Analysis of Microstructure and Low Permeability with 3D Digital Rock Modeling

The sandstone microstructure and permeability are important parameters for quantitative evaluation of groundwater/oil/gas resources and prediction of ﬂow rates of water/oil/gas. In this study, we applied seven low-permeability sandstone samples obtained from North China to research the microstructure and permeability based on digital core technology. Rock images were collected by X-ray microcomputed tomography ( μ CT), and then software (Avizo) was applied to analyze the microstructure and calculate the parameters such as porosity, connected porosity, average equivalent diameter, tortuosity, and shape factor. By introducing the shape factor into the Kozeny–Carman equation, we modiﬁed the Kozeny–Carman equation and found that the modiﬁed equation is a function of porosity, diameter of particles, tortuosity, and particle shape factor.


Introduction
e characteristics of low-permeability reservoirs are widely used to improve the water injection effect of lowpermeability oilfields or exploit oil and gas resources. e research of low permeability is essential to studying the reservoir characteristics. Reasonable characterization and quantitative analysis of the complex pore structures inside rock were the basis for solving many underground reservoir characteristics [1]. e research on the microstructure of the digital core was mainly based on the study of the 3D digital core. Knackstedt et al. [2] controlled the finite size effect, discretion error, and statistical fluctuations to obtain high-precision results by using X-micron CT scanners; Tian and Han [3] used X-ray CT to examine the evolution of concrete internal damage. Wang et al. [4] combined CT and the pore network model to analyze the influence of particle size on permeability in hydrate-containing porous media. Wang et al. [5] used fractal theory and X-ray CT imaging to carry out three-dimensional modeling and analysis of the coal pore structure. Cao et al. [6] used computed tomography data to analyze the capillary pressure of the dense sandstone based on a digital rock model. Qin et al. [7] used the fractal and multifractal methods such as box dimension and moment method combined with X-CT imaging techniques to analyze the structure of the volcanic vesicle. Yang et al. [8] used X-ray tomography to determine the distribution of residual oil in sandstone with different permeability.
Permeability was the key relationship between microstructure and macroscopic physical properties of porous media [9], and it was not only affected by the geometry or shape of the pore space but also by the topological structure [10]. e pore space connectivity of porous media was an important factor affecting the flow characteristics of tight sandstone [11]. e tortuosity of the pore structure has an important effect on macroscopic transport properties. Nakashima found that the diffusion rate and permeability decreased with the increase of tortuosity [12]. e pore size and mean coordination number also affected the overall permeability, and the permeability increased with the increase of the pore size distribution width [13,14]. Sueyoshi et al. also found that permeability mainly depends on the porosity and pore size of the flow path [15].
Various methods have been attempted to calculate permeability; Hu and Blunt used an improved maximum ball algorithm to study the overall network pore structure and calculate permeability [16]. A support vector regression (SVR) model was developed to evaluate the low porosity and permeability of sandstone reservoirs by Feng et al. [17]. Wei et al. [18] used the relevant parameters of the porous media to figure out the KC constant of the Kozeny-Carman equation, which thus contributes to the further study and analysis of the permeability. Kozeny-Carman (KC) equation expressed the relationship of permeability and porosity of porous media micropore structure [19][20][21]: where k is the permeability; φ is the porosity; d is the average particle radius; τ is the tortuosity; and c is the empirical constant. However, the traditional Kozeny-Carman equation was an empirical formula. For a heterogeneous system, the calculated permeability was large, 10 or more times than the actual [22]. Many modified equations were presented. Pape et al. [23] proposed an equation for sand and gravel: ( where r is the particle radius and D f is the fractal dimension of the medium. Civan [24] proposed an equation for general porous media: (3) Costa [25] proposed an expression suitable for porous rocks: Based on the fractal theory, Yu [26] derived the permeability expression through a curved capillary model.
where D T is the tortuosity fractal dimension, L 0 is the characteristic length, and λ 3+D T max is the maximum pore diameter.
Xu and Yu [27] also established a modified K-C equation based on the fractal theory: Among them, C f is as follows: Karacan [28] used the fractal method to calculate the porosity and permeability of mining, and the permeability expression was as follows: where A T is the total area; η max is the maximum diameter of the pore; η min is the minimum diameter of the pore; and D p is the fractal dimension of the pore. Xia et al. [29] used twelve three-dimensional digital cores of sandstone reservoir rocks to evaluate permeability in terms of fractal dimension, lacunarity, and succolarity. e permeability was rewritten as follows: where SU is the succolarity, indicating the ability of the fluid to move in the porous medium. Shen et al. [30] proposed the equivalent Kozeny-Carman equation to establish a permeability prediction model: In the formula, N � 2.44, and the unit of k is the same with the unit of d 2.44 , i.e., m 2.44 . is is different with the unit in the KC equation.
To sum up, digital rock cores (X-ray (CT) scanning) were used to obtain the structure of the rock cores, and some permeability prediction models have been proposed. e research was mainly on higher permeability, and few lowpermeability prediction models had been established. e low-permeability model generally contained empirical constants or modified permeability units.
is paper used X-ray CT scanning technology and Avizo software to study and calculate pore and particle structure parameters of low-permeability sandstones. Finally, the pore and particle structure parameters were used to predict permeability. In addition, the shape factor was introduced to modify the KC equation.

The Porosity and Permeability Obtained by Experiments
In this work, 7 natural sandstone samples were collected at depths of 430 m-735 m below the sea level of North China. ese 7 natural sandstone samples were abbreviated as GQ1, GQ2, GQ3, LA1, LA2, TL1, and TL2. Samples GQ1, GQ2, and GQ3 were from Gequan Mine, samples LA1 and LA2 were from Lu'an Mine, and TL1 and TL2 were from Tunlan Mine. e permeability and porosity were tested using the core company's high-and low-permeability meter CAT112 and a helium porosity meter from American Coretest company Phi220; the experiment results are shown in Table 1.
Based on the measured data, the map of sandstone porosity and permeability is shown in Figures 1 and 2.
As can be seen from Figure 1, the porosity of sandstone in Gequan Mine is high, while the porosity of the Lu'an samples is relatively low. e highest porosity of Tunlan Mine (TL1) is 5.66%, and the lowest porosity of Lu'an Mine (LA1) is 0.38%. Figure 2 shows the experiment permeability, and the permeability of Tunlan Mine (TL2) is the highest. Comparing Figures 1 and 2, it can be found that high porosity does not lead to high permeability; for example, the porosity of the samples GQ1, GQ2, GQ3, and TL1 is higher than that of other three samples (LA1, LA2, and TL2); however, the permeability values of the samples GQ1, GQ2, GQ3, TL1, LA1, and LA2 are relatively close, the porosity of Tunlan Mine (TL2) is 1.14%, but the permeability is 0.0870 mD; this is higher than other samples.

e Pore and Particle Diameters.
is paper used a desktop micrometer CT scanner (NanoVoxel-3000) to measure samples of approximately 1 cm. As an example, Figure 3 shows the sample GQ1 reconstructed images of three directions.
Avizo software was used to analyze the reconstructed 3D digital core. In order to better adjust the threshold so that the connected porosity is close to the measured porosity, a 3D digital core of 200 * 200 * 200 was selected as the target study area which was larger than REV, and the voxel edge length was 18.05 μm. In addition, 200 * 200 * 200 is also the study region in Figures 4-6.
Median filtering and interactive threshold segmentation were performed to obtain the three-dimensional pore structure of the core. Figure 4 shows the three-dimensional pore extraction process. e pores of rock samples were divided into connected pores and isolated pores, i.e., total pores � connected pores + isolated pores, as shown in Figure 5, the green parts are isolated pores, and the purple parts are connected pores. e GQ1 total porosity is 7.85%, where the isolated porosity is 3.34% and the connected porosity is 4.51%. It can be seen that the connected pore is about the experimentally measured data size. e porosity distribution of all rock samples is as follows (see Table 2).
To understand the distribution of pores with different pore sizes, the total pores were screened according to the pore size through the pore segmentation module of Avizo, and the pore size of the sample was obtained (as shown in Figure 6). e overall pore size distribution is shown in Figure 7. Figure 7 shows that the 40-80 μm diameters of the pore's aperture are the most widely distributed about 40%. e proportion of large pores is relatively small, and the pores larger than 400 μm are mainly connected pores. is was mainly due to the close connection of multiple pores, which resulted in the threshold segmentation as a whole. Finally, we obtained the sample GQ1's average pore diameter as 68 μm. e particle part was obtained after removing the pores in the sample (see Figure 8). rough the analysis of diameters, we can see that the particles with a diameter of 0-25 μm account for the largest proportion. e final analysis shows that the average particle diameter of the sample GQ1 is 35 μm. e diameter distributions of pores and particles for all rock samples are as follows (see Table 3).

Shape Factor Analysis of Rock Samples.
In reality, not all the pores/capillaries are spherical, and the shape of the particles has an effect on the permeability [31,32]. Nemec and Levec researched shape factor of trilobe and quadralobe particles and the effect on the permeability [33]. Safari et al. developed a porosity-permeability relationship for ellipsoidal grains [34].
We used the Label Analysis module of Avizo software (see Figure 9) to analyze the pores' shape factor. e calculation of the shape factor was based on the ideal sphere model. e calculation equation is where s is the shape factor, A is the area, and V is the volume of the pore. Because of the presence of partially elongated pores, the surface area of the pore becomes larger, resulting in a shape factor greater than 1. According to the demarcation point of the shape factor, the pores' shape factor was divided; GQ1's image as a sample is shown in Figure 10. e shape factors of the pores and particles of the average sample GQ1 are 2.03 and 1.24, respectively. Figure 11 shows the statistic of the GQ1's pores and particles' shape factor distribution in each section. Table 4 shows the average shape factor distribution of pores and particles for all rock samples.

Tortuosity Analysis of Rock Samples.
e ball-and-stick model can intuitively show the connection among the pores. e GQ1's ball-and-stick model was established as shown in Figure 12.
e ball-and-stick model showed the connection between the internally connected pores in the rock sample. For this reason, the tortuosity can be analyzed by the ball-andstick model. Tortuosity � path length/straight-line distance between two points.
e tortuosity of the pore for all samples can be shown as follows (see Table 5).

Kozeny-Carman Equation.
e Kozeny-Carman equation, which was first proposed by Kozeny [19] in 1927 and revised by Carman [20,21], was abbreviated as the KC equation. It was widely used in many fields to estimate and predict hydraulic conductivity, such as underground seepage, oil and gas field exploitation, chemical engineering,  biochemistry, and electrochemistry. According to the KC equation, the permeability k and porosity φ of the porous medium can be expressed as In the above formula, c and S are the Kozeny constant and the specific surface area of the solid phase, respectively. Considering the tortuosity effect (tortuosity τ), the KC equation can be further expressed as [35,36] Ortho Slice

Median Filter
Interactive Thresholding

Volume Rendering
Target study area

Geofluids
Kaviany [36] believed that if the value of t was approximately 21/2, then for spherical particles, c was 2.5. Although the KC equation was widely used, people have noticed its limitations. e value of c varied widely in different scenarios. erefore, we did not consider c as a key factor when establishing the permeability model.

Modification of the Kozeny-Carman (KC) Equation.
Introducing a shape factor into the Kozeny-Carman equation and based on the traditional KC permeability prediction model, a permeability prediction model with shape factor as an influencing factor is established:   8 Geofluids where s is the pore shape factor and n is the constant. After analyzing and discussing the cores with different shape factors, it can be found that the exponent of the shape factor was ±15.8. When the exponent of the shape factor was 15.8 for samples GQ2, GQ3, and LA2 and the exponent was −15.8 for samples GQ1, TL1, TL2, and LA1, the errors of calculated permeability and experiment permeability were small.
e obtained permeability prediction model is as follows:   Using formula (16), the theoretical permeability and relative error are calculated when n is ±15.8 (as shown in Table 6). e calculated permeability errors of the samples are within 15.6%, and these errors are acceptable in engineering applications. Table 6 shows that shape factors have an effect on the permeability; however, since only seven samples were used, the data were small. is exponent may be different from the test samples, so the exponent values of ±15.8 require further investigation.

Conclusion
is paper used X-ray CT scanning technology and Avizo software to study and calculate pore and particle structure parameters of low-permeability sandstones. Finally, the pore and particle structure parameters were used to predict permeability. In addition, the shape factor was introduced to modify the KC equation.
In this study, seven sandstone samples were collected from the Lu'an, Gequan, and Tunlan coal mines, China. e pore and particle geometric parameters of the samples were analyzed by CT experiments and Avizo software. e permeability and porosity were tested using the core company's high-and low-permeability meter CAT112 and a helium porosity meter from American Coretest company Phi220. e three-dimensional digital cores were analyzed by Avizo software, and a permeability prediction model was established. From these studies, the following conclusions may be drawn: (1) e microstructures of the pore and particle can be well captured using CT experiments on sandstones and rock cores, and the structures of samples can be effectively obtained using Avizo software. (2) rough experiments, it was found that the degree of correlation between porosity and permeability was not very high, and there were the cases of low porosity and high permeability. (3) e parameters such as porosity, connected porosity, average equivalent diameter, tortuosity, and shape factor can be calculated by Avizo software from the extracted volume. (4) It can be concluded that the shape factors have an effect on the permeability. By introducing the shape factor into the Kozeny-Carman equation, we modified the Kozeny-Carman equation.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.