstructure.Figure 1
Adsorption isotherms of shale samples with different particle sizes.


### 3.2. Comparison of the Pore Structures of the Sample with Different Particle Sizes

The surface area is calculated from the Brunauer-Emmett-Teller (BET) method [33], and the pore volume of the shale samples is derived from the BJH (Barrett, Joyner, and Halenda) method [34], which is based on the Kelvin equation. Figure 2(a) explains, as the particle size decreases (the mesh number increases), regardless of the sample, the BET-calculated surface area increases. In all samples, an abrupt change in the BET-calculated surface area when the particle size is decreased from <187.5 μm (80 mesh) to <150 μm (100 mesh) is found. This could be due to either creation of new surfaces or connection of isolated pores. The BET surface area of the sample in the 100 mesh interval is around twice the BET surface area of the sample in the 40 mesh interval. In addition, the BET surface area of sample 1 has increased from 1.82 m2/g (40 mesh) to 3.83 m2/g (100 mesh). Similarly, the pore volume also has increased as the particle size is decreased (Figure 2(b)). Considering sample 1 as an example, the pore volume in the 100 mesh particle size is 0.0074 cm3/g which is twice more than the pore volume in the 100 mesh particle size (0.0036 cm3/g). Furthermore, the sample that is crushed to 40 mesh size was measured with the smallest BET surface area and pore volume. Figure 2(c) exhibits the average pore size of the samples based on their particle sizes. It is deduced as the particle size decreases from 375 μm (40 mesh) to 250 μm (60 mesh), the average pore size shows an apparent change. On the other hand, further change in particle size beyond 250 μm (60 mesh) to 100 mesh, the average pore size does not show a meaningful trend, although fluctuations are observed.Figure 2
The effect of the particle size on the pore parameters.
(a)
BET surface area(b)
Pore volume(c)
Average pore sizeThe pore size distribution (PSD) curves of samples with different particle sizes are presented in Figure3 where a multimodal distribution is seen in all samples. Although there are discrepancies in thermal maturity, these samples still have similar peaks at certain pore sizes (e.g., the peak with the pore size between 30 and 40 nm). The importance of thermal maturity is due to the fact that at higher maturities, organic matter should have developed more organic pores and it is expected that change to be revealed in adsorption tests. A closer look explains that samples 1, 2, 3, and 5 have a discernable peak around 3 nm which is absent in sample 4. Liu et al. [35] applied the Gaussian distribution to distinguish the presence of various pore groups in shale samples based on the peaks of different pore sizes in a PSD curve. Considering their conclusions, Liu et al. [35], we further deconvoluted different possible pore groups in the PSD curve. On the one hand, it is found that as the particle size decreases, the number of pores in each group has increased following the Liu et al. [35] method. For example, in sample 1, highest frequencies of PSD of the 40 mesh particles are around 3, 5, 9, 17, and 34 nm, respectively, but staying the same in smaller particle sizes or higher meshes.Figure 3
The pore size distribution (PSD) of the samples with different particle sizes.


### 3.3. Comparison of the Fractal Dimensions of Samples with Different Particle Sizes

#### 3.3.1. Fractal Dimension

Based on the results by the FHH model, we calculated the fractal dimensions of all the samples based on different particle sizes. Figure4 displays an example of the fractal analysis of sample 1 with different particle sizes. The plots illustrates that a linear correlation exists between the lnQ and lnlnP0/P, indicating that the pore size distributions of sample 1 with different particle sizes all follow the fractal behavior. The summary of the fractal dimensions of all samples while considering variations in particle sizes can be seen in Figure 5. The fractal dimension values are all located in a reasonable range of fractal values (between 2 and 3) [36, 37]. However, comparing different samples, the effect that the particle size has on the fractal dimension results is not similar. In this regard, it is observed as the particle size decreases, the fractal dimension of sample 1, 2, 4, and 5 does not change significantly (Figure 4), whereas, in sample3, as the particle size decreases from <375 μm (40 mesh) to <250 μm (60 mesh), the fractal dimension demonstrates an evident decrease. On the contrary, as the particle size continues to decrease from <250 μm (60 mesh) to <150 μm (100 mesh), the fractal dimension does not change notably. Additionally, in a particle size range of less than 250 μm (60 mesh), the fractal dimensions comparing five samples follow the order: sample 3 > sample 1 > sample 4 > sample 2 > sample 5. This means as the particle size decreases, up to <150 μm, the order of fractal dimension change among these five samples remains the same (sample 3 has the largest fractal dimension and sample 5 the smallest value).Figure 4
The effect of the particle size on the fractal dimensions of the shale sample (sample 1).Figure 5
Summary of the fractal values of the samples with different particle sizes.


#### 3.3.2. Multifractal Dimension

Based on Section2.3, we calculated the generalized dimensions Dq from N2 adsorption of all the shale samples with different particle sizes. Here, in Dq values (in this study, q is from -10 to 10), D0 represents the singularity of nonempty boxes containing some value of porosity under successive finer partitions. D1 can characterize the concentration degree of the pore size distribution along the pore size intervals. D2 accounts for the behavior of the second sampling moments [31]. The Dq value of sample 1 with different particle sizes can be seen in Figure 6 where D0 > D1 > D2 order of change exists in sample 1. This is indicating that the pore size distribution of sample 1, considering the existence of different particle sizes, presents the multifractal behavior [31]. The difference between the D−10andD10 can be used to quantify the heterogeneity of the pore size distribution of the sample [38]. Based on this, the larger D−10–D10 value corresponds to a higher degree of heterogeneity in the pore size distribution. The D−10–D10 values of all studied samples with different particle sizes are calculated, and the results can be seen in Figure 7. Based on the results, as the particle size decreases, the heterogeneity index (D−10–D10) fluctuates with a slight decreasing trend. This infers that the change in particle size has some impact on the heterogeneity that is present in the pore structures. Comparing all samples, the heterogeneity index when particle size is <250 μm (60 mesh) is very close to the heterogeneity index when the particle size is <187.5 μm (80 mesh). Consequently, as the particle size further decreases to <150 μm (100 mesh), the heterogeneity index decreases, regardless of the sample number. In sample 1 as an example, the heterogeneity index decreases from 1.19 to 1.15 as the particle size is reduced from <187.5 μm (80 mesh) to <150 μm (100 mesh).Figure 6
The correlation betweenDq and q of sample 1.Figure 7
The effect of the particle size on the heterogeneity of the pore size distribution.


## 3.1. The Adsorption Isotherms of the Shale Samples with Different Particle Sizes

The adsorption isotherms that are obtained from the shale samples crushed into different particle sizes can be seen in Figure1. Although these shale samples are at different thermal maturity levels, the adsorption isotherms still shows some similarities considering their difference in particle size. For instance, in samples with smaller particle size, under the same relative pressure, the adsorption quantity is recorded higher. This indicates that the adsorption capacity of the sample has increased. Considering sample 1 as an example, as the particle size decreased from 40 mesh (<375 μm) to 100 mesh (<150 μm), the adsorption volume has increased from 4.04 cm3/g to 8.04 cm3/g. Based on the recommendations by the International Union of Pure and Applied Chemistry (IUPAC), the shape of the adsorption isotherms of the samples with different particle sizes all belong to type IV, showing the existence of mesopores (2-50 nm) and macropores (>50 nm) [32]. Moreover, hysteresis loop is observed between the adsorption and the desorption isotherm branch in all samples. The shape of the hysteresis loop points out that there is an abundance of slit-shape pores in these shale samples although being different in particle size [32]. Ultimately, as the particle size has changed, the adsorption isotherms are somehow affected which can be regarded as alteration in sample microstructure.Figure 1
Adsorption isotherms of shale samples with different particle sizes.


## 3.2. Comparison of the Pore Structures of the Sample with Different Particle Sizes

The surface area is calculated from the Brunauer-Emmett-Teller (BET) method [33], and the pore volume of the shale samples is derived from the BJH (Barrett, Joyner, and Halenda) method [34], which is based on the Kelvin equation. Figure 2(a) explains, as the particle size decreases (the mesh number increases), regardless of the sample, the BET-calculated surface area increases. In all samples, an abrupt change in the BET-calculated surface area when the particle size is decreased from <187.5 μm (80 mesh) to <150 μm (100 mesh) is found. This could be due to either creation of new surfaces or connection of isolated pores. The BET surface area of the sample in the 100 mesh interval is around twice the BET surface area of the sample in the 40 mesh interval. In addition, the BET surface area of sample 1 has increased from 1.82 m2/g (40 mesh) to 3.83 m2/g (100 mesh). Similarly, the pore volume also has increased as the particle size is decreased (Figure 2(b)). Considering sample 1 as an example, the pore volume in the 100 mesh particle size is 0.0074 cm3/g which is twice more than the pore volume in the 100 mesh particle size (0.0036 cm3/g). Furthermore, the sample that is crushed to 40 mesh size was measured with the smallest BET surface area and pore volume. Figure 2(c) exhibits the average pore size of the samples based on their particle sizes. It is deduced as the particle size decreases from 375 μm (40 mesh) to 250 μm (60 mesh), the average pore size shows an apparent change. On the other hand, further change in particle size beyond 250 μm (60 mesh) to 100 mesh, the average pore size does not show a meaningful trend, although fluctuations are observed.Figure 2
The effect of the particle size on the pore parameters.
(a)
BET surface area(b)
Pore volume(c)
Average pore sizeThe pore size distribution (PSD) curves of samples with different particle sizes are presented in Figure3 where a multimodal distribution is seen in all samples. Although there are discrepancies in thermal maturity, these samples still have similar peaks at certain pore sizes (e.g., the peak with the pore size between 30 and 40 nm). The importance of thermal maturity is due to the fact that at higher maturities, organic matter should have developed more organic pores and it is expected that change to be revealed in adsorption tests. A closer look explains that samples 1, 2, 3, and 5 have a discernable peak around 3 nm which is absent in sample 4. Liu et al. [35] applied the Gaussian distribution to distinguish the presence of various pore groups in shale samples based on the peaks of different pore sizes in a PSD curve. Considering their conclusions, Liu et al. [35], we further deconvoluted different possible pore groups in the PSD curve. On the one hand, it is found that as the particle size decreases, the number of pores in each group has increased following the Liu et al. [35] method. For example, in sample 1, highest frequencies of PSD of the 40 mesh particles are around 3, 5, 9, 17, and 34 nm, respectively, but staying the same in smaller particle sizes or higher meshes.Figure 3
The pore size distribution (PSD) of the samples with different particle sizes.


## 3.3. Comparison of the Fractal Dimensions of Samples with Different Particle Sizes

### 3.3.1. Fractal Dimension

Based on the results by the FHH model, we calculated the fractal dimensions of all the samples based on different particle sizes. Figure4 displays an example of the fractal analysis of sample 1 with different particle sizes. The plots illustrates that a linear correlation exists between the lnQ and lnlnP0/P, indicating that the pore size distributions of sample 1 with different particle sizes all follow the fractal behavior. The summary of the fractal dimensions of all samples while considering variations in particle sizes can be seen in Figure 5. The fractal dimension values are all located in a reasonable range of fractal values (between 2 and 3) [36, 37]. However, comparing different samples, the effect that the particle size has on the fractal dimension results is not similar. In this regard, it is observed as the particle size decreases, the fractal dimension of sample 1, 2, 4, and 5 does not change significantly (Figure 4), whereas, in sample3, as the particle size decreases from <375 μm (40 mesh) to <250 μm (60 mesh), the fractal dimension demonstrates an evident decrease. On the contrary, as the particle size continues to decrease from <250 μm (60 mesh) to <150 μm (100 mesh), the fractal dimension does not change notably. Additionally, in a particle size range of less than 250 μm (60 mesh), the fractal dimensions comparing five samples follow the order: sample 3 > sample 1 > sample 4 > sample 2 > sample 5. This means as the particle size decreases, up to <150 μm, the order of fractal dimension change among these five samples remains the same (sample 3 has the largest fractal dimension and sample 5 the smallest value).Figure 4
The effect of the particle size on the fractal dimensions of the shale sample (sample 1).Figure 5
Summary of the fractal values of the samples with different particle sizes.


### 3.3.2. Multifractal Dimension

Based on Section2.3, we calculated the generalized dimensions Dq from N2 adsorption of all the shale samples with different particle sizes. Here, in Dq values (in this study, q is from -10 to 10), D0 represents the singularity of nonempty boxes containing some value of porosity under successive finer partitions. D1 can characterize the concentration degree of the pore size distribution along the pore size intervals. D2 accounts for the behavior of the second sampling moments [31]. The Dq value of sample 1 with different particle sizes can be seen in Figure 6 where D0 > D1 > D2 order of change exists in sample 1. This is indicating that the pore size distribution of sample 1, considering the existence of different particle sizes, presents the multifractal behavior [31]. The difference between the D−10andD10 can be used to quantify the heterogeneity of the pore size distribution of the sample [38]. Based on this, the larger D−10–D10 value corresponds to a higher degree of heterogeneity in the pore size distribution. The D−10–D10 values of all studied samples with different particle sizes are calculated, and the results can be seen in Figure 7. Based on the results, as the particle size decreases, the heterogeneity index (D−10–D10) fluctuates with a slight decreasing trend. This infers that the change in particle size has some impact on the heterogeneity that is present in the pore structures. Comparing all samples, the heterogeneity index when particle size is <250 μm (60 mesh) is very close to the heterogeneity index when the particle size is <187.5 μm (80 mesh). Consequently, as the particle size further decreases to <150 μm (100 mesh), the heterogeneity index decreases, regardless of the sample number. In sample 1 as an example, the heterogeneity index decreases from 1.19 to 1.15 as the particle size is reduced from <187.5 μm (80 mesh) to <150 μm (100 mesh).Figure 6
The correlation betweenDq and q of sample 1.Figure 7
The effect of the particle size on the heterogeneity of the pore size distribution.


## 3.3.1. Fractal Dimension

Based on the results by the FHH model, we calculated the fractal dimensions of all the samples based on different particle sizes. Figure4 displays an example of the fractal analysis of sample 1 with different particle sizes. The plots illustrates that a linear correlation exists between the lnQ and lnlnP0/P, indicating that the pore size distributions of sample 1 with different particle sizes all follow the fractal behavior. The summary of the fractal dimensions of all samples while considering variations in particle sizes can be seen in Figure 5. The fractal dimension values are all located in a reasonable range of fractal values (between 2 and 3) [36, 37]. However, comparing different samples, the effect that the particle size has on the fractal dimension results is not similar. In this regard, it is observed as the particle size decreases, the fractal dimension of sample 1, 2, 4, and 5 does not change significantly (Figure 4), whereas, in sample3, as the particle size decreases from <375 μm (40 mesh) to <250 μm (60 mesh), the fractal dimension demonstrates an evident decrease. On the contrary, as the particle size continues to decrease from <250 μm (60 mesh) to <150 μm (100 mesh), the fractal dimension does not change notably. Additionally, in a particle size range of less than 250 μm (60 mesh), the fractal dimensions comparing five samples follow the order: sample 3 > sample 1 > sample 4 > sample 2 > sample 5. This means as the particle size decreases, up to <150 μm, the order of fractal dimension change among these five samples remains the same (sample 3 has the largest fractal dimension and sample 5 the smallest value).Figure 4
The effect of the particle size on the fractal dimensions of the shale sample (sample 1).Figure 5
Summary of the fractal values of the samples with different particle sizes.


## 3.3.2. Multifractal Dimension

Based on Section2.3, we calculated the generalized dimensions Dq from N2 adsorption of all the shale samples with different particle sizes. Here, in Dq values (in this study, q is from -10 to 10), D0 represents the singularity of nonempty boxes containing some value of porosity under successive finer partitions. D1 can characterize the concentration degree of the pore size distribution along the pore size intervals. D2 accounts for the behavior of the second sampling moments [31]. The Dq value of sample 1 with different particle sizes can be seen in Figure 6 where D0 > D1 > D2 order of change exists in sample 1. This is indicating that the pore size distribution of sample 1, considering the existence of different particle sizes, presents the multifractal behavior [31]. The difference between the D−10andD10 can be used to quantify the heterogeneity of the pore size distribution of the sample [38]. Based on this, the larger D−10–D10 value corresponds to a higher degree of heterogeneity in the pore size distribution. The D−10–D10 values of all studied samples with different particle sizes are calculated, and the results can be seen in Figure 7. Based on the results, as the particle size decreases, the heterogeneity index (D−10–D10) fluctuates with a slight decreasing trend. This infers that the change in particle size has some impact on the heterogeneity that is present in the pore structures. Comparing all samples, the heterogeneity index when particle size is <250 μm (60 mesh) is very close to the heterogeneity index when the particle size is <187.5 μm (80 mesh). Consequently, as the particle size further decreases to <150 μm (100 mesh), the heterogeneity index decreases, regardless of the sample number. In sample 1 as an example, the heterogeneity index decreases from 1.19 to 1.15 as the particle size is reduced from <187.5 μm (80 mesh) to <150 μm (100 mesh).Figure 6
The correlation betweenDq and q of sample 1.Figure 7
The effect of the particle size on the heterogeneity of the pore size distribution.


## 4. Conclusions

In this study, it was shown that crushing the samples to a smaller size could change how the pore structures of shale samples are interpreted. This was overall inferred due to creation of fractures, pore collapse, or generation of new pores and surface areas in the samples from the crushing process. The adsorption isotherms displayed that, as the particle size decreases, the adsorption capacity increases. In addition, the surface area and pore volume also increased. The BET surface area of the sample in the 100 mesh particle size can be twice the value of the sample in the 40 mesh range. The average pore size varied clearly when the particle sizes decreased from 40 mesh to 60 mesh. However, as the particle size decreased further to 100 mesh, the average pore size does not change significantly. Next, the pore size distribution curves of samples showed similar peak locations, indicating that crushing the sample from 40 mesh to 100 mesh does not introduce new pores.Ultimately, fractal analysis defined as the particle size decreases from 40 mesh to 60 mesh; the fractal dimension would vary (the reduction of the fraction dimension for sample 3 and the increase of the fractal dimension for sample 4). However, the fractal dimension remains almost constant as the particle size becomes finer from 60 mesh to 100 mesh. In this study, the surface area, pore volume, the average pore size, and the fractal dimension of the samples in the 40 mesh range would differ compared to other particle size range. In this regard, we conclude, if the particle size is too large, the adsorption result might not be reliable due to incomplete equilibration and slow gas diffusion. Ultimately, the multifractal dimension result exhibited that, as the particle size decreases from 80 mesh to 100 mesh, the heterogeneity index of the shale samples decreases, meaning the samples are becoming more homogeneous. Overall, from this study, considering the pore structure parameters (pore surface area, pore volume, average pore radius, pore complexity, and heterogeneity), it is recommended that a reasonable particle size for should be either 60 mesh (<250μm) or 80 mesh (<187.5 μm), and 60 mesh is considered as the reasonable size since finer scale requires additional effort.



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*Source: 8898142-2021-04-24.xml*